Research Contribution DOI:10.56994/ARMJ.012.00?.00?
Received: 2025; Accepted: 2025


On boundary points of minimal continuously Hutchinson invariant sets

Per Alexandersson Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden per.w.alexandersson@gmail.com ,  Nils Hemmingsson Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY, USA nils.hemmingsson@stonybrook.edu ,  Dmitry Novikov Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001 Israel dmitry.novikov@weizmann.ac.il ,  Boris Shapiro Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden shapiro@math.su.se  and  Guillaume Tahar Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, China guillaume.tahar@bimsa.cn
(Date: March 2, 2026)
Abstact.
A linear differential operator T=Q⁢(z)⁢dd⁢z+P⁢(z) with polynomial coefficients defines a continuous family of Hutchinson operators when acting on the space of positive powers of linear forms. In this context, T has a unique minimal Hutchinson-invariant set MC⁢HT in the complex plane. Using a geometric interpretation of its boundary in terms of envelops of certain families of rays, we subdivide this boundary into local and global arcs (the former being portions of integral curves of the rational vector field Q⁢(z)P⁢(z)⁢∂z), and singular points of different types which we classify below.

The latter decomposition of the boundary of MC⁢HT is largely determined by its intersection with the plane algebraic curve formed by the inflection points of trajectories of the field Q⁢(z)P⁢(z)⁢∂z. We provide an upper bound for the number of local arcs in terms of degrees of P and Q. As an application of our classification, we obtain a number of global geometric properties of minimal Hutchinson-invariant sets.

Key words and phrases:
Action of linear differential operators, TC⁢H-invariant subsets, minimal TC⁢H-invariant subset, rational vector fields
2020 Mathematics Subject Classification:
Primary 37F10, 37E35; Secondary 34C05

1 Introduction

Given a linear differential operator

T=Q⁢(z)⁢dd⁢z+P⁢(z) (1.1)

where P,Q are polynomials that are not identically vanishing, we say that a closed subset S⊂ℂ is continuously Hutchinson invariant for T (TC⁢H-invariant set for short) if for any u∈S and any arbitrary non-negative number t, the image T⁢(f) of the function f⁢(z)=(z−u)t either has all roots in S or vanishes identically. In [AHN+24], we have initiated the study of general topological properties of TC⁢H-invariant sets.

The main motivation for the present study that it covers an interesting and manageable special case of a more general inverse Pólya-Schur problem introduced in [ABS]. For the convenience of our readers, let us briefly recall what the Pólya–Schur problem/theory and its inverse are, see [CsCr, ABS].

The main question of the classical Pólya–Schur theory can be formulated as follows.

Problem 1.1.

Given a subset S⊂ℂ of the complex plane, describe the semigroup of all linear operators T:ℂ⁢[z]→ℂ⁢[z] sending any polynomial with roots in S to a polynomial with roots in S (or to 0).

Definition 1.2.

If an operator T has the latter property, then we say that S is a T-invariant set, or that T preserves S.

So far 1.1 has only been solved for the circular domains (i.e., images of the unit disk under Möbius transformations), their boundaries [BB], and more recently for strips [BCh]. Even a very similar case of the unit interval is still open at present. It seems that for a somewhat general class of subsets S⊂ℂ, 1.1 is currently out of reach of all existing methods.

In [ABS], the following inverse problem in the Pólya–Schur theory which seems both natural and more accessible than Problem 1.1 has been proposed.

Problem 1.3.

Given a linear operator T:ℂ⁢[x]→ℂ⁢[x], characterize all closed T-invariant subsets of the complex plane. Alternatively, find a sufficiently large class of T-invariant sets.

Paper [ABS] concentrates on the fundamental case when T is a linear finite order differential operator with polynomial coefficients and shows that under some weak assumptions on these coefficients, there exists a unique minimal T-invariant set (and its analogs when T acts on polynomials of degree greater than or equal to a given positive integer n). Many basic properties of T-invariant sets such as their convexity, compactness etc are discussed in [ABS] as well as the delicate connection of Problem 1.3 to the classical complex dynamics.

However effective criteria characterizing T-invariant sets and explicit description of the minimal T-invariant set in somewhat interesting cases are currently missing which motivated the consideration in [AHN+24] of the action of T on integer and positive powers of linear forms. This situation is still quite interesting and appears to be more tractable.

In particular, the following results have been obtained in [AHN+24]:

  • •

    provided that either P or Q is not a constant polynomial, there is a unique minimal continuously Hutchinson invariant set MC⁢HT for a given operator T (in what follows we will always assume that this condition is satisfied);

  • •

    the only TC⁢H-invariant set is the whole ℂ unless |deg⁡Q−deg⁡P|≤1;

  • •

    a complete characterization of operators T for which MC⁢HT has an empty interior has been obtained (see Section 2.1 for details).

In this paper, we will focus on operators whose minimal set MC⁢HT has a nonempty interior.

Definition 1.4.

For an operator T given by (1.1) with P and Q not vanishing identically, at each point z such that P⁢Q⁢(z)≠0, we define the associated ray r⁢(z) as the half-line {z+t⁢Q⁢(z)P⁢(z)|t∈ℝ+}.

Remarkably, TC⁢H-invariant sets (and, in particular, the minimal one) can be characterized in terms of associated rays.

Theorem 1.5 (Theorem 3.18 in [AHN+24]).

A closed subset S⊆ℂ is TC⁢H-invariant if and only if it satisfies the following two conditions:

  1. (1)

    S contains the roots of the polynomials P and Q;

  2. (2)

    for any point z∉S, the associated ray r⁢(z) is disjoint from S.

1.1 Main results

In the present paper, using Theorem 1.5, we provide a qualitative description of the boundary of minimal continuously Hutchinson invariant sets, including an exhaustive typology of its singular points. Our classification mainly depends on the intersection of the boundary ∂MC⁢HT with the curve of inflections ℑR of the field R⁢(z)⁢∂z=Q⁢(z)P⁢(z)⁢∂z.

Definition 1.6.

The curve of inflections ℑR of the vector field R⁢(z)⁢∂z is defined as the closure of the set of points satisfying Im⁡(R′)=0, see [AHN+24]. It is a real plane algebraic curve of degree at most d=3⁢deg⁡P+deg⁡Q−1 (in this paper, we will always have d≥3).

The curve of inflections splits the complex plane into inflection domains where the sign of Im⁡(R′) remains the same.

Points of ∂MC⁢HT outside its intersection with ℑR are classified with the help of two correspondences Γ and Δ sending the boundary ∂MC⁢HT to itself and defined as follows:

For a given point z of the boundary ∂MC⁢HT, Γ⁢(z) is essentially the intersection of MC⁢HT with the integral curve of the rational field R⁢(z)⁢∂z starting at z, where R⁢(z)=Q⁢(z)/P⁢(z). In contrast, Δ is the intersection of the associated ray r⁢(z) with the closure of ∂MC⁢HT in the compactification ℂ∪𝕊1 of the complex plane (see Section 2.2). Formal definitions of Γ and Δ are given in Definition 4.1. Qualitatively, the boundary ∂MC⁢HT is made of two kinds of arcs:

  • •

    local arcs which are integral curves of the field R⁢(z)⁢∂z (i.e. Δ⁢(z)=∅ and Γ⁢(z)≠∅);

  • •

    global arcs at each point z of which the associated ray r⁢(z) is tangent to ∂MC⁢HT elsewhere (i.e. Γ⁢(z)=∅ and Δ⁢(z)≠∅).

Local arcs are locally strictly convex real-analytic arcs (see Proposition 4.11). In contrast, global arcs (formed by points of global type) can fail to be C1.

Local arcs inherit an obvious orientation from the vector field R⁢(z)⁢∂z. Global arcs also have canonical orientation, but its definition requires some work (see Section 4.5.2).

Local and global arcs connect special singular points of ∂MC⁢HT which in most of the cases belong to the curve of inflections. The latter decomposes into three loci (singular, tangent and transverse), each determining its own variety of singular points.

Definition 1.7.

The curve of inflections ℑR of the field R⁢(z)⁢∂z decomposes into:

  • •

    the singular locus 𝔖R formed by the points where several branches of ℑR intersect;

  • •

    the tangency locus 𝔗R formed by the non-singular points where the field R⁢(z)⁢∂z is tangent to ℑR;

  • •

    transverse locus ℑR∗ formed by the non-singular points of ℑR where the field R⁢(z)⁢∂z is transverse to ℑR.

The singular and the tangency loci are given by algebraic conditions. Therefore their intersection with ∂MC⁢HT is controlled in terms of deg⁡P and deg⁡Q. On the contrary, many points of the boundary can belong to the transverse locus ℑR∗. We refine the definition of the correspondence Δ according to the value of R⁢(z)u−z (which, by definition, is a positive number).

Definition 1.8.

We define Δ⁢(z)=(r⁢(z)¯∖{z})∩MC⁢HT¯, where r⁢(z)¯,MC⁢HT¯ are closures of r⁢(z),MC⁢HT in the compactification ℂ∪𝕊1 of ℂ, respectively.

For any z∈ℑR∖𝒵⁢(P⁢Q), we have Δ⁢(z)=Δ−⁢(z)∪Δ0⁢(z)∪Δ+⁢(z) where u∈Δ⁢(z)∩ℂ belongs to:

  • •

    Δ−⁢(z) if R′⁢(z)≤−R⁢(z)u−z;

  • •

    Δ0⁢(z) if R′⁢(z)=−R⁢(z)u−z;

  • •

    Δ+⁢(z) if R′⁢(z)≥−R⁢(z)u−z,

and u∈Δ⁢(z)∩𝕊1 belongs to

  • •

    Δ−⁢(z) if R′⁢(z)≤0;

  • •

    Δ0⁢(z) if R′⁢(z)=0;

  • •

    Δ+⁢(z) if R′⁢(z)≥0.

In particular, if R′⁢(z)>0, then Δ−⁢(z)=∅.

The main result of the present paper is a classification of boundary points of minimal continuously Hutchinson sets.

Theorem 1.9.

For any linear differential operator T given by (1.1), any point z of the boundary ∂MC⁢HT of its minimal TC⁢H-invariant set belongs to one of the following types:

  • •

    roots of polynomials P and Q (at most deg⁡P+deg⁡Q of them);

  • •

    singular points of the curve of inflections (at most 2⁢d of them);

  • •

    tangency points between the curve of inflections and the field R⁢(z)⁢∂z:

    • –

      straight segments, half-lines and lines (contained in at most deg⁡P+deg⁡Q+1 lines);

    • –

      at most 2⁢d2 isolated points;

  • •

    points of the transverse locus ℑR∗ belonging to one of the four subclasses:

    • –

      bouncing type: Δ+⁢(z)≠∅ and Γ∪Δ−⁢(z)≠∅;

    • –

      switch type: Δ+⁢(z)≠∅ and Γ⁢(z)∪Δ−⁢(z)=∅;

    • –

      C1-inflection type: Δ+⁢(z)=∅, Δ−⁢(z)≠∅ and Γ⁢(z)=∅;

    • –

      C2-inflection type: Δ+⁢(z)=∅ and either Δ−⁢(z)=∅ or Γ⁢(z)≠∅.

  • •

    points not on the curve of inflections belonging to one of the three subclasses:

    • –

      local type: Γ⁢(z)≠∅ and Δ⁢(z)=∅;

    • –

      global type: Γ⁢(z)=∅ and Δ⁢(z)≠∅;

    • –

      extruding type: Γ⁢(z)≠∅ and Δ⁢(z)≠∅.

Here, d=3⁢deg⁡P+deg⁡Q−1.

There can be many singular points of bouncing, extruding, C1-inflection, C2-inflection and switch types (we do not have a polynomial bound of their number in terms of deg⁡P and deg⁡Q). An extensive description of their geometric features is given below:

  • •

    at points of extruding type, the boundary of ∂MC⁢HT is not convex and it switches from a global to a local arc (see Section 4.6 and Fig. 1);

  • •

    at points of bouncing type, ∂MC⁢HT hits the curve of inflections, but does not cross it. In a neighborhood of such a point, the boundary ∂MC⁢HT remains in the closure of the same inflection domain (see Section 5.2 and Fig. 1);

  • •

    at points of switch type, ∂MC⁢HT is strictly convex, crosses the curve of inflections and the boundary switches from a local to a global arc (see Section 5.5 and Fig. 1);

  • •

    at points of C1-inflection type, ∂MC⁢HT crosses the curve of inflections and it switches from a global to another global arc having the opposite orientation. At such a point the curvature of ∂MC⁢HT is discontinuous (see Section 5.4 and Fig. 1);

  • •

    at points of C2-inflection type, ∂MC⁢HT crosses the curve of inflections and the boundary switches from a global arc to a local arc. Besides, the curvature of ∂MC⁢HT is continuous at such a point (see Section 5.3 and Fig. 1).

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Figure 1. Top row from the left: Extruding type, Switch type. Bottom row from the left: C1-inflection type, C2-inflection type, Bouncing type. In the pictures, blue arcs are global arcs, red arcs are local arcs and the black arc is a germ of the curve of inflections. The pointed arrow is the associated ray indicating the support points, when applicable.

Our second main result is an upper bound on the number of points of C1-inflection, C2-inflection and switch type in terms of d=3⁢deg⁡P+deg⁡Q−1.

Theorem 1.10.

For any operator T given by (1.1), the numbers of points of switch, C1-inflection and C2-inflection type (respectively |𝒮|, |ℐ1| and |ℐ2|) in ∂MC⁢HT satisfy the following bounds:

|𝒮|≤e16⁢d⁢ln⁡(d)+46⁢d3;
2⁢|ℐ1|+|ℐ2|≤e16⁢d⁢ln⁡(d)+46⁢d3.
Corollary 1.11.

For any operator T given by (1.1), the boundary ∂MC⁢HT of the minimal set contains at most d16⁢d+d⁢(2⁢d+1) local arcs.

In the last section of the paper, we deduce many results about the global geometry of minimal sets from the classification of boundary points. In several cases, an exact description can be given in terms of local and global arcs. In particular, we can prove that in generic case, the minimal TC⁢H-invariant set is connected in ℂ.

Theorem 1.12.

For any linear differential operator T given by (1.1), the minimal continuously Hutchinson invariant set MC⁢HT is a connected subset of ℂ with the possible exception of the case when R⁢(z) is of the form λ+μz+o⁢(z−1) with λ∈ℂ∗ and μ/λ∈ℝ.

In this later case, (unless both P and Q are constants and then there is no reasonable notion of a minimal set), MC⁢HT is formed by at most 12⁢deg⁡P+12⁢deg⁡Q connected components.

1.2 Organization of the paper

  • •

    In Section 2, we provide the basic background information on Hutchinson invariant sets developed in [AHN+24], including the results about their asymptotic geometry.

  • •

    In Section 3, we describe the local geometry around singular points of the vector field R⁢(z)⁢∂z in terms of their local degree and principal value. We also describe the main properties of the curve of inflections defined by the equation I⁢m⁢(R′)=0 and we also introduce the notion of horns.

  • •

    In Section 4, we describe boundary points in the complement to the curve of inflections, introducing Γ− and Δ− correspondences.

  • •

    In Section 5, we classify boundary points in the generic locus of the curve of inflections, proving Theorems 1.9 and  1.10 (in Sections 5.6 and 5.8.3 respectively). Corollary 1.11 is also proved in Section 5.8.3.

  • •

    In Section 6, we apply the latter results to get precise descriptions of minimal sets in several cases. Theorem 1.12 is proven in Section 6.4.

Acknowledgments. The third author is supported by the Israel Science Foundation (grants No. 1167/17 and 1347/23), by funding received from the MINERVA Stifting with the funds from the BMBF of the Federal Republic of Germany, and from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 802107). The fourth author wants to acknowledge the financial support of his research provided by the Swedish Research Council grant 2021-04900 and the hospitality of the Weizmann institute of Science in April 2021 and January 2024 when substantial part of the project was carried out. He is sincerely grateful to Beijing Institute of Mathematical Sciences and Applications for the support of his sabbatical leave in the Fall of 2023. The fifth author would like to thank France Lerner for valuable remarks about the terminology related to singular points.

2 Preliminary results and basic properties of MC⁢HT

The following notation will be important throughout this text.

Notation 2.1.

Given an operator T as in (1.1), we define p∞,q∞∈ℂ∗, and p,q∈ℕ so that

P⁢(z)=p∞⁢zp+o⁢(zp);
Q⁢(z)=q∞⁢zq+o⁢(zq).

Furthermore, we set λ=q∞p∞∈ℂ∗ and ϕ∞=arg⁡(λ).

Similarly, for any point α∈ℂ, we have R⁢(z)=rα⁢(z−α)mα+o⁢(|z−α|mα) with rα≠0 and mα∈ℤ. We denote by ϕα the argument of rα.

Remark 2.2.

Observe that frequently used affine changes of the variable z are applied to the vector field R⁢(z)⁢∂z and not to the rational function R⁢(z) itself.

2.1 Regularity of the minimal set

For an operator T as in (1.1), its minimal set MC⁢HT can be of three possible types:

  • •

    regular if MC⁢HT coincides with the closure of its interior;

  • •

    fully irregular if MC⁢HT has empty interior;

  • •

    partially irregular if MC⁢HT has nonempty interior but is not regular.

Actually, irregularity is related to specific reality conditions. The characterization of operators for which MC⁢HT is fully irregular is contained in Theorem 1.15 of [AHN+24].

Theorem 2.3.

For an operator T as in (1.1), the minimal set MC⁢HT is fully irregular in the following cases:

  • •

    R⁢(z)=λ for some λ∈ℂ∗;

  • •

    R⁢(z)=λ⁢(z−α) for some λ∉ℝ<0, α∈ℂ and deg⁡Q=1;

  • •

    R⁢(z)=λ⁢(z−α) for some λ∈ℝ>0, α∈ℂ and deg⁡Q≥2;

  • •

    operators satisfying the following conditions (up to an affine change of variable):

    • –

      R⁢(z) is real on ℝ;

    • –

      roots of P and Q are real, simple and interlacing (i.e. the roots of P and Q alternate along the real axis);

    • –

      |deg⁡Q−deg⁡P|≤1;

    • –

      if deg⁡Q−deg⁡P=1, then λ∈ℝ>0.

In any other case, MC⁢HT has a nonempty interior.

In this paper, we will always assume that MC⁢HT has a nonempty interior.

Remark 2.4.

If deg⁡P+deg⁡Q≤1, then MC⁢HT is either totally irregular or coincides with ℂ (see Theorem 1.15 of [AHN+24]). Therefore, our operators will always satisfy deg⁡P+deg⁡Q≥2. This implies in particular that d=3⁢deg⁡P+deg⁡Q−1 satisfies d≥3.

Referring to the closure of the interior of MC⁢HT as the regular locus and its complement in MC⁢HT as the irregular locus, we observe that the latter is contained in very specific lines of the plane.

Definition 2.5.

For a given rational function R⁢(z), a line Λ is called R-invariant if for any z∈Λ such that R⁢(z) is defined, we have z+R⁢(z)∈Λ.

In particular, up to an affine change of variable, we can assume Λ=ℝ and thus R⁢(z) is a real rational function. Besides, a R-invariant line is automatically an irreducible component of the curve of inflections ℑR.

Definition 2.6.

For an operator T whose minimal set MC⁢HT is not fully irregular, a tail is a semi-open straight segment [α,β[ in MC⁢HT satisfying the following conditions:

  • •

    the segment ]α,β] belongs to an R-invariant line;

  • •

    for any z∈]α,β], β−αR⁢(z)∈ℝ>0;

  • •

    for any z∈]α,β], z is disjoint from the regular locus of MC⁢HT;

  • •

    α belongs to the regular locus of MC⁢HT;

  • •

    β∈𝒵⁢(P⁢Q);

  • •

    β is a root of the same multiplicity for both P and Q.

In particular, every tail belongs to a R-invariant line.

The following fact has been proven in Corollary 7.8 of [AHN+24].

Theorem 2.7.

For an operator T whose minimal set MC⁢HT is not fully irregular, the irregular locus of MC⁢HT is a (possibly empty) finite union of tails.

In particular, if P and Q have no common roots, then the minimal set of the corresponding operator is either regular or fully irregular.

2.2 Extended complex plane

Following Theorem 1.5, TC⁢H-invariant sets are characterized by the position of the associated rays starting in their complements. Let us introduce a certain compactification111Notice that the most frequently used compactification of ℂ is ℂ¯=ℂ⁢P1. of ℂ which comes very handy in our considerations. We baptise it the extended complex plane ℂ∪𝕊1⊃ℂ.

The extended complex plane ℂ∪𝕊1 is set-theoretically the disjoint union of ℂ and 𝕊1 endowed with the topology defined by the following basis of neighborhoods:

  • •

    for a point x∈ℂ, we choose the usual open neighborhoods of x in ℂ;

  • •

    for a direction θ∈𝕊1, we choose open neighborhoods of the form I∪C⁢(z,I) where I is an open interval of 𝕊1 containing θ and C⁢(z,I) is an open cone with apex z∈ℂ whose opening (i.e. the interval of directions) is I.

Definition 2.8.

Given R⁢(z) as above, let p∈ℂ be a non-singular point of R⁢(z). We define σ⁢(p) as the argument of R⁢(p). We think of σ⁢(p) as a point of the circle at infinity.

One can easily see that 𝕊1 of the extended plane ℂ∪𝕊1 can be identified with the above circle at infinity. The extended plane is compact and homeomorphic to a closed disk. In particular, usual straight lines in ℂ have compact closures in ℂ∪𝕊1. (Below we will make no distinction between a real line in ℂ and its closure in ℂ∪𝕊1). Open half-planes in ℂ∪𝕊1 are, by definition, connected components of the complement to a line.

Given a TC⁢H-invariant set S⊂ℂ, we denote by S¯ its closure in the extended plane ℂ∪𝕊1.

The following result, but with a slightly different formulation, has been proved in Lemma 4.4 of [AHN+24].

Lemma 2.9.

Given an TC⁢H-invariant set S⊂ℂ, let α:[0,1]→ℂ be such that:

  • •

    ∀t∈(0,1), αt∈Sc;

  • •

    σ⁢(α0)≠σ⁢(α1);

  • •

    σ⁢(α) is homotopic to the positive arc from σ⁢(α0) to σ⁢(α1) in the circle at infinity via a homotopy H⁢(t,x):[0,1]×[0,1] such that H⁢(0,x0)=σ⁢(α⁢(0)), H⁢(1,x0)=σ⁢(α⁢(1)) for all x0∈[0,1].

If X denotes the connected component of in ℂ∪𝕊1 containing the interval ]σ(α0),σ(α1)[ in the complement of r⁢(α0)∪α∪r⁢(α1), then X⊂Sc.

2.3 Integral curves

Another result has been proved in Proposition A.2 of [AHN+24].

Proposition 2.10.

Given a TC⁢H-invariant set S⊂ℂ and some point z0∈S, if there is a positively oriented integral curve γ:[0,ϵ[→ℂ of the vector field R⁢(z)⁢∂z such that limt→ϵγ⁢(t)=z0, then for any t∈[0,ϵ], γ⁢(t)∈S.

When referring to the proposition above, we say that the bounded backward trajectories of R⁢(z)⁢∂z of points in any invariant set S belongs to S.

2.4 Root trails

For any point u∈ℂ, the root trail 𝔱⁢𝔯u of u is the closure of the set of points z such that the associated ray r⁢(z) contains u. Except for the trivial cases described in Section 3 of [AHN+24], root trails are plane real-analytic curves. By definition, the root trail of any point of MC⁢HT is also contained in MC⁢HT. Furthermore, for any fixed u∈ℂ, we defined a t-trace (corresponding to u) as any continuous function γ⁢(t) such that

Q⁢(γ⁢(t))+(γ⁢(t)−u)⁢P⁢(γ⁢(t))=0

for all t≥0. That is, any t-trace γ⁢(t) is a concatenation of parts of 𝔱⁢𝔯u such that the resulting curve is continuous for any t≥0.

Lemma 2.11.

Consider a linear differential operator T given by (1.1) and some point u∈ℂ. Assuming that R⁢(z) is not of the form λ⁢(z−u), then

(i) for any point u∈ℂ and any point z0∉𝒵⁢(P⁢Q) such that z0∈𝔱⁢𝔯u and R⁢(z0)+(u−z0)⁢R′⁢(z0)≠0, the root trail 𝔱⁢𝔯u has a unique branch passing through z0 and its tangent slope is the argument of R2⁢(z0)R⁢(z0)+(u−z0)⁢R′⁢(z0) (mod π).

(ii) If R⁢(z0)+(u−z0)⁢R′⁢(z0)=0 and m≥2 is the smallest integer such that R(m)⁢(z0)≠0, then 𝔱⁢𝔯u has m intersecting branches at z. Their tangent slopes are:

θ0m+k⁢πm,

where θ0 is the argument of R⁢(z0)R(m)⁢(z0) and k∈ℤ/m⁢ℤ.

Before proving Lemma 2.11 we prove the next two Lemmas.

Lemma 2.12.

If γ⁢(t) is smooth planar curve, γ⁢(0)=z0, and γ˙⁢(t)=G⁢(γ⁢(t)) for some function G holomorphic and non-vanishing at z0 then the sign of the curvature of γ⁢(t) at z0 coincides with the sign of Im⁡G′⁢(0).

Indeed, then γ¨⁢(t)=G′⁢(γ⁢(t))⋅γ˙⁢(t). By definition, the sign of the curvature of γ⁢(t) at z0 coincides with the sign of Im⁡γ¨⁢(t)γ˙⁢(t)|t=0=Im⁡G′⁢(0).

Lemma 2.13.

Let F be a function holomorphic at z0 with F⁢(z0)∈ℝ and let m=ordz0⁡(F−F⁢(z0)). Then the germ of IF={Im⁡F=0} at z0 consists of m smooth branches with tangent slopes θ0m+k⁢πm, k∈ℤ/m⁢ℤ, where θ0=−arg⁡F(m)⁢(z0).

If m=1 and γ⁢(t) is a parameterization of IF such that F⁢(γ⁢(t))≡F⁢(z0)+t then the sign of the curvature of γ⁢(t) coincides with the sign of −Im⁡[F′′⁢(z0)(F′)2⁢(z0)].

Proof.

Indeed, we have F⁢(z)=a0+am⁢(z−z0)m+…, a0∈ℝ, so the branches of IF are tangent to the m lines satisfying equation Im⁡am⁢(z−z0)m=0, which have slopes as stated.

For the second claim, note that γ˙⁢(t)=1F′⁢(γ⁢(t)), so the claim follows from Lemma 2.12. ∎

Remark 2.14.

Similar results hold for F having a pole at z0 by considering 1F.

Proof of Lemma 2.11.

Note that by definition

𝔱⁢𝔯u={z∈ℂ⁢ s.t. ⁢R⁢(z)u−z∈ℝ+}⊂{Im⁡R⁢(z)u−z=0},

and Lemma 2.11 follows from the Lemma 2.13 with F⁢(z)=R⁢(z)u−z and the fact that arg⁡R⁢(z0)=arg⁡(u−z0). ∎

Remark 2.15.

The condition R⁢(z0)+(u−z0)⁢R′⁢(z0)=0 means that the point u=z0−R⁢(z0)R′⁢(z0) is obtained as the the first iteration of Newton’s method of approximating roots of R⁢(z) with the starting point z0.

When u is a point at infinity in the extended plane ℂ∪𝕊1, the root trail 𝔱⁢𝔯u of u is the closure of the points z where the argument of R⁢(z) coincides with u.

Lemma 2.16.

Consider a linear differential operator T given by (1.1) such that R⁢(z) is not constant. For any point u at infinity and any point z0∉𝒵⁢(P⁢Q) such that z0∈𝔱⁢𝔯u, provided R′⁢(z0)≠0, the root trail 𝔱⁢𝔯u has a unique branch passing through z0 and its tangent slope is the argument of R⁢(z0)R′⁢(z0) (mod π).

If R′⁢(z0)=0 and m≥2 is the smallest integer such that R(m)⁢(z0)≠0, then 𝔱⁢𝔯u has m intersecting branches at z. Their tangent slopes are:

θ0m+k⁢πm,

where θ0 is the argument of R⁢(z0)R(m)⁢(z0) and k∈ℤ/m⁢ℤ.

Proof.

In this case 𝔱⁢𝔯u⊂{Im⁡(R⁢(z)/R⁢(z0))=0} and the claim follows again from Lemma 2.13 ∎

Remark 2.17.

From Lemmas 2.11 and 2.16 it immediately follows that if a root trail 𝔱⁢𝔯u can have m≥2 branches at some point z0, then z0 belongs to the curve of inflections ℑR (because R⁢(z0) and u−z0 are real colinear).

Besides, if m≥3, then R(k)⁢(z0)=0 for 2≤k≤m−1 and z0 is a singular point of ℑR.

2.4.1. Concavity of root trails

Proposition 2.18.

Let u be a point of the extended plane ℂ∪𝕊1 and z0 be a point of 𝔱⁢𝔯u such that z0∉𝒵⁢(P⁢Q)∪ℑR and z0≠u. We denote by L the tangent line to 𝔱⁢𝔯u at z0. We define f⁢(z,u) to be:

  • •

    (R′′(z0)(u−z0)2+2R′(z0)(u−z0)+2R(z0)](u−z0)(R′⁢(z0)⁢(u−z0)+R⁢(z0))2 if u∈ℂ;

  • •

    R′′⁢(z0)⁢R⁢(z0)R′⁢(z0)2 if u is a point at infinity.

Then the germ of 𝔱⁢𝔯u at z0 belongs to

(i) the same half-plane bounded by L as the associated ray r⁢(z0) if Im⁡(f) and Im⁡(R′⁢(z0)) have opposite signs.

(ii) They belong to distinct half-planes bounded by L if Im⁡(f) and Im⁡(R′⁢(z0)) have the same sign.

Finally, 𝔱⁢𝔯u has an inflection point at z0 if Im⁡(f)=0.

Proof.

Let Fu⁢(z)=R⁢(z)u−z for u∈𝒞 and Fu⁢(z)=u−1⁢R⁢(z) for u∈𝕊1 so that 𝔱⁢𝔯u={Im⁡Fu⁢(z)=0}. Let c=Fu′⁢(z0). We have

L={z0+c−1⁢ℝ}={z|Im⁡(c⁢(z−z0))=0}.

If γ⁢(t) is a local parameterization of (t⁢r)u at z0 such that Fu⁢(γ⁢(t))=Fu⁢(z0)+t then γ˙⁢(0)=c−1. Moreover, (t⁢r)u⊂L+={Im⁡c⁢(z−z0)>0} if the curvature of γ⁢(t) is positive and (t⁢r)u⊂L−={Im⁡c⁢(z−z0)<0} otherwise.

The tangent ray r⁢(z0)={z0+R⁢(z0)⁢ℝ+} lies in L+ if Im⁡R⁢(z0)⁢c>0 and in L− otherwise.

By Lemma 2.13 the sign of curvature of γ⁢(t) is opposite to the sign of Im⁡[Fu′′⁢(z0)(Fu′)2⁢(z0)].

For u∈ℂ we have c=Fu′⁢(z0)=R′⁢(z0)⁢(u−z0)+R⁢(z0)(u−z0)2 and

Fu′′⁢(z0)=−R′′⁢(z0)⁢(u−z0)2+2⁢R′⁢(z0)⁢(u−z0)+2⁢R⁢(z0)(u−z0)3,

so we are interested in signs of

Im⁡R⁢(z0)⁢c=Im⁡R⁢(z0)⁢R′⁢(z0)⁢(u−z0)+R⁢(z0)(u−z0)2=Im⁡R′⁢(z0)

(recall that R⁢(z0)u−z0>0) and

Im⁡Fu′′⁢(z0)(Fu′)2⁢(z0)=−Im⁡(R′′(z0)(u−z0)2+2R′(z0)(u−z0)+2R(z0)](u−z0)(R′⁢(z0)⁢(u−z0)+R⁢(z0))2. (2.1)

For u∈𝕊1 we have c=u−1⁢R′⁢(z0) and we are interested in the signs of Im⁡R′⁢(z0) and Im⁡R′′⁢(z0)⁢R⁢(z0)(R′)2⁢(z0).

∎

In the transverse locus ℑR∗ of the curve of inflections, the concavity of root trails with respect to the line containing the associated ray depends on the sign of some geometrically meaningful real function.

Proposition 2.19.

Consider a point z0∈ℑR∗∖𝒵⁢(P⁢Q) and some point u∈ℂ∪𝕊1. Assume that R⁢(z0)+R′⁢(z0)⁢(u−z0)≠0 (or R′⁢(z0)≠0 if u is a point at infinity). Let L be the line containing the associated ray r⁢(z0).

The germ of 𝔱⁢𝔯u at z0 and the positive germ γz0+ of the integral curve of the field R⁢(z)⁢∂z starting at z0 belong to the same open half-plane bounded by L if R′⁢(z0)+R⁢(z0)/(u−z0) is negative (R′⁢(z0)<0 if u is a point at infinity).

The germ of 𝔱⁢𝔯u at z0 and γz0+ belong to opposite open half-planes bounded by L if R′⁢(z0)+R⁢(z0)/(u−z0) is positive (R′⁢(z0)>0 if u is a point at infinity).

Proof.

Without loss of generality, we assume that z0=0 and R⁢(z)=1+R′⁢(0)⁢z+(a+b⁢i)⁢z2+o⁢(z2) with R′⁢(0)∈ℝ, a∈ℝ and b∈ℝ>0 (b≠0 because z0=0 belongs to the transverse locus of the curve of inflections). Necessarily u>0. Since b>0, γ0+ belongs to the upper half-plane.

By Lemma 2.16, 𝔱⁢𝔯u has a unique branch at 0 tangent to ℝ. Let Fu⁢(z)=R⁢(z)u−z0 for u∈r⁢(z0) and Fu⁢(z)=R⁢(z) for u∈𝕊1, so 𝔱⁢𝔯u={Im⁡Fu⁢(z)=0}. Choose a parameterization γ⁢(t) of this branch in such a way that Fu⁢(γ⁢(t))=Fu⁢(z0)+t. Then

γ˙⁢(0)=1Fu′⁢(0)=u−z0R′⁢(0)+R⁢(0)u−z0 or γ˙⁢(0)=1R′⁢(0)

for u∈ℂ or u∈𝕊1 being a point at infinity, respectively. Therefore γ˙⁢(0)>0 if R′⁢(0)+R⁢(0)/(u−z0)>0 (resp. R′⁢(0)>0) and γ˙⁢(0)<0 otherwise.

By (2.1) the sign of the curvature of γ⁢(t) at 0 is opposite to the sign of Im⁡R′′⁢(0)=Im⁡b>0, i.e. is negative. Thus γ⁢(t) lies in the lower half-plane (i.e. not in the same half-plane as γ0+) if R′⁢(0)+R⁢(0)/(u−z0)>0 is positive and in the same half-plane as γ0+ if R′⁢(0)+R⁢(0)/(u−z0)<0 (R′⁢(0)>0 and R′⁢(0)<0 resp. for u∈𝕊1). Since R′⁢(z0)∈ℝ, the number R′⁢(z0)+R⁢(z0)/(u−z0) is invariant under the maps z↦a⁢z+b and z↦z¯ used for normalization, and the claim follows. ∎

2.4.2. Root trails and connected components of the minimal set

When deg⁡Q−deg⁡P=0, root trails provide a bound on the number of connected components of the minimal set (in all other cases, it is known that MC⁢HT is connected).

Proposition 2.20.

Consider a linear differential operator T given by (1.1) and satisfying deg⁡Q−deg⁡P=0. Any connected component C of MC⁢HT satisfies the following conditions:

  • •

    C contains at least one root of P;

  • •

    C contains at least one root of Q;

  • •

    the sum of orders of zeros and poles of R⁢(z) in C vanishes.

Proof.

We assume that a connected component C of MC⁢HT is disjoint from 𝒵⁢(P). Note that deg⁡Q−deg⁡P=0 implies that the union of the zeros of t⁢Q⁢(z)+P⁢(z)⁢(z−u) for any u∈ℂ, T>0 and t∈[0,T] is bounded.

Hence, for any u∈MC⁢HT∖C, the root trail of u is disjoint from C, as otherwise there would be points in the complement of MC⁢HT belonging to the root trail of u. Since MC⁢HT coincides with the TC⁢H-extension of any point in MC⁢HT (see Lemma 2.2 of [AHN+24]), it follows that C cannot belong to the minimal invariant set.

Suppose now that there is a component for which the sums of orders of the zeros of Q does not equal the sums of orders of the zeros of P. Then there is a component C such that the sums of the orders of the zeros of P, say d0 is strictly greater than the sums of the orders of the zeros of Q, say d1. Taking u∈C we have that for all t, the zeros of t⁢Q⁢(z)+P⁢(z)⁢(z−u) belonging to C have total degree d0+1. However, when sending t→0, d1 of these zeros tend to the zeros of Q belonging to C and at most one tend to ∞. This implies that at least d0−d1>0 of the end points of the root trail of u does not belong to C, a contradiction. ∎

We prove now that the interior (MC⁢HT)∘ of the minimal set satisfies (outside zeroes and poles of R⁢(z)) a weak property of local connectedness.

Lemma 2.21.

For any linear differential operator T given by (1.1) we consider a point α of the boundary ∂MC⁢HT that is neither a zero nor a pole of R⁢(z). For any sufficiently small neighborhood V of α, the connected component MV of V∩MC⁢HT containing α has connected interior.

Proof.

If α does not belong to the regular locus of MC⁢HT, then MV is a line segment and hence has empty interior.

We have R⁢(z)=rα+o⁢(z−α) for some rα∈ℂ∗. Then, a continuity argument immediately shows that MV has connected interior, as otherwise points in the complement of MC⁢HT would have associated rays intersecting MC⁢HT.

∎

In the following, we prove that the closure of a connected component of the interior of MC⁢HT cannot be disjoint from 𝒵⁢(P).

Lemma 2.22.

For any linear differential operator T given by (1.1), one of the following statements holds:

  1. (1)

    MC⁢HT is fully irregular;

  2. (2)

    MC⁢HT=ℂ;

  3. (3)

    the closure of any connected component of the interior (MC⁢HT)∘ of the minimal set contains a root of P⁢(z);

  4. (4)

    the closure of any connected component of the interior (MC⁢HT)∘ of the minimal set contains an endpoint of a tail.

Proof.

We suppose that we are not in the case (1), (2). Besides, we assume the existence of a connected component C of the interior (MC⁢HT)∘ of the minimal set whose closure is disjoint from 𝒵⁢(P), contradicting statement (3).

We first prove that C cannot be the only connected component of (MC⁢HT)∘. Indeed, roots of P⁢(z) that do not belong to the regular locus of MC⁢HT (the closure of the interior) belong to tails (see Theorem 2.7) and they are not zeros or poles of R⁢(z). Besides, MC⁢HT is assumed to be distinct from ℂ. Consequently, we have |deg⁡Q−deg⁡P|≤1. The only case where the regular locus of MC⁢HT can be disjoint from 𝒵⁢(P) is when R⁢(z) is of the form λ or λ⁢(z−α). In the first case, MC⁢HT is known to be totally irregular. In the second case, either λ∈ℝ>0 (and MC⁢HT is totally irregular, see Theorem 2.3) or λ∉ℝ>0 and MC⁢HT has no tails (and P⁢(z) has no root at all). We assume therefore that the interior MC⁢HT has several connected components.

We denote by A the set of points of C that belong to the closure of another connected component of (MC⁢HT)∘. By assumption, these are not these points are not roots of P⁢(z) and Lemma 2.21 shows that each of them is a zero of R⁢(z).

Since MC⁢HT is minimal, there is a point u∈MC⁢HT∖C¯ and a point z0∈𝔱⁢𝔯u∩C. As the root trail 𝔱⁢𝔯u changes continuously in u, u may be chosen outside A. Since |deg⁡Q−deg⁡P|≤1, the zeros of t⁢Q⁢(z)+P⁢(z)⁢(z−u) as t→0 tends to 𝒵⁢(P)∪{u}. Further, we can assume that γ⁢(t) does not equal ∞ for some finite t, as this would imply deg⁡Q−deg⁡P=1 and λ<0, in which case MC⁢HT is equal to ℂ. Hence, the minimal set MC⁢HT therefore contains a continuous path γ⁢(t) from an element of 𝒵⁢(P)∪{u} to z0 such that γ⁢(t) solves t⁢Q⁢(γ⁢(t))+P⁢(γ⁢(t))⁢(γ⁢(t)−u)=0.

The path γ has to enter the component C and can do so either through a tail or an element of A. The path γ cannot contain any element α∈A because the equations Q⁢(α)=0 and t⁢Q⁢(α)+P⁢(α)⁢(α−u)=0 (for some t>0) imply P⁢(α)=0, contradicting our assumption. Our assumption that neither (1), (2) nor (3) was satisfied thus implies (4). ∎

2.5 Asymptotic geometry of Hutchinson invariant sets

Let us recall the results of [AHN+24] concerning minimal Hutchinson invariant sets (see Theorems 1.11 and 1.12 of [AHN+24]).

Theorem 2.23.

For any operator T as in (1.1) with a minimal set MC⁢HT having a nonempty interior, MC⁢HT is:

  • •

    a compact contractible subset of ℂ if deg⁡Q−deg⁡P=1, and R⁢e⁢(λ)≥0;

  • •

    a noncompact non-trivial subset of ℂ if deg⁡Q−deg⁡P=0 or −1;

  • •

    trivial, i.e. equal to ℂ otherwise.

Besides, the closure MC⁢HT¯ in the extended plane ℂ∪𝕊1 is contractible, connected and compact.

Thus, the only interesting cases for the description of ∂MC⁢HT are those for which the values of deg⁡Q−deg⁡P are 1, 0 or −1. In the latter two cases, we have more precise results given below.

2.5.1. deg⁡Q−deg⁡P=−1

The following statement has been proved in Corollary 6.2 of [AHN+24].

Proposition 2.24.

For an operator T as in (1.1) such that deg⁡Q−deg⁡P=−1. Then the complement of its minimal Hutchinson invariant set MC⁢HT in ℂ has exactly two connected components X1,X2. Each Xi contains infinite cones whose intervals of directions are arbitrarily close to (ϕ∞−π2,ϕ∞+π2) and (ϕ∞+π2,ϕ∞+3⁢π2) respectively.

2.5.2. deg⁡Q−deg⁡P=0

The following statement has been proven in Corollary 6.4 of [AHN+24].

Proposition 2.25.

Take any operator T as in (1.1) such that deg⁡Q−deg⁡P=0. Then for any ϵ>0, there exists an open cone 𝒞 whose interval of directions is arbitrary close to (ϕ∞+π,ϕ∞+π) and such that MC⁢HT is contained in 𝒞.

3 Local analysis of the boundary of MC⁢HT

We consider an operator T as in (1.1) whose minimal set MC⁢HT has a nonempty interior.

Notation 3.1.

For any point α∈∂MC⁢HT, we define rα∈ℂ∗, mα∈ℤ so that

R⁢(z)=Q⁢(z)P⁢(z)=rα⁢(z−α)mα+o⁢(|z−α|mα). (3.1)

We also define ϕα=a⁢r⁢g⁢(rα) and dα:𝕊1⟶𝕊1 where dα⁢(θ)=ϕα+mα⁢θ.

3.1 Description of a tangent cone

Definition 3.2.

For any α∈∂MC⁢HT, we define 𝒦α as the subset of 𝕊1 formed by directions θ such that there is a sequence (zn)n∈ℕ satisfying the following conditions:

  • •

    for any n∈ℕ, zn∈(MC⁢HT)c;

  • •

    zn⟶α;

  • •

    a⁢r⁢g⁢(zn−α)⟶θ.

We also define ℒα as the subset of 𝕊1 formed by directions θ such that the half-line α+ei⁢θ⁢ℝ+ does not intersect the interior of MC⁢HT.

Lemma 3.3.

For any α∈∂MC⁢HT, the following statements hold:

  1. (1)

    𝒦α and ℒα are nonempty closed subsets of 𝕊1;

  2. (2)

    ℒα⊂𝒦α;

  3. (3)

    for any θ∈𝒦α, d⁢(θ)∈ℒα. In particular, Kα is invariant under dα;

  4. (4)

    for any θ∈𝒦α, there exists a closed interval J⊂𝒦α of length at most π containing both θ and dα⁢(θ);

  5. (5)

    ℒα≠𝕊1.

Proof.

From Definition 3.2 it immediately follows that 𝒦α and ℒα are closed subsets of 𝕊1.

If α∈∂MC⁢HT, then we can find a sequence of points in the complement of MC⁢HT approaching α. By compactness of 𝕊1, we can choose a subsequence for which the arguments converge to some limit. Thus 𝒦α is nonempty.

Then, for any θ∈𝒦α, we have a sequence (zn)n∈ℕ in the complement of MC⁢HT accumulating to α with the limit slope θ. The associated rays r⁢(zn) accumulate to α+ei⁢dα⁢(θ)⁢ℝ+. Since none of them intersects the interior of MC⁢HT, the half-line α+ei⁢dα⁢(θ)⁢ℝ+ does not intersect it either and dα⁢(θ)∈ℒα.

Besides, in the case where θ≠dα⁢(θ), (up to taking a subsequence of (zn)n∈ℕ, there is a closed interval J⊂𝕊1 such that:

  • •

    the endpoints of J are θ and dα⁢(θ);

  • •

    the length of J is at most π;

  • •

    for any η in the interior of J, there is a bound N⁢(η) such that for any n≥N⁢(η), the associated ray r⁢(zn) intersects the half-line α+ei⁢η⁢ℝ+ at some point Pη,n.

Existence of sequences (Pη,n)n≥N⁢(η) proves that for any η∈J, one has η∈𝒦α.

Finally, ℒα≠𝕊1 because in this case, MC⁢HT would have empty interior. ∎

Remark 3.4.

Note that in the case θ=dα⁢(θ), the interval J is a singleton {θ}.

Let us deduce local description of 𝒦α and ℒα depending on the local invariants of α.

Corollary 3.5.

For any α∈∂MC⁢HT, the following statements hold:

  • •

    if |mα|≥2, then 𝒦α=ℒα and they are contained in the finite set of arguments satisfying θ≡ϕα1−mα⁢[2⁢π1−mα];

  • •

    if mα=1, then ϕα=0 and 𝒦α=ℒα;

  • •

    if mα=0, then ϕα∈ℒα;

  • •

    if mα=−1, then 𝒦α=ℒα and these sets are formed by at most two intervals, each of length at most π and having their midpoints at ϕα2 and ϕα2+π.

Proof.

We consider maximal interval J in 𝒦α (which is non-empty by Lemma 3.3). The images of J under the iterated action of dα belong to ℒα.

If |mα|≥2, then J is a singleton since otherwise the union of its iterates would coincide with 𝕊1 (contradicting Lemma 3.3). Thus J has to be a fixed point of the map dα.

If mα=1 and ϕα≠0, then J coincides with 𝕊1 because no other connected subset of the circle is preserved under the action of nontrivial rotation. Therefore dα is the identity map.

If mα=0, then for any θ∈𝒦α, dα⁢(θ)=ϕα. Therefore ϕα∈ℒα.

If mα=−1, then J is invariant under the action of θ↦ϕα−θ. Thus, either ϕα2 or ϕα2+π is the bisector of J. If J is of length strictly bigger than π, then Lemma 3.3 shows that its complement (of length strictly smaller than π) is also contained in ℒα. Therefore Lα=𝕊1 which is a contradiction. ∎

We obtain a bound on the number of petals of MC⁢HT that can be attached to a boundary point.

Corollary 3.6.

For any linear differential operator T given by (1.1) we consider a point α of the boundary ∂MC⁢HT. Then for any sufficiently small open subset V⊂ℂ the interior of the connected component MV of V∩MC⁢HT containing α has at most:

  • •

    |1−mα| connected components if mα≠1;

  • •

    deg⁡P connected components if mα=1

where R⁢(z)=λ⁢(z−α)mα+o⁢((z−α)mα) with λ∈ℂ∗ and mα∈ℤ.

Proof.

If α is not a zero or a pole of R⁢(z), then Lemma 2.21 proves the statement. Besides, if mα∉{0,1}, Corollary 3.5 proves that α is in the closure of at most 1−mα components.

In the remaining cases, α is a simple zero of R⁢(z). If α is also a root of degree d of P, then it is a root of degree d+1 of Q.

We can divide P and Q by (z−α)d while keeping the same minimal set MC⁢HT (because in this case 𝒵⁢(P⁢Q) remains unchanged). Consequently, we can assume that α is not a root of P. Lemma 2.22 proves that for any connected component C of (MC⁢HT)∘ such that α is in the closure of C, either some root of P⁢(z) belongs to the closure of C or some tail is attached to C. If α is in the closure of several connected components of (MC⁢HT)∘, then a same root of P cannot be in the closure of two of them because MC⁢HT¯ would fail to be contractible. Similarly a given tail is attached to only one connected component of (MC⁢HT)∘ (and contains at least one root of P). Therefore, α is in the closure of at most deg⁡P components. ∎

3.2 Curve of inflections

In §A.3 of [AHN+24] we introduced the curve of inflections ℑR of an analytic vector field R⁢(z)⁢∂z. By definition, it is the closure in ℂ of the subset of ℂ∖𝒵⁢(P⁢Q) at each point of which the integral curve of the vector field R⁢(z)⁢∂z passing through this point has zero curvature. Here and throughout, 𝒵⁢(F) denotes the set of zeros of the function F. Below we provide some additional information about ℑR.

For an operator T for which R⁢(z) is not of the form λ or λ⁢(z−α) for some λ∈ℂ∗ and α∈ℂ, the function R′⁢(z) is a non-constant rational function. Therefore the curve of inflections ℑR of R⁢(z)⁢∂z (which is defined as the closure of the set of points for which I⁢m⁢(R′⁢(z))=0) is a real plane algebraic curve.

We first characterize the points at which several local branches of the curve of inflections intersect.

Lemma 3.7.

A point z0∈ℑR belongs to exactly m≥2 local branches of ℑR in the following cases:

  1. (1)

    z0 is a critical point of R′⁢(z) of order m−1 (including zeroes of order m of R⁢(z));

  2. (2)

    z0 is a pole of R⁢(z) of order m−1.

The 2⁢m limit slopes of the local branches at z0 form a regular 2⁢m-gon in 𝕊1.

Proof.

This follows immediately from Lemma 2.13. ∎

Corollary 3.8.

The curve of inflections ℑR has at most 4⁢deg⁡P+deg⁡Q−2 singular points.

Proof.

There are at most deg⁡P poles of R⁢(z) and the critical points of R′⁢(z) are the zeroes of R′′⁢(z). ∎

Lemma 3.9.

Let F⁢(z):ℂ→ℂ⁢P1 be a non-constant rational function of degree d. Then the real algebraic curve Γ={z∈ℂ|Im⁡F⁢(z)=0}¯ is non-empty, has at most d connected components and has exactly d connected components for generic F.

Proof.

Clearly, as F−1⁢(x)≠∅ for any x∈ℝ∖{F⁢(∞)}, Γ≠∅ as well.

By the open mapping theorem, the map F:Γ¯→ℝ⁢P1, where Γ¯ is the closure of Γ in ℂ⁢P1, is onto on each connected component of Γ¯. Since F has degree d this means that Γ has at most d components.

Note that the ramification points of F:Γ¯→ℝ⁢P1 coincide with the ramification points of F:ℂ⁢P1→ℂ⁢P1 lying on Γ¯. Thus if the ramification values of F are not in ℝ⁢P1 then the former map is an unramified cover of degree d, so has exactly d connected components. This means that the bound is sharp. ∎

3.2.1. Inflection domains

Definition 3.10.

The curve of inflections ℑR subdivides ℂ into two open (not necessarily connected) domains: ℑ+ given by I⁢m⁢(R′⁢(z))>0 and ℑ− given by I⁢m⁢(R′⁢(z))<0.

Observe that in ℑ+ (resp. ℑ−), the integral curves of the vector field R⁢(z)⁢∂z are turning counterclockwise (resp. clockwise).

3.2.2. Circle at infinity

Consider the closure of the curve of inflections ℑR in the extended complex plane ℂ∪𝕊1.

Lemma 3.11.

The intersection ℑR∩𝕊1 is:

  • •

    is empty if deg⁡Q−deg⁡P=1 and λ∉ℝ;

  • •

    coincides with the set {ϕ∞2,ϕ∞2+π2,ϕ∞2+π,ϕ∞2+3⁢π2} if deg⁡Q−deg⁡P=−1.

In the remaining two cases:

  • •

    deg⁡Q−deg⁡P=1 and λ∈ℝ;

  • •

    deg⁡Q−deg⁡P=0; or

  • •

    deg⁡Q−deg⁡P∉{−1,0,1}

the set ℑR∩𝕊1 consists of 2⁢k points forming a regular 2⁢k-gon for some k satisfying k≤max⁡{deg⁡P,deg⁡Q}+1.

Proof.

If k=deg⁡Q−deg⁡P∈ℤ∖{0,1}, then R′⁢(z) has an expansion of the form k⁢λk⁢zk−1+o⁢(zk−1) near ∞ from which the characterization of the infinite branches of the real locus of R′⁢(z) follows by Lemma 2.13 applied to either R′⁢(z) or to 1R′⁢(z) depending on whether k>0 or k<0 (clearly both have the same real locus outside their poles).

If deg⁡Q−deg⁡P=0, then R⁢(z) has an expansion λ+Azk+o⁢(z−k) for some A∈ℂ∗ and k∈ℕ∗ near ∞. (The case when R⁢(z) is constant is ruled out by the genericity assumptions). Therefore R′⁢(z) has an expansion −A⁢kzk+1+o⁢(z−k−1). We conclude that ℑR has 2⁢k infinite branches whose limit directions form a regular 2⁢k-gon.

If deg⁡Q−deg⁡P=1, then R⁢(z) has an expansion λ⁢z+A+B⁢z−k+o⁢(z−k) for some A∈ℂ, B∈ℂ∗, and k∈ℕ∗. (The case when R⁢(z) is a linear function is ruled out by the genericity assumptions). We obtain that R′⁢(z) is of the form λ−B⁢kzk+1+o⁢(z−k−1). Consequently, unless λ is real, the curve of inflections ℑR is compact in ℂ. If λ is real, the infinite branches of ℑR are asymptotically the same as that of the real locus of −k⁢Bzk+1. Therefore ℑR has 2⁢k infinite branches whose limit directions form a regular 2⁢k-gon.

In these last two cases, we have R′⁢(z)=Mzk+1+o⁢(z−k−1) for some M∈ℂ∗ and k≥1. The number k is the ramification index of either R−λ⁢z (for deg⁡Q−deg⁡P=1) or R (for deg⁡Q−deg⁡P=0) at infinity, thus K cannot be bigger than the degree max⁡{deg⁡P,deg⁡Q} of R. Therefore k≤max⁡{deg⁡P,deg⁡Q}+1.

∎

3.2.3. Singularities of the vector field

Next we deduce from Corollary 3.5 a proof of the statement that any root of P⁢(z) or Q⁢(s) belonging to ∂MC⁢HT automatically belongs to the curve of inflections.

Corollary 3.12.

Consider an operator T as in (1.1) such that MC⁢HT does not coincide with ℂ and has a nonempty interior. Let α be a zero or a pole of R⁢(z) such that α∈∂MC⁢HT. Then α also belongs to the curve of inflections ℑR. Additionally, the number of local branches of ℑR at α equals:

  • •

    a+1 if α is a pole of order a≥1;

  • •

    a−1 if α is a zero of order a≥2;

  • •

    some integer b≥1 if α is a simple zero.

Proof.

The statement is proved by direct computation of I⁢m⁢(R′) in case of a pole or a zero of order a≥2. If α is a simple zero of R⁢(z), then we have R⁢(α+ϵ)=R′⁢(α)⁢ϵ+o⁢(ϵ). If α∈∂MC⁢HT, then ϕα=a⁢r⁢g⁢(R′⁢(α))=0 (see Corollary 3.5). Thus α∈ℑR.

Unless R⁢(z) is linear, R⁢(z) is of the form R′⁢(α)⁢(z−α)+M⁢(z−α)d+o⁢(|z−α|d) for some d≥2 and M∈ℂ∗. Thus R′⁢(z)=R′⁢(α)+M⁢d⁢(z−α)d−1+o⁢(|z−α|d−1). Consequently, the number of local branches of the equation I⁢m⁢(R′)=0 equals d−1.

If R⁢(z)=λ⁢(z−α), then R⁢e⁢(λ)≥0 (otherwise MC⁢HT=ℂ) and I⁢m⁢(λ)≠0 (otherwise MC⁢HT is totally irregular). It follows that I⁢m⁢(R′⁢(z)) is a non-vanishing constant and the curve of inflections is empty. In this case, ℑR does not contain any zero or pole of R⁢(z) on the boundary of MC⁢HT. ∎

3.2.4. Tangency locus

Definition 3.13.

For the rational vector field R⁢(z)⁢∂z, the tangency locus 𝔗R is the subset of the curve of inflections ℑR where R⁢(z)⁢∂z is tangent to some branch of ℑR.

Proposition 3.14.

For an operator T as in (1.1), the tangency locus 𝔗R is the union of:

  • •

    at most max⁡{deg⁡Q,deg⁡P}+1 lines and;

  • •

    at most 2⁢(3⁢deg⁡P+deg⁡Q−1)2 points.

Proof.

For any point z∈𝒯R, an immediate computation involving the Taylor expansion of R′⁢(z) proves that z belongs to the intersection of the curve of inflections (given by Im⁡(R′)=0) with a real plane algebraic curve given by the equation Im⁡(R′′⁢R)=0. Indeed, the tangent line to ℑR at some z0∈ℑR is given by the equation R′′⁢(z0)⋅(z−z0)∈ℝ, and the associated ray direction is R⁢(z0). The degrees of these two curves are respectively deg⁡Q+3⁢deg⁡P−1 and 2⁢deg⁡Q+6⁢deg⁡P−2. Therefore, Bézout’s theorem implies that 𝔗R∩∂MC⁢HT contains at most 2⁢(deg⁡Q+3⁢deg⁡P−1)2 such points and some irreducible components corresponding to the common factors of the two equations.

By definition of the tangency locus these irreducible components are the integral curves of R⁢(z)⁢∂z contained in the curve of inflections. Such integral curves have identically vanishing curvature and therefore they are segments of straight lines. Therefore the relevant irreducible components are straight lines. But ℑR intersects 𝕊1 at most 2⁢max⁡{deg⁡Q,deg⁡P}+2 points by Lemma 3.11. Thus the number of the lines is at most max⁡{deg⁡Q,deg⁡P}+1. ∎

We deduce an estimate on the number of connected components of the transverse locus ℑR∗ of the curve of inflections. Denote d=3⁢deg⁡P+deg⁡Q−1=deg⁡ℑR.

Corollary 3.15.

For an operator T as in (1.1), the transverse locus ℑR∗ of the curve of inflections is formed by at most 2⁢d2+6⁢d+2 connected components.

Proof.

A connected component of ℑR∗ is either a smooth closed loop (so a connected component of ℑR) or an arc joining points at infinity, singular points of ℑR∗ or isolated points of the tangency locus.

Following Proposition 3.14, the tangent locus contains at most 2⁢d2 isolated points. Each of them is the endpoint of two arcs of the transverse locus.

Lemma 3.11 proves that at most 2⁢max⁡{deg⁡P,deg⁡Q}+2 arcs of the transverse locus go to infinity.

Lemma 3.7 provides the analog result for the multiple points of the curve of inflections. In the ”worst” case, poles of R⁢(z) and critical points of R′⁢(z) are simple. At most four arcs of the transverse locus are incident to such points. There are at most 4⁢deg⁡P+deg⁡Q−2≤2⁢d such points (see Corollary 3.8) so they are incident to at most 4⁢d arcs.

Adding these bounds, we obtain an upper bound 4⁢d2+10⁢d+4 on the number of ends of non-compact connected components of the transverse locus, i.e. there are at most 2⁢d2+5⁢d+2 non-compact connected components. By Lemma  3.9 the number of the compact connected components (loops) of ℑR is at most d, which gives the required upper bound. ∎

Corollary 3.16.

On each connected component of the transverse locus ℑR∗, the sign of Im⁡(R′′⁢R) remains constant. If Im⁡(R′′⁢R) is positive (resp. negative), then for any point z of the component, the associated ray r⁢(z) points towards ℑ+ (resp. ℑ−).

Proof.

Any regular point z of the curve of inflections satisfying Im⁡(R′′⁢(z)⁢R⁢(z))=0 belongs to the tangency locus (see the proof of Proposition 3.14). A direct computation proves the rest of the claim. ∎

3.3 Horns

In this section, we introduce some curvilinear triangles called horns and find conditions under which we can conclude that they do not belong to the minimal set MC⁢HT. Our aim is to prove that some parts of the boundary of the minimal sets are portions of integral curves of the vector field R⁢(z)⁢∂z.

3.3.1. Definitions

Recall that σ⁢(q) is the argument of R⁢(q), i.e. σ⁢(q)=Im⁡log⁡R⁢(q) and r⁢(q)=q+R⁢(q)⁢ℝ+ is the associated ray.

Definition 3.17.

Assume that a segment γpp′ of the positive trajectory of R⁢(z)⁢∂z starting at p∉𝒵⁢(P⁢Q) and ending at p′ doesn’t intersect the curve of inflections except possibly at p. Assume that the total variation of σ along γpp′ is less than π/2.

We define the horn △p′′p′p at p as an open curvilinear triangle formed by γpp′ and tangents to this trajectory at p and p′ intersecting at a point p′′.

Definition 3.18.

A horn △p′′p′p is called small positive (resp. small negative) if

  1. (1)

    for any point u∈△p′′p′p, the argument σ⁢(u+t⁢R⁢(u)) is monotone increasing (resp. decreasing) in the variable t as long as t≥0 and u+t⁢R⁢(u)∈△p′′p′p

  2. (2)

    for any two points u,v∈△p′′p′p, the scalar product (R⁢(u),R⁢(v)) is positive.

A horn △p′′p′p is called small if it is either small positive or small negative.

Remark 3.19.

A small positive horn becomes a small negative one after conjugation, i.e. after replacing R⁢(z) with R⁢(z¯)¯. Indeed,

(R⁢(u),R⁢(v))=Re⁡R⁢(u)⁢R⁢(v)¯

remains the same after the conjugation, and

d⁢σ⁢(u+t⁢R⁢(u))d⁢t⁢(t)=Im⁡R′⁢(u+t⁢R⁢(u))R⁢(u+t⁢R⁢(u))⁢R⁢(u)

changes sign.

Lemma 3.20.

The curve of inflections (given by Im⁡R′=0) does not intersect small horns.

Proof.

We have that dσ(u+tR(u)d⁢t|t=0=Im⁡R′⁢(u)≥0. Assume that we have the equality at some u∈△p′′p′p. Since △p′′p′p is open and R′ is an open map, this assumption will imply that dσ(u+tR(u)d⁢t|t=0 changes sign in △p′′p′p, which contradicts the smallness assumptions. ∎

We define the cone complementary to △p′′p′p (in short, the complementary cone) to be the open cone ∠p′′ with the apex p′′ bounded by part of the ray r⁢(p) starting at p′′ and by the ray extending the segment p′⁢p′′.

Lemma 3.21.

Consider a point p which neither belongs to 𝒵⁢(P⁢Q) nor to the interior of MC⁢HT. Assume that the integral curve γ of the vector field R⁢(z)⁢∂z containing p is not a straight line. Then there exists a horn △p′′p′p such that both △p′′p′p and its complementary cone ∠p′′ do not intersect MC⁢HT.

Proof.

Let D⁢(p)={|z−p|<δ𝒵⁢(p)=12⁢dist⁡(p,𝒵⁢(P⁢Q))}.

First, assume that p∉MC⁢HT. Then by definition, r⁢(p)⊂MC⁢HTc.

Choose some δ>0 and define p0=p and

pi=pi−1+δ⁢R⁢(pi−1)∈r⁢(pi)⊂MC⁢HTc∩D⁢(p),i=1,…,N=N⁢(δ)=O⁢(δ𝒵⁢(p)δ),

(we stop when pN+1∉D⁢(p)).

The broken line γ^ppN=∪i=1N[pi−1,pi]⊂MC⁢HTc∩D⁢(p) is the Euler approximation to the positive trajectory γp+ of R⁢(z)⁢∂z starting from p and converges to it (more exact, to the connected component γpp′⊂D⁢(p) of γp+∩D⁢(p) containing p) as δ→0. Thus γpp′⊂MC⁢HTc¯. Repeating this argument for all p~∉MC⁢HT sufficiently close to p we see that

γpp′⊂(MC⁢HTc¯)o=MC⁢HTc. (3.2)

If γpp′ is a subset of the curve of inflections then it is a part of a straight line, which is excluded by our assumption. Thus we can assume that for p′ sufficiently close to p the curve γpp′ intersects the curve of inflections only at p. Therefore γpp′ is convex and, choosing p′ closer to p if needed, we can assume that γpp′ is of angle smaller than π. Therefore

(⋃s∈γpp′r⁢(s))∘=∠p′′⁢⋃△p′′p′p⊂MC⁢HTc. (3.3)

Second, assume that p∈∂MC⁢HT and let γpp′ be a part of the connected piece of γp+∩D⁢(p) containing p such that γpp′ is convex and of angle smaller than π/2. Let pi∉MC⁢HT be a sequence of points tending to p and take pi′∈γpi+ such that γpipi′ converges to γpp′. By analyticity this convergence is uniform in C1 sense as well. Therefore

(⋃s∈γpp′r⁢(s))∘⊂⋃i(⋃s∈γpipi′r⁢(s))∘⊂MC⁢HTc, (3.4)

which finishes the proof.

∎

3.3.2. Small horns exist

Proposition 3.22.

For any point p∉𝒵⁢(P⁢Q) such that the trajectory γ⁢(p) of R starting at p is not a straight line, there exists a small horn △p′′p′p.

Proof.

Using an affine change of variables we can assume that p=0 and R⁢(0)=1. By assumption R⁢(z) is not a real rational function. Let

R⁢(u)=1+ρ⁢(u)+i⁢b⁢um+O⁢(um+1),b>0,ρ∈ℝ⁢[u],m≥1 (3.5)

be the Taylor expansion of R⁢(z) at 0 (the case m=1 is covered by Lemma 3.23). Here we can assume that b>0 by replacing R⁢(z) by R⁢(z¯)¯, if necessary.

First, we consider the case m=1, i.e. p∉ℑR.

Lemma 3.23.

For every compact set K not intersecting the curve of inflections ℑR, there is a δ=δ⁢(K)>0 such that for every p∈K, there is a small horn △p′′p′p of diameter greater than δ.

Proof.

Indeed, for any p∈K the function Re⁡R⁢(u)⁢R⁢(v)¯ is positive and Im⁡R′⁢(u)R⁢(u)⁢R⁢(v) are is non-zero at (p,p)∈ℂ2, so this remains true for all (u,v)∈ℂ2 such that dist⁡((p,p),(u,v))<δ=δ⁢(p) by continuity. This means that any △p′′p′p⊂Uδ⁢(p)⁢(p) is a small horn. The uniform lower bound follows from the continuity of δ⁢(p). ∎

From now on we assume that m≥2. Our next goal is to find the asymptotics of γ0 near 0 and the △p′′p′0. We abuse notation by writing the germ of γ0 as γ0={x+i⁢γ0⁢(x),x>0}.

Lemma 3.24.
γ0⁢(x)=bm+1⁢xm+1+O⁢(xm+2) (3.6)

and

△p′′p′0⊂{0<x<ϵ,0<y<γ0(x)}. (3.7)
Proof.

Note that

γ0⊂{Im⁡F=0}, where ⁢F′=1R,F⁢(0)=0. (3.8)

Indeed,

dd⁢t⁢Im⁡F⁢(γ0⁢(t))=Im⁡dd⁢t⁢F⁢(γ0⁢(t))=Im⁡(F′⋅γ0˙⁢(t))=0.

Now,

1R=11+ρ⁢(u)−i⁢b⁢um(1+ρ⁢(u))2+O⁢(um+1), (3.9)

so

F⁢(u)=u+ρ~⁢(u)−i⁢bm+1⁢um+1+O⁢(um+2),ρ~∈ℝ⁢[u].

For u=x+i⁢y we get

Im⁡F⁢(u)=y⁢(1+o⁢(1))−bm+1⁢xm+1+O⁢(um+2).

Recalling that γ0 is tangent to the real axis, we have y=o⁢(x). Therefore

{Im⁡F=0}={x+i⁢y:y=bm+1⁢xm+1+O⁢(xm+2)}⊂{y≥0}

near the origin, and the claim of the Lemma follows since r⁢(0)=ℝ+. ∎

Next, we have to check the two conditions in Definition 3.18 for △p′′p′0 with p′ sufficiently close to 0. The second condition is easy: since R⁢(0)=1 then the scalar product (R⁢(u),R⁢(v)) is positive for all u,v∈△p′′p′0 by continuity.

To check the first condition set u=x1+i⁢y1,v=x2+i⁢y2=u+t⁢R⁢(u)∈△p′′p′0 with t>0. By the second property of the small horns, we have x2>x1. By (3.7) we have yi=O⁢(xim+1). Combining (3.9) and

R′⁢(v)=ρ′⁢(v)+i⁢m⁢b⁢vm−1+O⁢(vm), (3.10)

we get

R′⁢(v)R⁢(v) =(ρ′⁢(x2)+i⁢m⁢b⁢x2m−1+O⁢(x2m))⁢1+ρ⁢(x2)−i⁢b⁢x2m+O⁢(x2m+1)(1+ρ⁢(x1))2
=ρ′⁢(x2)⁢(1+ρ⁢(x2))+i⁢m⁢b⁢x2m−1+O⁢(x2m)(1+ρ⁢(x2))2.

Thus, using (3.5), we get for Φ⁢(u,v)=(1+ρ⁢(x2))2⁢Im⁡R⁢(u)⁢R′⁢(v)R⁢(v) the equation

Φ =Im⁡([1+ρ⁢(x1)+i⁢b⁢x1m+O⁢(x1m+1)]⋅[ρ′⁢(x2)⁢(1+ρ⁢(x2))+i⁢m⁢b⁢x2m−1+O⁢(x2m)])
=m⁢b⁢x2m−1+O⁢(x2m)>0, (3.11)

where we use x1≤x2. This proves the first requirement of Definition 3.18. ∎

Corollary 3.25.

The germ of ℑR at p cannot lie between γp+ and r⁢(p).

Proof.

This would mean that this germ lies inside △p′′p′p which is impossible by Proposition 3.22 and Lemma 3.20. ∎

3.3.3. Removing small horns

We will use the following general Lemma

Lemma 3.26.

Assume that for some open set U⊂ℂ∖𝒵⁢(P⁢Q) and every point u∈U, the associated ray r⁢(u) lies in the union U∪(MC⁢HT)c. Then MC⁢HT∩U=∅.

Proof.

Indeed, if not then MC⁢HT∖U⊊MC⁢HT will be again invariant, which contradicts minimality of MC⁢HT. ∎

The crucial property of small horns is the following Lemma.

Lemma 3.27.

For any v∈△p′′p′p, one has r⁢(v)⊂△p′′p′p∪∠p′′.

Proof.

We prove the statement assuming that the small horn △p′′p′p is positive, the negative case will follow by conjugation.

Figure 2. Removing small horns.

Let u∈γpp′ be a point such that v∈r⁢(u). By definition of small horns, we have σ⁢(p)<σ⁢(u)<σ⁢(v), see Fig. 2.

The ray rv does not intersect γpp′. Indeed, assume that the ray r⁢(v) intersects γpp′ at a point s. Then at the intersection point the slope of γpp′ should be smaller than the slope of r⁢(v), i.e. σ⁢(s)<σ⁢(v) which contradicts the requirement that the slope is monotone increasing along the segment joining v and s.

Also r⁢(v) cannot intersect p⁢p′′ since σ⁢(v)>σ⁢(u) and σ⁢(u)>σ⁢(p), where u∈γpp′ such that v∈r⁢(u).

Thus r⁢(v) leaves △p′′p′p and enters ∠p′′ at some point of p′′⁢p′ with the slope σ⁢(p)<σ⁢(v)<σ⁢(p′). Thus r⁢(v) never leaves ∠p′′. ∎

Proposition 3.28.

Assume that △p′′p′p is a small horn and ∠p′′⊂MC⁢HTc. Then p is not in the interior of MC⁢HT.

Proof.

Follows from Lemmas 3.26 and 3.27. ∎

4 Boundary arcs

Recall that we consider an operator T whose minimal set MC⁢HT is different from ℂ and has a nonempty interior. We want to describe its boundary in combinatorial and dynamical terms. To do this, we introduce two set-valued functions.

Recall that in our terminology, MC⁢HT¯ is the closure of MC⁢HT in the extended plane ℂ∪𝕊1.

4.1 The correspondences Γ and Δ

Definition 4.1.

For any x∈∂MC⁢HT∖𝒵⁢(P⁢Q), we define:

  • •

    Γ⁢(x)={y∈γx+|y≠x}∩MC⁢HT¯ where γx+ is the positive trajectory of the vector field R⁢(z)⁢∂z starting at x;

  • •

    Δ⁢(x)={y∈r⁢(x)|y≠x}∩MC⁢HT¯.

Note that if y∈Γ⁢(x) or y∈Δ⁢(x), and x∈∂MC⁢HT, then y∈∂MC⁢HT as well.

Using correspondences Γ and Δ, we split the set of boundary points of MC⁢HT disjoint from the curve of inflections into the following three types.

Definition 4.2.

A point of ∂MC⁢HT∖(𝒵⁢(P⁢Q)∪ℑR) is a point of:

  • •

    local type if Γ⁢(z)≠∅ and Δ⁢(z)=∅;

  • •

    global type if Γ⁢(z)=∅ and Δ⁢(z)≠∅;

  • •

    extruding type if Γ⁢(z)≠∅ and Δ⁢(z)≠∅.

By Proposition 4.7 these are the only possibilities for points in ∂MC⁢HT∖ℑR.

4.2 Support lines

In this (sub)section We prove that for a given point z, the condition Δ⁢(z)≠∅ means that the associated ray r⁢(z) is a support line of MC⁢HT¯.

For any oriented support line of MC⁢HT¯, we define the co-orientation of its support in the following way. The support point x is:

  • •

    a direct support point if the standard orientation of ∂MC⁢HT and the orientation of the support line agree at x;

  • •

    an indirect support point otherwise.

In particular, if the support line is the positively oriented real axis, a support point x is called direct if the intersection of MC⁢HT with a neighborhood of x is contained in the upper half-plane (see Figure 3 for examples of indirect support points).

Definition 4.3.

Consider z∈ℂ such that:

  • •

    z does not belong to the tangency locus 𝒯R of the curve of inflections ℑR;

  • •

    z is not a root of P or Q.

Then we say that z∈𝔈+ (resp. 𝔈−) if the associated ray r⁢(z) is pointing inside the inflection domain ℑ+ (resp. ℑ−). This includes z∈ℑ+ (resp. ℑ−).

Refer to caption
Figure 3. The point where the red arrow is tangent to MC⁢HT is an indirect support point. The circular arrow indicates that the black point belongs to 𝔈+.
Lemma 4.4.

Consider z∈∂MC⁢HT∖𝒵⁢(P⁢Q) such that z∈𝔈+ (resp. 𝔈−). If y∈Δ⁢(z), then y is an indirect support point (resp. a direct support point).

Proof.

Without loss of generality, we can assume that z∈𝔈+, z=0, r⁢(z)=ℝ>0 and y=1. This implies that γ0 lies in the upper half-plane. By Lemma 3.21 there is a neighborhood V of y such that V∩(MC⁢HT¯)∘ is contained in the lower half-plane. Therefore y is an indirect support point. ∎

Lemma 4.5.

Take x,y∈∂MC⁢HT such that:

  • •

    x,y∈𝔈−∪𝔈+

  • •

    the associated rays r⁢(x) and r⁢(y) intersect at some point m∈ℂ;

  • •

    σ(y)∈]σ(x)−π,σ(x)[.

Then the open cone Γ with apex m and the interval of directions ]σ(y),σ(x)[ is disjoint from (MC⁢HT)∘ and there are the following subcases:

  • •

    either y∈𝔈+ or Δ⁢(y)⊂[y,m];

  • •

    either x∈𝔈− or Δ⁢(x)⊂[x,m].

Proof.

The path formed by the concatenation of segments [x,m] and [m,y] can be approached by a family paths joining x and y or a family of paths joining y and x whose interior points are disjoint from MC⁢HT. Lemma 2.9 applies to one of these families of paths so Γ is disjoint from (MC⁢HT)∘.

Then, we assume by contradiction that y∈𝔈− and some point z∈Δ⁢(y) does not belong to [y,m]. Since Γ is disjoint from (MC⁢HT)∘, it follows that z is an indirect support point of the line containing r⁢(y) which contradicts Lemma 4.4. Consequently either Δ⁢(y)⊂[y,m] or y∉𝔈−.

An analogous argument proves that either x∈𝔈− or Δ⁢(x)⊂[x,m]. ∎

4.3 Local arcs

In this section, we prove that local points (see Proposition 4.7) form local arcs.

Definition 4.6.

A local arc of ∂MC⁢HT is a maximal open arc of an integral curve of vector field R⁢(z)⁢∂z that contains only local points. In particular, it is disjoint from 𝒵⁢(P⁢Q) and ℑR.

Local arcs are oriented by the vector field R⁢(z)⁢∂z.

Using the geometry of horns (see Section 3.3), we can show that every local point actually belongs to a local arc of ∂MC⁢HT.

Proposition 4.7.

Consider a point p∈∂MC⁢HT and such that Δ⁢(p)=∅ and p∉𝒵⁢(P⁢Q)∪ℑR. Then, the germ of the integral curve γp of R⁢(z)⁢∂z passing through p belongs to ∂MC⁢HT.

Without loss of generality, we assume that p=0, r⁢(p)=ℝ+ and p∈𝔈+, so γpp′ lies in the upper half-plane. The proof consists of two steps illustrated by Figure 4 and Figure 5 respectively.

Lemma 4.8.

MC⁢HT lies above the integral curve γp of R⁢(z)⁢∂z passing through p.

Proof: see Fig. 4.

By Lemma 3.21 there exists p′∈γp+ such that the union △p′′p′p∪∠p′′ is outside of MC⁢HT. Let q′∈γpp′, q′≠p,p′, and let q′′∈ℝ+ be the intersection point of ℝ+ with the line tangent to γpp′ at q′. By Proposition 3.22 we can assume that △q′′q′p is a small horn at p, △q′′q′p⊂△p′′p′p. Clearly, σ⁢(q′)<σ⁢(p′).

The condition Δ⁢(p)=∅ implies that q′′∈MC⁢HTc. Moreover, as +∞∉Δ⁢(0), there is an open sector S with vertex on ℝ, containing [q′′,+∞) and disjoint from MC⁢HT.

For a point p~ sufficiently close to p and lying below γp consider a horn △q~′′q~′p~ with vertices q~′ and q~′′ close to q′ and q′′, respectively. By continuity, the part of r⁢(p~) starting from q~′′ lies in S. Also, q~′ lies in the horn △p′′p′p, so r⁢(q~′)∩MC⁢HT=∅ by Lemma 3.21.

Thus the complementary cone ∠q~′′ of p~ with vertex q~′′ lies outside of MC⁢HT. Therefore by Proposition 3.28 p~∉MC⁢HT∘, so p~∈MC⁢HTc¯. As this remains true for any point in a sufficiently small neighborhood of p~, we conclude p~∈(MC⁢HTc¯)∘=MC⁢HTc. Thus near p the set MC⁢HT lies above γp.

Figure 4. MC⁢HT lies above the trajectory γpp′.

∎

Lemma 4.9.

The boundary ∂MC⁢HT coincides with the integral curve γp in a neighborhood of p.

Proof: See Figure 5.

Lemma 4.8 and its proof implies that p lies on the boundary of a sector S with a vertex s∈ℝ, s≠p, and disjoint from MC⁢HT.

Recall that by Lemma 3.23 there is a lower bound δ on the size of small horns for all points close to p.

Assume that a point q∉MC⁢HT close to p lies above γp on a distance much smaller than δ and let △q′′q′q be its horn (necessarily small) of size δ/2. Both △q′′q′q and ∠q′′ lie outside of MC⁢HT.

Let p~ be a point on γq close to q and in the negative direction from q, let p~′′ be the intersection of r⁢(p~) and the line q′⁢q′′. The horn △p~′′q′p~ is smaller than δ, so is small.

The ray p~′′⁢q′ lies outside of MC⁢HT∘. Moreover, as long as the ray p′′+R⁢(p~)⁢ℝ+⊂r⁢(p~) lies outside MC⁢HT∘ we have ∠p~′′⊂MC⁢HTc, so p~∉MC⁢HT∘ by Proposition 3.28.

These arguments work for all points p~ sufficiently close to p and with slope σ⁢(p~) exceeding some negative number (namely the slope of the second side of S), in particular, for points slightly above γp−, the negative trajectory of γp. But p lies in the horn of size δ of such a point, which means that p∉MC⁢HT by Lemma 3.21, a contradiction.

Figure 5. γpp′ is boundary of MC⁢HT

∎

Local analysis of horns (see Section 3.3) leads to the following results about the correspondence Γ.

Corollary 4.10.

Consider z∈∂MC⁢HT such that z∉𝒵⁢(P⁢Q)∪ℑR. If Γ⁢(z)≠∅ then z is either the starting point or a point of a local arc.

Proof.

We just have to prove that for some y∈Γ⁢(z) such that y is close enough to z, the arc α of integral curve between z and y belongs to ∂MC⁢HT. This follows from Lemma 3.21. ∎

Proposition 4.11.

Any local arc is a locally strictly convex real-analytic curve. Its orientation coincides with the standard topological orientation of ∂MC⁢HT if it is contained in ℑ+ (and with the opposite orientation otherwise).

Proof.

As any integral curve of a real-analytic vector field, a local arc is a real-analytic curve in ℝ2. The arc has to be locally convex because otherwise, the associated ray (which is contained in the tangent line) at some point would cross the interior of MC⁢HT. Besides, direct computation shows that the curvature of an integral curve becomes zero only at points belonging to ℑR. ∎

Let us check that a local arc of ∂MC⁢HT cannot end inside an inflection domain. It cannot be periodic either.

Proposition 4.12.

Every local arc has an endpoint that belongs to 𝒵⁢(P⁢Q)∪ℑR.

Besides, if such an endpoint belongs to 𝒵⁢(P⁢Q), it is either a regular point or a simple pole of R⁢(z).

Proof.

Assume that the local arc γ is periodic and doesn’t intersect 𝒵⁢(P⁢Q)∪ℑR. Then following Proposition 4.11, γ is a strictly convex closed loop disjoint from 𝒵⁢(P⁢Q) and MC⁢HT is a strictly convex compact domain bounded by γ (in particular γ encompasses every point of 𝒵⁢(P⁢Q)). A neighborhood of γ is foliated by periodic integral curves γt of the vector field R⁢(z)⁢∂z that are also disjoint from 𝒵⁢(P⁢Q) and ℑR, so strictly convex as well. Each of them cuts out a strictly convex compact domain 𝒟t. For each point z in the complement of some 𝒟t, r⁢(z) remains disjoint from 𝒟t, which by Lemma 3.26 contradicts the minimality of MC⁢HT.

Now, we show that a local arc cannot go to infinity. When |deg⁡Q−deg⁡P|≤1, integral curves going to infinity enter the cones disjoint from MC⁢HT and never leave them (see Section 2.5) and otherwise MC⁢HT is trivial.

In the remaining cases, Poincaré-Bendixson theorem proves that a local arc γ has an ending point y∈∂MC⁢HT. We assume by contradiction that y∉𝒵⁢(P⁢Q)∪ℑR. We consider an arc β formed by a portion of the integral curve ending at y and a portion of the associated ray r⁢(y). Provided that β remains in the same inflection domain as y, the family of associated rays starting at the points of the arc β sweeps out a domain containing a cone (see Lemmas 2.9 and 3.21). Therefore, we have Δ⁢(y)=∅. Proposition 4.7 then proves that the local arc can be continued in a neighborhood of y.

If y∈𝒵⁢(P⁢Q) and is a zero or a pole of R⁢(z), then ℒy contains an interval of length at least π (see Definition 3.2). Corollary 3.5 proves that y is either a simple pole or a simple zero satisfying ϕy=0. In the latter case, y is a repelling singular point of R⁢(z)⁢∂z and therefore cannot be the endpoint of a local arc. ∎

As we will see in Section 4.6, in contrast with the case of ending points, a local arc can start inside an inflection domain at a point of extruding type.

4.4 Local connectedness of MC⁢HT

Here we show that MC⁢HT is locally connected, away from the part of the tangency locus that is formed by straight lines.

Lemma 4.13.

MC⁢HT is locally connected outside of ℑR∪𝒵⁢(P⁢Q).

Proof.

If z∈∂MC⁢HT is a point of local type, then the boundary locally coincides with the integral curve passing through z by 4.9. Next, let z∈∂MC⁢HT∖(𝒵⁢(P⁢Q)∪ℑR) with Δ⁢(z)≠0. Let y∈Δ⁢(z). As z∉𝒵⁢(P⁢Q)∪ℑR, by or 2.11 2.16 there is a unique germ of 𝔱⁢𝔯y passing through z, and it does so transversely to the integral curve of R⁢(z)⁢∂z passing through z. Then taking the backward trajectories of R⁢(z)⁢∂z of points in 𝔱⁢𝔯y, all points on one side of 𝔱⁢𝔯y near z belong to MC⁢HT. Taking now as a neighborhood basis a family of decreasing curvilinear quadrilaterals with two of the sides being trajectories of R⁢(z)⁢∂z and two sides being smooth curves on either side of 𝔱⁢𝔯y, it follows that all points on the other side of 𝔱⁢𝔯y have backward trajectories of R⁢(z)⁢∂z intersecting 𝔱⁢𝔯y inside these neighborhoods, provided they are sufficiently small. As for any x∈MC⁢HT, its backward trajectory belongs to MC⁢HT, it follows that MC⁢HT is locally connected at z. ∎

Lemma 4.14.

MC⁢HT is locally connected at zeros and poles of R.

Proof.

If z0 is a pole of R one can show using Proposition 3.12 in [AHN+24] and 3.5 that MC⁢HT is locally connected at z0. Next, for z0 a zero of R it follows from the same corollary and by using the local portrait of R⁢(z)⁢∂z near z0. ∎

Lemma 4.15.

MC⁢HT is locally connected at all z, such that γz is not a straight line.

Proof.

Since the integral curve γz of vector field R⁢(z)⁢∂z containing z is not a straight line, some germ of the negative part γz− lies outside of ℑR. Recall that if y∈MC⁢HT then the negative part γy− of γy necessarily lies in MC⁢HT.

Let γz−⁢(ϵ)⊂ℑRc be the part of the negative trajectory γz− lying strictly between z and y=gR−ϵ⁢(z) where gRt is the flow of R⁢(z)⁢∂z. By Lemma 4.13 there is a neighborhood Vy of y of size smaller than ϵ such that Vy∩MC⁢HT is connected. Now, let Uz=∪0≤t≤ϵgRt⁢(Vy). Clearly Uz is a neighborhood of z. We claim that Uz∩MC⁢HT is connected. Indeed, if w∈Uz∩MC⁢HT then w=gRt⁢(w′) for some w′∈Vy∩MC⁢HT, 0≤t≤ϵ. Since w′ lies in the same connected component of Uz∩MC⁢HT as y and w and w′ are jointed by γw−⊂Uz∩MC⁢HT this means that Uz∩MC⁢HT is connected. As ϵ can be chosen arbitrarily small, the statement follows. ∎

We denote by ℒ the union of all R-invariant lines.

Corollary 4.16.

∂MC⁢HT∖ℒR is parametrizable.

By Carathéodory’s theorem, the boundary of an open set is parametrizable if its boundary is locally connected. For each z∈∂MC⁢HT∖ℒR we have by 4.14 and Proposition 4.15 a neighborhood basis 𝒩⁢(z) consisting of sets such that U∩MC⁢HT is connected. Define U⁢(z)∈𝒩⁢(z) to be a set of the form U⁢(z)⊂B⁢(z,12⁢d⁢(z,ℒR)). The union

⋃z∈∂MC⁢HT∖ℒRU⁢(z)

is an open cover of ∂MC⁢HT∖ℒR and being a subset of ℂ, it has a countable subcover

⋃n∈ℕU⁢(zn).

For each zn,U⁢(zn)∩MC⁢HT¯ is locally connected and by 2.21 and the fact that all irregular points are contained in ℒR, its boundary is a Jordan curve away from the poles and zeros of R⁢(z). Hence ∂U⁢(zn)∩∂MC⁢HT¯ is parametrizable by Carathéodery’s theorem, injectively away from the zeros and poles of R⁢(z). We start with a z0 and use this parametrization of ∂U⁢(zn)∩∂MC⁢HT¯. Then for the smallest n=n1 such that U⁢(zn)∩U⁢(z0)≠∅,∂MC⁢HT∩U⁢(zn)⊄∂MC⁢HT∩U⁢(z0), we glue together the parametrizations of ∂(U⁢(zn1)∖U⁢(z0))∩∂MC⁢HT¯ with that of ∂U⁢(z0)∩∂MC⁢HT¯ along the end points of the parametrizations. We then have a parametrization of (U⁢(z0)∪U⁢(zn1))∩∂MC⁢HT≔ℬ1. We then take the smallest n=n2 such that U⁢(zn2)∩(U⁢(z0)∪U⁢(zn1))≠∅,∂MC⁢HT∩U⁢(zn)⊄ℬ1 and in the same way find a parametrization of ℬ2≔(U⁢(z0)∪U⁢(zn1)∪U⁢(zn2))∩∂MC⁢HT. We proceed in this way inductively to get a parametrization of ∂MC⁢HT∖ℒR, potentially pinched at the zeros and poles of R⁢(z) (but not anywhere else).

4.5 Global arcs

4.5.1. Additional results about correspondence Δ

Lemma 4.17.

Consider z∈∂MC⁢HT∖𝒵⁢(P⁢Q) such that z∈𝔈+ (resp. 𝔈−). If y∈∂MC⁢HT and y∈Δ⁢(z), then one of the following statements holds:

  • •

    y∈𝒵⁢(P⁢Q)∪ℑR;

  • •

    y∈ℑ− (resp. ℑ+).

Proof.

Without loss of generality, we assume that z=0, r⁢(z)=ℝ+ and z∈𝔈+.

We consider y∈Δ⁢(z) such that y∉𝒵⁢(P⁢Q)∪ℑR. If Δ⁢(y)=∅, then Proposition 4.7 shows that y belongs to a local arc. Besides, y is an indirect support point of the associated ray r⁢(z) (see Lemma 4.4). If y∈ℑ+, then the associated rays starting from a germ of the local arc at y sweep out a neighborhood of z and we get a contradiction. Therefore y∈ℑ−.

Now we consider the case where Δ⁢(y)≠∅ and assume by contradiction that y∈ℑ+. If I⁢m⁢(R⁢(y))>0, then Lemma 4.5 provides an immediate contradiction. If I⁢m⁢(R⁢(y))<0, then ℒy (see Definition 3.2) contains an interval of length strictly larger than π such that σ⁢(y) is one of the ends. It follows from Corollary 3.5 that y is a simple zero of R⁢(z) (and therefore y∈𝒵⁢(P⁢Q)).

If r⁢(y)=y+ℝ−, then for some small ϵ>0, points of the interval ]−ϵ,ϵ[ are disjoint from the interior of MC⁢HT. Associated rays starting from the points of ]−ϵ,ϵ[ sweep out an open cone containing a neighborhood of y. This contradicts the assumption y∈∂MC⁢HT. Therefore, r⁢(y)=y+ℝ+ and r⁢(y)⊂r⁢(z). In this case, for some small ϵ′>0, points of ]y−ϵ′,y[ are disjoint from the interior of MC⁢HT and their associated rays will sweep out a neighborhood of y if y∈ℑ+. Therefore, in that case we get that y∈ℑ−. Similar result holds for z∈𝔈−. ∎

Definition 4.18.

For any point z∈∂MC⁢HT such that s∉𝒵⁢(P⁢Q)∪ℑR and Δ⁢(z)≠∅, we define Δm⁢i⁢n⁢(z) (resp. Δm⁢a⁢x⁢(z)) as the infimum (resp. the supremum) in Δ⁢(z) of the order induced by the orientation of the associated ray r⁢(z).

Besides, we define Lz=|Δm⁢i⁢n⁢(z)−z|.

Lemma 4.19.

For any z∈∂MC⁢HT such that z∉𝒵⁢(P⁢Q)∪ℑR and Δ⁢(z)≠∅, we have Δm⁢i⁢n⁢(z)≠z and Lz≠0.

Proof.

Without loss of generality, we assume that z=0, z∈ℑ+ and z⁢(x)=ℝ>0. For any small enough real positive ϵ, we have R⁢e⁢(R⁢(ϵ)),I⁢m⁢(R⁢(ϵ))>0 and ϵ∈ℑ+. If such an ϵ belongs to Δ⁢(z), then it contradicts Lemma 4.17. ∎

Since MC⁢HT¯ is compact in ℂ∪𝕊1, it follows immediately that for any z, Δm⁢i⁢n⁢(z) is actually a point of ∂MC⁢HT¯.

Definition 4.20.

For any point z∈∂MC⁢HT such that z∉𝒵⁢(P⁢Q)∪ℑR and Δ⁢(z)≠∅, we define 𝒰⁢(z) as the connected component of (MC⁢HT)c∖[z,Δm⁢i⁢n⁢(z)] incident to:

  • •

    the right side of [z,Δm⁢i⁢n⁢(z)] if z∈ℑ+;

  • •

    the left side of [z,Δm⁢i⁢n⁢(z)] if z∈ℑ−.

i.e. in the half-plane bounded by r⁢(z) different to that containing the germ of the trajectory of R⁢(z)⁢∂z starting at z.

Lemma 4.21.

Consider z∈∂MC⁢HT such that z∉𝒵⁢(P⁢Q)∪ℑR, Δ⁢(z)≠∅ and z∈ℑ+ (resp. ℑ−). For any y∈∂MC⁢HT∩∂𝒰⁢(z) such that y∈ℑ+ (resp. ℑ−) and Δ⁢(y)≠∅, we have 𝒰⁢(y)⊂𝒰⁢(z).

Besides, if y≠z, we have 𝒰⁢(y)⊊𝒰⁢(z).

Proof.

By connectedness of MC⁢HT in the case deg⁡Q−deg⁡P=±1 and the asymptotic geometry of MC⁢HT in the case deg⁡Q−deg⁡P=0, it follows that the associated ray r⁢(y) intersect the associated ray r⁢(z). Applying Lemma 4.5 to r⁢(z) and r⁢(y), we see that Δ⁢(y)⊂∂𝒰⁢(z). Thus 𝒰⁢(y)⊂𝒰⁢(z).

By connectedness of MC⁢HT in the case deg⁡Q−deg⁡P=±1 and the asymptotic geometry of MC⁢HT in the case deg⁡Q−deg⁡P=0, it follows that the associated ray r⁢(y) intersects the associated ray r⁢(z). Applying Lemma 4.5 to r⁢(z) and r⁢(y), we see that Δ⁢(y)⊂∂𝒰⁢(z). Thus 𝒰⁢(y)⊂𝒰⁢(z).

When 𝒰⁢(y)=𝒰⁢(z), the associated ray r⁢(y) has to coincide with r⁢(z) (with the same orientation since y and z belong to the same inflection domain). It follows that y=z. ∎

4.5.2. Orientation of global arcs

By 4.16, the following notion is well-defined.

Definition 4.22.

A global arc in ∂MC⁢HT is a maximal open connected arc formed by points of global type.

Furthermore, for a global arc α defined on (tm⁢i⁢n,tm⁢a⁢x), its end point is defined as long as

ω+⁢(α)≔⋂t0∈(tm⁢i⁢n,tm⁢a⁢x){α⁢(t):t>t0}¯

is a singleton (and equals this element). The starting point is analogously defined if

ω−⁢(α)≔⋂t0∈(tm⁢i⁢n,tm⁢a⁢x){α⁢(t):t<t0}¯

is a singleton. If ω+⁢(α) is not a singleton, then it can only contain points contained in R-invariant lines, again by 4.16 and similarily for ω−⁢(α). Regardless if they are singletons or not, we call the sets ω±⁢(α) end accumulation and the start accumulation. In case they are in fact singletons, we will also call them end and starting points respectively. We have a geometrically meaningful way to define orientation on global arcs.

Lemma 4.23.

Any global arc (αt)t∈I can be oriented in such a way that for t′>t, we have:

  • •

    αt∈∂MC⁢HT∩∂𝒰⁢(αt′);

  • •

    𝒰⁢(αt′)⊃𝒰⁢(αt).

In particular, in ℑ+, the orientation of global arcs coincides with the standard topological orientation of ∂MC⁢HT (it coincides with the opposite orientation in ℑ−).

In particular, a global arc is an interval, i.e. it cannot be a closed loop.

Refer to caption
Figure 6. Two associated rays from the same global arc.
Proof.

Removal of αt from α cuts the arc into two pieces, one of which is contained in ∂𝒰⁢(αt) (see Figure 6). Lemma 4.21 then proves the inclusion of the sets of the form 𝒰⁢(αt) as t sweeps out the interval I which provides a meaningful orientation on the global arc. ∎

Lemma 4.24.

Along a global arc α, the function σ⁢(z)=arg⁡(R⁢(z)) is a monotone mapping of α to an interval in 𝕊1 with length at most π.

Besides, if σ⁢(αt)=σ⁢(αt′) for some t>t′, then Δ⁢(αt) coincides with the point at infinity σ⁢(αt)=σ⁢(αt′) that also belongs to Δ⁢(αt′).

Proof.

Consider two points αt and αt′ of a global arc satisfying t>t′ for the canonical orientation. By Lemma 4.23, αt′∈∂𝒰⁢(αt).

Without loss of generality, we assume that α is contained in ℑ+, αt=0 and r⁢(αt)=ℝ+. If σ(αt′)∈[−π,0[, any associated ray starting in a small enough neighborhood of αt′ will cross MC⁢HT. If σ⁢(αt′)=0, then the interior of the strip bounded by r⁢(αt),r⁢(αt′) and the portion of global arc between αt and αt′ is disjoint from MC⁢HT. It follows that Δ⁢(αt) contains only the point at infinity. In the remaining case, we have σ(αt′)∈]0,π[. ∎

Proposition 4.25.

Consider z∈∂MC⁢HT such that z∉𝒵⁢(P⁢Q)∪ℑR and Δ⁢(z)≠∅. Then, z is either the endpoint or a point of a global arc.

Proof.

We consider an arbitrarily small open arc α of ∂MC⁢HT∩𝒰⁢(z) ending at z. By assumptions, α is disjoint from 𝒵⁢(P⁢Q)∪ℑR. If some point y∈α satisfies Γ⁢(y)≠∅, then α partially coincides with a local arc. Since the ending point of any local arc belongs to 𝒵⁢(P⁢Q)∪ℑR (see Proposition 4.12), comparison of the orientation of local arcs and the orientation of ∂MC⁢HT in a given inflection domain (see Lemma 4.11) proves that z also belongs to this local arc. This is a contradiction. Therefore, any point y in the arc α satisfies Γ⁢(y)=∅. Proposition 4.7 then implies that each point of the arc α satisfies Δ⁢(y)≠∅ and is thus a point of global type. Therefore, z is either the endpoint or a point of a global arc containing α. ∎

Proposition 4.26.

If a point z∈∂MC⁢HT satisfies:

  • •

    z∉𝒵⁢(P⁢Q)∪ℑR;

  • •

    Δ⁢(z)≠∅;

  • •

    Γ⁢(z)=∅;

then z belongs to a global arc.

Proof.

Following Proposition 4.25, z is either the ending point or a point of a global arc. We consider a connected neighborhood V of z in ∂MC⁢HT that is disjoint from 𝒵⁢(P⁢Q)∪ℑR. Without loss of generality, we assume that V belongs to ℑ+.

We consider a point y∈V such that the oriented arc from y to z in ∂MC⁢HT has the same orientation as the standard topological orientation of the boundary. If Γ⁢(y)≠∅, then y is either a point or the starting point of a local arc (Corollary 4.10) that can be continued til z (see Proposition 4.12) since V is disjoint from 𝒵⁢(P⁢Q)∪ℑR. Therefore, Γ⁢(y)=∅ and it follows then from Proposition 4.7 that Δ⁢(y)≠∅. Thus, any such point y is a global point belonging to global arc α.

Then, we consider points y∈V such that the oriented arc from y to z in ∂MC⁢HT has the opposite orientation as the standard topological orientation of the boundary. If such point y satisfies Γ⁢(y)≠∅, then it is a point or the starting point of a local arc. Since V is connected and disjoint from 𝒵⁢(P⁢Q)∪ℑR, it contains at most one local arc starting at a point of extruding type. The complement of the closure of this local arc in V coincides with global arc β. By hypothesis, z is not a point of extruding type so it belongs to a global arc β. ∎

Proposition 4.27.

If z0∈ℂ is the endpoint of a global arc α and is neither a zero nor a pole of R⁢(z), then Δ⁢(z0)≠∅.

Proof.

We assume that α is parameterized by the interval ]0,1[ (with the correct orientation) and α⁢(t)→z0 as t→1. For any n≥2, we pick a point βn∈Δ⁢(α⁢(1−1/n)). Since ℂ∪𝕊1 is compact, the sequence (βn)n≥2 has an accumulation point β. Since z0 is the endpoint of α, the point β cannot coincide with z0 (see Lemma 4.23). It follows that a family of associated rays accumulates on a half-line starting at z0 and containing β. Since arg⁡(R⁢(z)) is continuous in a neighborhood of z0, we get that this half-line coincides with the associated ray r⁢(z0). ∎

Definition 4.28.

A point z∈∂MC⁢HT is a non-convexity point if there is a cone 𝒞 at z of angle strictly bigger than π and a neighborhood V of z such that 𝒞∩V⊂MC⁢HT.

For a point z0 for which Δ⁢(z0) consists of a single point u satisfying the condition R⁢(z0)+(u−z0)⁢R′⁢(z0)≠0, Lemma 2.11 proves that: (i) the root trail 𝔱⁢𝔯u has a unique branch at z0, (ii) it is contained in MC⁢HT, and (iii) its tangent slope is the argument of R2⁢(z0)R⁢(z0)+(u−z0)⁢R′⁢(z0) (mod π). These lemma yields that if at some point z0, Δ⁢(z0) contains more than one point, then ∂MC⁢HT cannot be smooth at z0:

Proposition 4.29.

At a point z∉ℑR∪𝒵⁢(P⁢Q) such that Δ⁢(z) contains at least two points, the boundary ∂MC⁢HT has a non-convexity point.

Proof.

First assume that Δ⁢(z) contains two points u,v both of which are not points at infinity. Lemma 2.11 proves that z belongs to two distinct root trails. Assuming that R⁢(z)+(u−z)⁢R′⁢(z) and R⁢(z)+(v−z)⁢R′⁢(z) are nonzero, the tangent slopes of these root trails at z are determined by the argument of R2⁢(z)R⁢(z)+(u−z)⁢R′⁢(z) and R2⁢(z)R⁢(z)+(v−z)⁢R′⁢(z). By hypothesis, we have I⁢m⁢(R′⁢(z))≠0 and these two branches intersect transversely at z and the claim follows, taking the backward trajectories of the two root-trails. If R⁢(z)+(u−z)⁢R′⁢(z)=0, then two branches of the root trail intersect transversely.

In the remaining case, Δ⁢(z) contains exactly one point u satisfying the condition R⁢(z)+(u−z)⁢R′⁢(z)≠0 and a point σ⁢(z) at infinity. Then the root trail of u at z has a slope given by the argument of R2⁢(z)R⁢(z)+(u−z)⁢R′⁢(z) (or R⁢(z)R′⁢(z) if u is at infinity, see Lemma 2.16). Similarly, R′⁢(z)∉ℝ so these curves intersect transversely at z. Summarizing we see that in all possible cases, the boundary ∂MC⁢HT has a non-convexity point. ∎

4.6 Points of extruding type

Outside the local and the global arcs, the only singular boundary points in the complement of 𝒵⁢(P⁢Q)∪ℑR which can occur are points of extruding type.

Proposition 4.30.

Let z be a point of extruding type in ∂MC⁢HT. Then z is both the endpoint of a global arc and the starting point of a local arc.

The boundary ∂MC⁢HT is not C1 at z and z is a non-convexity point.

Proof.

See Fig. 7 below. By definition of the correspondence Γ, and local considerations of R, z is the starting point of a local arc. Propositions 4.25 and  4.26 show that z is the ending point of a global arc.

For any point u∈Δ⁢(z), the root trail 𝔱⁢𝔯u has a unique local branch at z and its tangent direction is the argument of R2⁢(z)R⁢(z)+(u−z)⁢R′⁢(z) (mod π), see Lemma 2.11. Indeed, R⁢(z)+(u−z)⁢R′⁢(z)≠0 because u−z is real collinear to R⁢(z) while Im⁡(R′⁢(z))≠0. Since z∉ℑR, this branch transversely intersects the integral curve of R⁢(z)⁢∂z containing z. Both of these branches are (semi)analytic curves contained in MC⁢HT and the associated rays of the points lying on the negative part of γz intersect 𝔱⁢𝔯u⊂MC⁢HT. Thus the negative part of γz is disjoint from ∂MC⁢HT. ∎

Refer to caption
Figure 7. Negative part of γz cannot be on the boundary as 𝔱⁢𝔯u⊂MC⁢HT.

4.7 Boundary arcs in inflection domains

Proposition 4.31.

For any connected component 𝒟 of the complement of the curve of inflections ℑR, ∂MC⁢HT∩𝒟 is a union of disjoint topological arcs. In each of them, local and global arcs have the same orientation. If Im⁡(R′) is positive (resp. negative) in 𝒟 then the latter orientation coincides with (is opposite to) the topological orientation of ∂MC⁢HT.

Proof.

The statement about orientation follows from Proposition 4.11 and Lemma 4.23. Proposition 4.30 shows that a point of extruding type is incident to a local and a global arcs. It remains to prove that any point of 𝒵⁢(P⁢Q)∩𝒟 is incident to at most two arcs.

Such a point z0 is neither a zero nor a pole of R⁢(z), see Corollary 3.12. Therefore, 3.6 together with the fact that all irregular points are contained in ℐR proves the statement. ∎

5 Singular boundary points on the curve of inflections

At points belonging to the curve of inflections the boundary ∂MC⁢HT can display more complicated behaviours. In this section, we classify boundary points that belong to the transverse locus ℑR∗ of the curve of inflections (see Definition 1.7). For the following definition, recall the 1.8.

Definition 5.1.

A point of ∂MC⁢HT∖𝒵⁢(P⁢Q) belonging to the transverse locus ℑR∗ is a point of:

  • •

    bouncing type if Δ+≠∅ and Γ∪Δ−≠∅;

  • •

    switch type if Δ+≠∅ and Γ∪Δ−=∅;

  • •

    C1-inflection type if Δ+=∅, Δ−≠∅ and Γ=∅;

  • •

    C2-inflection type if Δ+=∅ and either Δ−=∅ or Γ≠∅.

5.1 Horns at points of the transverse locus

At a point p∈ℑR∗, the curve of inflections is smooth and the vector field R⁢(z)⁢∂z is transversal to it. This means that by (3.5) we can assume that

R⁢(u)=1+ρ⁢u+(a+i⁢b)⁢u2+… (5.1)

where we assumed that p=0. The condition Im⁡R′⁢(0)=0 means that ρ∈ℝ, and the transversality condition is equivalent to b≠0. Without loss of generality we can assume that b>0. In other words, m=2 in (3.5) which implies that the integral curves locally look like cubic curves with inflections at these points.

We define the diameter of a horn △p′′p′p to be the least upper bound t0>0 of all t′>0 such that there is u∈△p′′p′p such that u+t⁢R⁢(u)∈△p′′p′p for all t∈(0,t′).

Lemma 5.2.

For p∈ℑR∗, there exists a neighborhood Ω of p and ϵ>0 such that for all points u∈Ω+¯, where Ω+=ℑ+∩Ω, there exists a small horn of diameter greater than ϵ.

Proof.

Without loss of generality we assume p=0. Consider the function

T⁢(u,t)=d⁢σ⁢(u+t⁢R⁢(u))d⁢t⁢(t)=Im⁡[R′⁢(u+t⁢R⁢(u))R⁢(u+t⁢R⁢(u))⁢R⁢(u)]

defined in ℂu×ℝt. Note that by definition ℑ+={T⁢(u,0)>0}. Since T⁢(0,t)=2⁢b⁢t+O⁢(t2), we have ∂T∂t⁢(0,0)=2⁢b>0. Therefore ∂∂t⁢T⁢(u,t)>b>0 in some sufficiently small neighborhood U~×(−ϵ~,ϵ~) of (0,0). Let U~+=ℑ+∩U~. By definition of ℑ+, we have U~+×{0}⊂{T>0}. Taken together, this implies that U~+×[0,ϵ~]⊂{T>0} for some ϵ~>0. This means that the argument of R⁢(u+t⁢R⁢(u)) is monotone increasing for 0<t<ϵ~ and for every u∈U~+.

By transversality of ℑR and R at 0 we have

∂∂t⁢T⁢(gRt⁢(0)+s⁢R⁢(gRt⁢(0)),0)|s=t=0=∂∂s⁢T⁢(gRt⁢(0)+s⁢R⁢(gRt⁢(0)),0)|s=t=0>0,

where gRt is the flow of R⁢(z)⁢∂z.

Thus there is a neighborhood Ω⊂U~ of 0 and ϵ<ϵ~ such that T⁢(gRt⁢(u)+s⁢R⁢(gRt⁢(u)),0)>0 as soon as u∈Ω+=ℑ+∩Ω and t,s∈[0,ϵ]. Decreasing Ω,ϵ if needed, we can assume that

gRt⁢(u)+s⁢R⁢(gRt⁢(u))⊂U~+for all ⁢u∈Ω+⁢ and⁢t,s∈[0,ϵ].

Thus for every u∈Ω+, any horn of diameter at most ϵ lies in U~+ and is therefore a small horn.

If u∈Ω+¯ then the horn △u′′u′u of diameter ϵ lies in a union of small horns △u~′′u~′u~ of diameter ϵ, where u~∈△u′′u′u. Thus △u′′u′u is a small horn as well.

∎

5.2 Points of bouncing type

Proposition 5.3.

Let z be a point of bouncing type in ∂MC⁢HT:

Δ+⁢(z)≠∅andΓ⁢(z)∪Δ−⁢(z)≠∅.

Then z is the ending point of a global arc and also the starting point of another arc which can either be local or global.

Further, z is is a point of nonconvexity and for small enough neighborhoods V of z, one has V∩∂MC⁢HT∩ℑR={z}.

Proof.

Without loss of generality, we can assume that z=0, R⁢(0)=1, and R′⁢(0)∈ℝ∖{0}. Since z belongs to ℑR∗, we get R′′⁢(0)=a+b⁢i with a∈ℝ and b∈ℝ∗. Without loss of generality, we can assume that b>0 and therefore γz lies in the upper half-plane.

Let u+∈Δ+⁢(0)≠∅. If u+∉Δ0⁢(0) then the germ of 𝔱⁢𝔯u+ at 0 is smooth and tangent to ℝ at 0 and contained in the lower half-plane, see Lemma 2.11 and Proposition 2.19). Otherwise it consists of two branches transversal to ℝ and orthogonal one to another. Denote by α+⊂𝔱⁢𝔯u+⊂MC⁢HT the arc starting at 0 and lying in the lower right quadrant.

Similarly, since Γ⁢(0)∪Δ−⁢(0) is nonempty there exists an arc α− (either portion of an integral curve γ0 or a root trail 𝔱⁢𝔯u− of a u−∈Δ−⁢(0)) starting at 0, contained in the upper right quadrant, and belonging to MC⁢HT.

Denote by α=α−∪α+, and let V be a small neighborhood of 0 such that α cuts V into two parts. The part V− of V to the left of α is entirely contained in MC⁢HT (as r⁢(u) intersects α⊂MC⁢HT for any u∈V−). This domain contains the intersection of V with a cone with vertex at 0 and of angle strictly larger than π. It also contains all the intersection of V with ℑR excepted for the point 0, see Lemmas 2.11, 2.16 and Remark 2.17.

It remains to prove that in a neighborhood of z, ∂MC⁢HT is formed by exactly two arcs.

Lemma 5.4.

Assume that Γ⁢(0)=∅ and Δ−⁢(0)≠∅. Then 0 is a starting point of a global arc.

Proof.

By Lemma 3.21 the points lying below the forward trajectory γ0+ of 0 are not in MC⁢HT. Let q be a point lying slightly above γ0 and near 0 such that q∉MC⁢HT (it exists since otherwise γ0+⊂∂MC⁢HT). Denote by p~=p~⁢(q) the first point on the backward trajectory γq− starting from q such that r⁢(p~)∩MC⁢HT≠∅, in particular p~∈MC⁢HT and therefore γp~−⊂MC⁢HT. This point exists since the point of intersection of γq− with α has this property by definition of α.

Let γp~q be the closed piece of trajectory of R joining p~ and q. By definition of p~, all point of the (evidently closed) set γp~q∩MC⁢HT except p~ are necessarily in ∂MC⁢HT and of local type. Proposition 4.7 then implies γp~q∩∂MC⁢HT={p~}: indeed, otherwise the set γp~q∩MC⁢HT cannot be closed.

The trajectory γ0+ is convex, so taking q sufficiently close to γ0+ we can assume that r⁢(q)⊂MC⁢HTc intersect γ0+ and therefore all trajectories of R⁢∂z close to and above γ0+. The points p~⁢(q′), where q′∈r⁢(q) lies above γ0+, form a global arc of ∂MC⁢HT starting at 0 and lying between α− and γ0 in the upper right quadrant.

∎

If Γ⁢(0)≠∅, then a local arc whose germ is contained in the upper half-plane starts at 0.

In both cases the portion of the boundary ∂MC⁢HT in a neighborhood of 0 in the lower right quadrant is a global arc ending at z. Indeed, let q∈△z′′z′z⊂MC⁢HTc and denote again by p~=p~⁢(q) the first point on γq such that r⁢(p~)∩MC⁢HT≠∅, in particular p~∈MC⁢HT. As before, it lies on γq− between q and γq−∩α. Repeating the arguments of Lemma 5.4 we see that γp~q∩MC⁢HT={p~} and the points p~ form a global arc of ∂MC⁢HT lying in the lower half-plane and ending at z.

Refer to caption
Figure 8. Bouncing type

∎

5.3 Points of C2-inflection type

Recall that a point z∈∂MC⁢HT is called a point of C2-inflection type if Δ+⁢(z)=∅ and either Δ−⁢(z)=∅ or Γ⁢(z)≠∅.

Proposition 5.5.

Consider a point p of ∂MC⁢HT∖𝒵⁢(P⁢Q) belonging to the transverse locus ℑR∗. If Δ⁢(p)=∅, then Γ⁢(p)≠∅ and p is the starting point of a local arc.

Proof.

We keep the previous normalization p=0, R⁢(0)=1, Im⁡R′′⁢(0)=b>0. The positive trajectory γ0+ cuts ℑ+ into two parts, one containing the convex hull of γ0 (denoted by Ω++) and another one denoted by Ω+−.

The arguments of Lemma 4.8 (using Lemma 5.2 instead of Lemma 3.23) can be repeated word by word as long as p~∈ℑR∗∩∂Ω++, see Figure 4. This proves that MC⁢HT does not intersect Ω+−. Together with Δ⁢(0)=∅ this implies that some small sector S−={−ϵ<arg⁡z<0} does not intersect MC⁢HT.

Now, assume by contradiction, that there exists q∈Ω++∖MC⁢HT. Shrinking Ω++ if needed we can repeat the arguments of Lemma 4.9 as long as p~∈ℑ+, see Figure 5, to conclude that Ω++∩MC⁢HT=∅.

Therefore there is a neighbourhood U+ of 0 in ℑ+ which is disjoint from MC⁢HT. We can assume that U+ is the intersection of ℑ+ with a small disk centered at 0.

For sufficiently small ϵ>0, the set U=U+∪{−ϵ<arg⁡z<ϵ} is disjoint from MC⁢HT; the part lying in the lower half-plane is in U+∪S− and the part lying in the upper half-plane is in U+∪△p′′p′0∪∠p′′.

Now, take a small neighborhood U− of 0 in Ω− bounded by a convex curve transversal to R. For any u∈U−, the ray r⁢(u), being close to R+, lies inside U∪U−. By Lemma 3.26, this implies that U−⊂MC⁢HTc, and therefore 0∉MC⁢HT, a contradiction. Thus Ω++⊂MC⁢HT and γ0⊂∂MC⁢HT. ∎

Proposition 5.6.

Consider a point p of C2-inflection type. Then there is a neighborhood V of p in which ∂MC⁢HT is formed by:

  • •

    a portion of a local arc γp parameterized by an interval [0,ϵ[, ϵ>0 with γp⁢(0)=p;

  • •

    a portion of a global arc γ~p parameterized by [0,ϵ[ and such that γ~p⁢(0)=p and Δ⁢(γ~p⁢(t))={γp⁢(t)}.

In particular, p is simultaneously the starting point of a local arc and the starting point of a global arc.

Proof.

We again assume that p=0 and R⁢(0)=1. Following the definition of a point of C2-inflection type (see Theorem 1.9), we have that Δ+⁢(0)=∅ and either Δ−⁢(0)=∅ or Γ⁢(0)≠∅. By Proposition 5.5, if Δ−⁢(0)=∅, then Γ⁢(0) is also nonempty. Therefore, in both cases 0 is the starting point of a local arc γ0 and we deduce the shape of the boundary close to 0 from the assumption Δ+⁢(0)=∅.

The curve γ0 divides the domain ℑ+ into two parts, and as before we denote the one containing a horn of 0 by Ω+−.

Lemma 5.7.

Ω+−∩MC⁢HT=∅.

Proof.

At first, consider the case R′⁢(0)<0. Set I−≔∂Ω+−∩ℑR∖{0}. We claim that r⁢(u)∩MC⁢HT=∅ for all u∈I− sufficiently close to 0.

Set ρ≔−(R′⁢(0))−1. By assumption MC⁢HT∩ℝ+=Δ−⁢(0) is a compact subset of (ρ,+∞], so MC⁢HT∩ℝ+⊂[ρ′,+∞], ρ′>ρ. Again, by closedness of MC⁢HT and by Lemma 3.21 we can assume that for any ϵ>0 and all sufficiently small 0<δ′<δ′⁢(ϵ)

MC⁢HT∩{|Im⁡z|<δ′}⊂{Re⁡z>ρ′−ϵ,Im⁡z≤0}∪{Re⁡z<ϵ}. (5.2)

This means that MC⁢HTc contains not only the horn △p′′p′0 but also the box

Π={−δ′<Imz≤0,ϵ<Rez<ρ+ϵ}

(we take ϵ<ρ′−ρ2).

For all u∈I−, the slope σ⁢(u) is positive:

Im⁡R⁢(u)=R′⁢(0)⁢Im⁡u+O⁢(u2)>0, (5.3)

as Im⁡u<0 and Re⁡u=O⁢(Im⁡u) by transversality of ℑR and r⁢(0). Moreover, by (5.3) we have Im⁡(u+t⁢R⁢(u))=0 for t=ρ+O⁢(u), thus r⁢(u)∩ℝ+=ρ+O⁢(u). Therefore for any ϵ>0 and for any point u∈I− sufficiently close to 0, the ray r⁢(u) has an arbitrarily small slope and r⁢(u)∩ℝ+∈(ρ−ϵ,ρ+ϵ). Therefore

r⁢(u)∩{Im⁡z>0}⊂△p′′p′0∪∠p′′⊂MC⁢HTc (5.4)

for all u∈I− sufficiently close to 0.

Let u∈I−, |Im⁡u|<δ′, and take u′′∈r⁢(u) with Re⁡u′′=ϵ. If ϵ is sufficiently small then by Lemma 5.2 we can assume that the horn △u′′u′u is small.

The complementary cone ∠u′′ lies above r⁢(u) and to the right of {Re⁡z>ϵ}. Thus

∠u′′∩{Im⁡z<0}⊂Π

and therefore ∠u′′∩{Im⁡z≤0}⊂MC⁢HTc. Choosing ϵ sufficiently small we can assume that the angle of the sector ∠u′′ is small enough to conclude from (5.4) that ∠u′′∩{Im⁡z>0}⊂MC⁢HTc as well. This implies that u∈MC⁢HTc and therefore Ω+−⊂MC⁢HTc as well.

Refer to caption
Figure 9. ∠u′′ lies in the complement to MC⁢HT

If R′⁢(0)≥0 then Δ−⁢(0)⊂Δ+⁢(0)=∅, so Δ⁢(0)=∅. Then the arguments of Lemma 4.8 are applicable for all p~∈Ω+−, which proves the required claim in this case as well.

∎

Let γ~0 be the set of points in ℑ− whose associated rays are tangent to the positive trajectory γ0 of R⁢(z)⁢∂z starting at 0.

Lemma 5.8.

For R as in (5.1) the local arc γ0 is described by the relation y⁢(x)=b3⁢x3+o⁢(x3), x≥0 and the curve γ~0 is described by y⁢(x)=5⁢b3⁢x3+o⁢(x3), x≤0.

Proof.

Using (3.6) for (5.1) we see that

γ0⁢(t)=t+o⁢(t)+i⁢(b3⁢t3+O⁢(t4))

with a∈ℝ and therefore the slope σ⁢(γ0⁢(t))=b⁢t2+O⁢(t3).

The point u∈γ~0 whose associated ray r⁢(u) is tangent to γ0 at γ0⁢(t) has the form

u=γ0⁢(t)−s⁢R⁢(γ0⁢(t))=t−s+o⁢(t)+i⁢(−s⁢b⁢t2+b3⁢t3+O⁢(t4))

with the condition σ⁢(u)=σ⁢(γ0⁢(t))=b⁢t2+O⁢(t3). The latter condition means that s=2⁢t+o⁢(t) and therefore

u=−t+o⁢(t)−i⁢(53⁢b⁢t3+O⁢(t4)).

∎

Lemma 5.9.

For any u∈γ~0, the ray r⁢(u) does not intersect γ~0∪γ0 between u and the point of tangency z=z⁢(u) of r⁢(u) and γ0.

Proof.

By Lemma 5.8, there is function γ⁢(x) such that γ0∪γ~0={y=γ⁢(x)}, with γ⁢(x)=b3⁢x3+o⁢(x3) for x>0 and γ⁢(x)=5⁢b3⁢x3+o⁢(x3) for x<0. Both expressions, being power series, can be differentiated and produce asymptotic formulae for γ′⁢(x),γ′′⁢(x) as well. In particular, γ′′⁢(x) is continuous and monotone on the interval [Re⁡α,Re⁡z].

Assume r⁢(u)⊂{y=k⁢x+b}. By construction, γ^=γ⁢(x)−k⁢x−b vanishes at Re⁡u and has a double zero at Re⁡z. Any other point of intersection of r⁢(u) and γ~0∪γ0 will imply the existence of another zero of γ^ on [Re⁡u,Re⁡z], thus γ^ will have four zeros on this interval counting multiplicities. By Rolle’s Theorem this will imply the existence of two zeros of γ^′′=γ′′ on [Re⁡u,Re⁡z], which contradicts monotonicity of γ′′. ∎

By Lemma 5.8 the curve γ~0 is tangent to ℝ, thus transversal to ℑR and divides Ω− into two parts. Denote by Ω−− the closed part consisting of points whose associated rays do not intersect γ0 and let Ω−+ denotes the second part. Clearly Ω−+⊂MC⁢HT.

Corollary 5.10.

Let u∈γ~0 and set u+≔r⁢(u)∩ℑR and r+⁢(u)≔r⁢(u)∖Ω−−=u++R⁢(u)⁢ℝ+. Then r+⁢(u)∩(MC⁢HT)∘=∅.

Proof.

Indeed, the piece of r+⁢(u) between u+ and the point of tangency z=z⁢(u) of r⁢(u) and γ0 lies in Ω+− by Lemma 5.9, and the remaining piece coincides with r⁢(z). Thus the claim follows from Lemmas 5.7 and  3.21. ∎

Lemma 5.11.

For any u∈Ω−−∖γ~0, one has

  1. (1)

    r⁢(u)∩γ~0=∅,

  2. (2)

    r⁢(u)∖Ω−−⊂MC⁢HTc.

Proof.

Let γαβ be the piece of trajectory of R⁢(z)⁢∂z containing u and with endpoints α=α⁢(u)∈γ~0 and β=β⁢(u)∈ℑR: by Lemma 5.8 γ~0 is transversal to the trajectories of R⁢(z)⁢∂z near 0 except γ0. Let D⁢(u)=∪z∈γαβr⁢(z) be the domain sweeped by the rays associated to the points of γαβ. Since γαβ is convex, ∂D⁢(u)=r⁢(α)∪γαβ∪r⁢(β). By Lemmas 5.9,  5.8, and  5.7, ∂D⁢(u)∩γ~0={α}, so r⁢(u)∩γ~0=∅.

The boundary of D+⁢(u)=D⁢(u)∖Ω−− consists of r⁢(β), the piece of ℑR lying between β and the point α+ and r+⁢(α) which do not intersect MC⁢HT∘ by Lemma 5.7 and Corollary 5.10. This implies the second claim of the Lemma. ∎

Proof of Proposition 5.6.

Take some u∈Ω−−∘ and let Ω−−⁢(u) be the curvilinear triangle bounded by γαβ, γ~0 and ℑ. The ray r⁢(u) does not cross γαβ by convexity and does not cross γ~0 by Lemma 5.11. Therefore it leaves the domain Ω−−⁢(u) through ℑ, with r⁢(u)∖Ω−−⁢(u)⊂MC⁢HTc Lemma 5.11. Thus r⁢(u)⊂Ω−−⁢(u)∪MC⁢HTc.

As Ω−−⁢(u′)⊂Ω−−⁢(u) for any u′∈Ω−−⁢(u), this means that r⁢(u′)⊂Ω−−⁢(u)∪MC⁢HTc for all u′∈Ω−−⁢(u), and therefore Ω−−∩MC⁢HT=∅ by Lemma 3.26.

As Ω−+⊂MC⁢HT, we see that γ~0⊂∂MC⁢HT.

∎

Refer to caption
Figure 10. A point in the transverse locus of ∂MC⁢HT∩ℑR with empty Δ correspondence is the starting point of a global arc.

∎

5.4 Points of C1-inflection type

Recall that a point p∈∂MC⁢HT∖𝒵⁢(P⁢Q) belonging to the transverse locus ℑR∗ is called a point of C1-inflection type if Δ+⁢(p)=Γ⁢(p)=∅ and Δ−⁢(p)≠∅.

Proposition 5.12.

A point p∈∂MC⁢HT∩ℑR∗ of C1−inflection type is the starting point of two global arcs (one in each of the incident inflection domains).

Proof.

We assume that p=0, R⁢(0)=1 and γ0⊂ℑ+ lies in the upper half-plane. We use z=x+i⁢y notations.

Exactly as in the case of bouncing type, the conditions Δ−≠∅ and Γ=∅ imply that 0 is a starting point of a global arc, see Lemma 5.4. Denote this arc by η={y=ξ⁢(x)}. The arguments of Lemma 5.4 show that η lies between 𝔱⁢𝔯+∞ and 𝔱⁢𝔯u∞, where u∞=supΔ−. Thus η has second order tangency with ℝ+. Let k⁢(x) be the point of intersection of r⁢((x,ξ⁢(x))) with ℝ+. One can easily see that k⁢(x) is an increasing function, and k⁢(x)→u∞ as x→0. This means that limx→0+ξ⁢(x)/x2 exists and is positive.

We repeat the arguments of the C2-inflection case above one by one, with γ0 replaced by η. The Lemma 5.7 can be repeated verbatim (necessarily R′⁢(0)<0), and we get Ω+−⊂MC⁢HT.

Let η~⊂ℑ− be the curve of points bounding (the germ at 0 of) the domain Ω−− consisting of points of ℑ− whose associated rays do not intersect η. In particular, r⁢(u) is a supporting line to η for all u∈η~. If γuβ is a trajectory of R⁢(z)⁢∂z starting at u∈η~ and ending at β∈ℑR then γuβ⊂Ω−− by convexity of γuβ, as in Lemma 5.11.

Lemma 5.13.

For any u∈η~ the ray r⁢(u) does not intersect η~ except at u itself.

Proof.

Note first that η lies above r⁢(u) for any α∈η~. Assume the opposite to the claim of the lemma, and let u′∈r⁢(u)∩η~ be such a point of intersection. If σ⁢(u′)<σ⁢(u) then the ray r⁢(u′) lies below r⁢(u) and therefore cannot intersect η. Similarly, if σ⁢(u′)>σ⁢(u) then r⁢(u) cannot intersect η. Therefore σ⁢(u′)=σ⁢(u), so u′ is a point of tangency of r⁢(u) and R⁢(z)⁢∂z just like u.

Now, recall that 0∈η lies above r⁢(u). Moreover, as u∈ℑ−, the germ of r⁢(u) at u lies in Ω−−: for v∈r⁢(u) close to u we have σ⁢(v)<σ⁢(u), so r⁢(v) doesn’t intersect η. Thus the germ of η~ at u lies above r⁢(u) as well. Thus both ends of the part η~u0 of η~ between u and 0 lie above r⁢(u), so η~u0 intersects r⁢(u) in even number of points not including u, i.e. at least at three points including u, all of them the points of tangency of r⁢(α) and R⁢(z)⁢∂z.

As u→0 the line r⁢(u) converges to ℝ and the points of tangency necessarily converge to 0 as well. This means that R⁢∂z has point of tangency of order at least 3 with ℝ at 0. However, p∈ℑR∗ means that Im⁡R′′⁢(0)≠0, so the order of tangency of R⁢∂z with ℝ at 0 is exactly 2 (as Im⁡R⁢(0)=Im⁡R′⁢(0)=0). This contradiction proves the Lemma.

∎

The same arguments as in Lemma 5.11 and in the proof of Proposition 5.6 now show that Ω−− satisfies the conditions of Lemma 3.26 and is therefore disjoint from MC⁢HT. ∎

5.5 Points of switch type

Recall that a point p∈∂MC⁢HT∖𝒵⁢(P⁢Q) belonging to the transverse locus ℑR∗ is called a point of switch type if Δ+⁢(p)≠∅ and Γ⁢(p)∪Δ−⁢(p)=∅.

Proposition 5.14.

The negative part γ0−⁢(t) of the integral curve γ0⁢(t) is a part of the boundary ∂MC⁢HT.

As before, we assume that p=0, R⁢(0)=1 and Im⁡R′′⁢(0)>0. The trajectory γ0⁢(t) and the curve ℑR divide U into four domains Ω±,±, with Ω+− intersecting the △p′′p′0.

Lemma 5.15.

In the above notation, Ω++⊂MC⁢HTc.

Proof.

Recall that by Lemma 3.21 the union H⁢(0)=△p′′p′0∪∠p′′ does not intersect MC⁢HT.

Assume first that R′⁢(0)≤0 and set ρ≔−(R′⁢(0))−1∈ℝ+∪{+∞}. The set Δ⁢(0) is a relatively closed subset of (0,ρ] not containing ρ (as Δ−⁢(0)=∅). Thus there exists a ρ′<ρ such that the ray [ρ′,+∞] is disjoint from MC⁢HT. As both [ρ′,+∞] and MC⁢HT¯ are closed in ℂ∪𝕊1 there is some neighborhood U~ of [ρ′,+∞] in ℂ∪𝕊1 disjoint from MC⁢HT¯. Choose an open sector S⊂U~ with the ray (ρ′,+∞) as bisector, necessarily disjoint from MC⁢HT.

By shrinking Ω±± we can assume that |σ⁢(z)|<|σ⁢(S)| for all z∈Ω±±, where ±σ⁢(S) are the slopes of the sides of S. Moreover, we restrict ourselves to a neighborhood U of 0 so small that rays r⁢(z) do not intersect the interval [0,ρ′] for z∈U∩{Re⁡z≥0} and z≠0: this is possible since the trails of these points lie in the lower half-plane. In particular, for any z∈Ω++⊂U∩{Re⁡z≥0} the ray r⁢(z) intersects the boundary of H⁢(0)∪S only once at γ0+⁢(t).

The same conclusion holds in the case R′⁢(0)>0. In this case σ⁢(z)>0 for all z∈U, Im⁡z>0.

Since Γ⁢(0)=∅ there exists a point q∈Ω++∖MC⁢HT. We can now repeat the arguments of Lemma 4.9 for U: the crucial fact used in Lemma 4.9 was that r⁢(p~) doesn’t intersect MC⁢HT, and this holds for points of Ω++. Therefore Ω++∩MC⁢HT=∅.

∎

Proposition 5.16.

Take p=γ0⁢(t) for some t<0 sufficiently close to 0. The open curvilinear triangle H⁢(p)⊂Ω−+ bounded by r⁢(p), the curve γ0−⁢(t) and the inflection curve lies outside MC⁢HT.

Denote x=Re⁡z, y=Im⁡z.

Lemma 5.17.

One has ∂∂x⁢σ⁢(x+i⁢y)<0 as long as z=x+i⁢y lies in a sufficiently small sector {|z|<δ,x<0,|y|<−ϵ⁢x} for some ϵ,δ>0 depending on the rational function R⁢(z) only.

Proof.

Recall that R⁢(z)=1+a⁢z+b⁢z2+.., with Im⁡b>0. Then

log⁡R=a⁢z+(b−a22)⁢z2+…

and

(log⁡R)′=a+(2⁢b−a2)⁢z+…

Therefore

∂∂xσ(x+iy)=∂∂xIm(logR)=Im(logR)′=Im(2b−a2)z+o(z)<0

if |arg⁡z−π|<π−arg⁡(2⁢b−a2) and |z| is sufficiently small (note that 0<arg⁡(2⁢b−a2)<π as Im⁡b>0 and a∈ℝ). ∎

Lemma 5.18.

The associated ray r⁢(z) does not intersect γ0−⁢(t) for any z∈H⁢(p).

Proof.

First consider the case Im⁡z<0. As γ0−⁢(t) is tangent to ℝ, we can assume that this part of H⁢(p) satisfies the conditions of Lemma 5.17, so σ⁢(z)>σ⁢(z+)>0, where z+=z+t∈γ0−⁢(t). Thus the ray r⁢(z) lies in the half-plane bounded by the line tangent to γ0−⁢(t) at z+ and containing z. Therefore r⁢(z) does not intersect γ0−⁢(t): by convexity of γ0−⁢(t)it lies in the other half-plane bounded by the line tangent to γ0−⁢(t) at z+.

Now, assume that Im⁡z≥0. Recall that we have chosen U so small that for any z∈U, Im⁡z>0, the intersection r⁢(z)∩ℝ⊂(ρ′,+∞)⊂ℝ+. Thus r⁢(z)∩{Im⁡w<0}⊂{Re⁡z>ρ′>0} which is disjoint from γ−. ∎

Lemma 5.19.

The associated ray r⁢(z) does not intersect r⁢(p) for any z∈H⁢(p).

Proof.

Let γz− be a part of the integral curve of R⁢∂z ending at z and let z′∈γz−∩r⁢(p) be the first (from z) point where γz− enters H⁢(p). This point must necessarily lie on r⁢(p) because γz− does not intersect neither the curve of inflection points ℑR nor γ− by uniqueness of solutions of ODE. Thus σ⁢(z′)≤σ⁢(p).

Assume now that the ray r⁢(z) intersects r⁢(p). Then σ⁢(z)>σ⁢(p)≥σ⁢(z′) as z lies below r⁢(p). Therefore the slope σ⁢(w), w∈γz−, is not monotone, implying that γz− has an inflection point. But this is impossible since γz− does not intersect the curve of inflections. ∎

Proof of Proposition 5.16.

By minimality, it is enough to prove that r⁢(z)⊂H⁢(p)∪MC⁢HTc for any z∈H⁢(p). By Lemmas 5.18, 5.19 the ray r⁢(z) does not intersect r⁢(p) and γ0−⁢(t). Thus r⁢(z) leaves H⁢(p) through the curve of inflections ℑR with a small slope.

If the slope is positive then r⁢(z)∖H⁢(p)⊂U++∪H⁢(0). In particular, this is the case for all z∈H⁢(p), Im⁡z<0 by Lemma 5.17.

Assume now that σ⁢(z)<0 (and therefore Im⁡z>0). Recall that we have chosen U so small that for any z∈U, Im⁡z>0, the intersection r⁢(z)∩ℝ⊂(s′,+∞)⊂ℝ+. Thus r⁢(z)∖H⁢(p)⊂Ω++∪H⁢(0)∪S.

Taken together, r⁢(z)⊂H⁢(p)∪MC⁢HTc for all z∈H⁢(p). Therefore H⁢(p)⊂MC⁢HTc by Lemma 3.26.

∎

Proposition 5.20.

Consider a point p∈∂MC⁢HT of switch type. Then there is a neighborhood V of p such that MC⁢HT∩V is contained in a half-disk centered at p. Besides, no neighborhood of p in MC⁢HT can be contained in a cone centered at p with an angle strictly smaller than π. A point of switch type is the ending point of both a local and a global arcs.

Proof.

We essentially repeat the arguments of Lemma 4.8. Take u∈Δ⁢(0)≠∅ and let 𝔱⁢𝔯u+ be a germ of the branch of 𝔱⁢𝔯u⊂MC⁢HT lying in the lower-right quadrant. For any q∈𝔱⁢𝔯u+ let z=z⁢(q)∈γq+ be the last point such that r⁢(z)∩MC⁢HT≠∅. This point exists since γq+≠∅ eventually enters △p′′p′0⊂MC⁢HTc. As in Lemma 5.4, z∈∂MC⁢HT and therefore Im⁡z<0, as otherwise z∈△p′′p′0. The points z⁢(q) form a global arc ending at 0. ∎

5.6 Classification of boundary points

Here, we summarize several results obtained in the previous sections to finalize our classification of boundary points on ∂MC⁢HT.

Proof of Theorem 1.9.

It follows from Corollary 3.8 that there are at most 4⁢deg⁡P+deg⁡Q−2≤2⁢d singular points in the curve of inflections (d=3⁢deg⁡P+deg⁡Q−1). Proposition 3.14 proves that the tangency locus 𝔗R is formed by at most 2⁢d2 isolated points and d lines.

The classification of points in ℑR∗ is trivial. If Δ+ is nonempty, then a point is of bouncing or switch type depending on whether Γ∪Δ− is empty or not. If Δ+ is empty, then a point is of C1-inflection or C2-inflection type depending whether the conjunction of Δ−≠∅ and Γ=∅ is satisfied or not.

Finally, for points outside 𝒵⁢(P⁢Q)∪ℑR, we just have to check that Γ and Δ cannot both be empty. This is proved in Proposition 4.7. ∎

5.7 Estimates concerning local and global arcs

We introduce the following notations:

  • •

    |ℒ| is the number of local arcs;

  • •

    |𝒢| is the number of global arcs;

  • •

    |ℬ| is the number of points of bouncing type;

  • •

    |ℰ| is the number of points of extruding type;

  • •

    |ℐ1| is the number of points of C1−inflection type;

  • •

    |ℐ2| is the number of points of C2−inflection type;

  • •

    |𝒮| is the number of points of switch type.

We prove that the number of points of switch type provides an estimate for the number of local arcs (up to an error term depending only on deg⁡P and deg⁡Q).

Lemma 5.21.

In the boundary ∂MC⁢HT, the ending point of every local arc, except at most d⁢(2⁢d+1) of them, is a point of switch type where d=3⁢deg⁡P+deg⁡Q−1. Conversely every point of switch type is the endpoint of some local arc.
In other words, we have |𝒮|≤|ℒ|≤|𝒮|+d⁢(2⁢d+1).

Proof.

Proposition 5.20 proves that every point of switch type is the endpoint of some local arc. It remains to list all possible endpoints for local arcs.

Following Proposition 4.12, every local arc has an endpoint that either belongs to 𝒵⁢(P⁢Q) or to ℑR. For any point α which is the endpoint of a local arc, ℒα contains an interval of length at most π. It follows from Corollary 3.5 that such a point is either a simple pole of R⁢(z) or a point which is neither a zero or a pole of R⁢(z). Only two local arcs can have the same simple pole as their endpoint. Any other point is the endpoint of at most one local arc (because only one integral curve passes through such a point). Consequently, at most 3⁢deg⁡P+deg⁡Q local arcs have an endpoint in 𝒵⁢(P⁢Q).

It remains to count local arcs one endpoint of which belongs to ℑR∖𝒵⁢(P⁢Q). Any such point is incident to a unique integral curve which implies that it can be the endpoint of only one local arc. If such a point belongs to the transverse locus of the curve of inflections, then it is a point of switch type (see Propositions 5.3, 5.6, 5.12 and 5.20). There are |𝒮| of them. Following Proposition 3.14, the tangency locus of ℑR is formed by at most 2⁢d2 points and d straight lines where d=3⁢deg⁡P+deg⁡Q−1. As arg⁡(R⁢(z)) is constant on each such line, they contain at most one endpoint of a local arc. Thus |ℒ|≤|𝒮|+d⁢(2⁢d+1).

∎

Similarly, we prove an estimate on the number of global arcs that do not start at a point of the transverse locus of the curve of inflections.

Lemma 5.22.

In the boundary ∂MC⁢HT, the starting point of every global arc, except at most 12⁢d+5⁢d2 of them, is either a point of C1-inflection, C2-inflection or of bouncing type.
Besides, we have 2⁢|ℐ1|+|ℐ2|≤|𝒢|≤|ℬ|+2⁢|ℐ1|+|ℐ2|+12⁢d+5⁢d2.

Proof.

Proposition 5.6 show that every point of C1-inflection is the starting point of a global arc while Proposition 5.12 show that every point of C1-inflection is the starting point of two global arcs. It follows that 2⁢|ℐ1|+|ℐ2|≤|𝒢|.

Then, we list every possible starting point for a global arc (Lemma 4.23 proves that global arcs cannot be closed loops).

Since points of extruding type are not starting points of global arcs (see Proposition 4.30), every global arc either starts at a point at infinity or at a point of 𝒵⁢(P⁢Q)∪ℑR.

We first count the number of global arcs which can start at infinity. If deg⁡Q−deg⁡P=1, then by Theorem 2.23 we know that MC⁢HT is compact. If deg⁡Q−deg⁡P=−1, then Proposition 6.9 proves that MC⁢HT has only one connected component while its complement has two connected components. Therefore, we have at most four infinite global arcs in this case. If deg⁡Q−deg⁡P=0, the complement of MC⁢HT is connected so each connected component has at most two infinite global arcs. Following Proposition 2.20, MC⁢HT has at most deg⁡P+deg⁡Q connected components. Therefore the number of global arcs starting at infinity is at most 2⁢deg⁡P+2⁢deg⁡Q. In the only case where 4>2⁢deg⁡P+2⁢deg⁡Q while deg⁡Q−deg⁡P=−1, Q⁢(z) is constant while deg⁡P=1. In this case, MC⁢HT is a straight line (see Proposition 6.7).

Let us consider the points of the transverse locus of ℑR. Each point of C1−inflection type is the starting point of exactly two global arcs (see Proposition 5.12). Each point of bouncing or C2−inflection type is the starting point of exactly one global arc (see Propositions 5.3 and 5.6). No global arc starts at a point of switch type (see Proposition 5.20).

Now let us consider the tangency locus of ℑR. It is formed by at most 2⁢d2 points and deg⁡P+deg⁡Q+1 R-invariant lines (see Definition 2.5). For each such line, at most 4 global arcs can have start accumulation ω−⁢(α) belonging to it, because each line has two sides and rays have two possible directions. Otherwise, the rays starting from these global arcs intersect the interior of MC⁢HT near the other global arcs. Using Corollaries 3.6, we conclude that each of the 2⁢d2 remaining points of the tangent locus is the starting point of at most two global arcs. For the same reasons, each singular point of ℑR that does not belong to 𝒵⁢(P⁢Q) is the starting point of at most two global arcs. There are 4⁢deg⁡P+deg⁡Q−2≤2⁢d such points (see Corollary 3.8).

It remains to estimate the number of global arcs that can start at a root α of P⁢(z) or Q⁢(z) in terms of the local degree mα of R⁢(z) in α. Corollary 3.6 proves that α is the starting point of at most:

  • •

    two arcs if mα=0;

  • •

    2⁢(1−mα) arcs if mα≤−1;

  • •

    2⁢deg⁡P arcs if mα≥1.

Consequently, in the worst case scenario, the roots of P⁢(z) and Q⁢(z) are simple and disjoint so at most 2⁢deg⁡P⁢(deg⁡P+2) global arcs can start at these points.

Therefore, the number of global arcs whose starting point does not belong to ℑR∗ is at most (2⁢deg⁡P+2⁢deg⁡Q)+4⁢(deg⁡P+deg⁡Q+1)+4⁢d2+4⁢d+2⁢deg⁡P⁢(deg⁡P+2).

If deg⁡P=0, then deg⁡Q=1 (otherwise MC⁢HT is trivial) and MC⁢HT is fully irregular (and has therefore no global arcs) so we can replace the obtained bound by the slightly weaker (but more practical) upper bound 12⁢d+5⁢d2. ∎

5.8 Long arcs

In order to prove Theorem 1.10, we introduce a new decomposition of the boundary ∂MC⁢HT.

Definition 5.23.

For any linear differential operator T given by (1.1), we define the long arcs as the maximal arcs formed by gluing local and global arcs along points of extruding or bouncing type (see Sections 4.6 and 5.2).

In particular, a long arc belongs to the closure of a unique inflection domain. Consequently, local and global arcs of a same long arc share the same orientation (see Section 4.5.2). This defines the orientation a long arc.

5.8.1. Estimates concerning long arcs

Drawing on the estimates of Section 5.7, we prove that the number of long arcs corresponds to the number of intersections between ∂MC⁢HT and the transverse locus of ℑR that are not of bouncing type (in other words, where the boundary of the minimal set crosses the curve of inflections).

Lemma 5.24.

Every long arc except at most 28⁢d2+52⁢d of them goes from a point of switch type to a point of C1-inflection or C2-inflection type. The number |𝒜| of long arcs satisfies the following inequalities:

2⁢|𝒮|≤|𝒜|≤2⁢|𝒮|+26⁢d+14⁢d2;
4⁢|ℐ1|+2⁢|ℐ2|≤|𝒜|≤4⁢|ℐ1|+2⁢|ℐ2|+26⁢d+14⁢d2.
Proof.

Inequality 2⁢|𝒮|≤|𝒜| follows from the fact that every point of switch type is the ending point of two long arcs. Similarly, points of C1-inflection or C2-inflection type are the starting points of two long arcs so we obtain 2⁢|ℐ1|+2⁢|ℐ2|≤|𝒜|.

We denote by |𝒜L| the number of long arcs containing a local arc. We already know that the endpoint of a local arc cannot be a point of extruding or bouncing type. Therefore, the endpoint of a local arc contained in a long arc is also the endpoint of the long arc. We deduce then from Lemma 5.21 that the endpoints of all but at most d⁢(2⁢d+1) these long arcs containing a local arc are points of switch type: |𝒜L|≤|𝒮|+d⁢(2⁢d+1).

Then, denote by |𝒜G| the number of long arcs that do not contain a local arc. The start accumulations of these long arcs are in particular start accumulations of a global arc and cannot be points of bouncing type. We deduce from the proof of Lemma 5.22 that the start accumulation of these long arcs, except at most 12⁢d+5⁢d2 of them are in fact starting points and they are points of C1-inflection or C2-inflection type. In other words, we have |𝒜G|≤2⁢|ℐ1|+|ℐ2|+12⁢d+5⁢d2.

Finally, we deduce that the total number of long arcs satisfies |𝒜|≤|𝒮|+2⁢|ℐ1|+|ℐ2|+13⁢d+7⁢d2. Combining this inequality with 2⁢|𝒮|≤|𝒜|, we obtain that |𝒮|≤2⁢|ℐ1|+|ℐ2|+13⁢d+7⁢d2 and therefore |𝒜|≤4⁢|ℐ1|+2⁢|ℐ2|+26⁢d+14⁢d2. We obtain similarly that |𝒜|≤2⁢|𝒮|+26⁢d+14⁢d2. This also provides a bound on the number of long arcs that do not go from a point of switch type to a point of C1-inflection or C2-inflection type. ∎

5.8.2. Admissible long arcs

We will refer to a long arc going from a point of switch type to a point of C1-inflection or C2-inflection type (or the opposite) as an admissible long arc.

Definition 5.25.

We associate to each admissible long arc α a combinatorial symbol sα that contains the following information:

  • •

    the connected component of the transverse locus ℑR∗ containing the starting point of α;

  • •

    the connected component of ℑR∗ containing the endpoint of α;

  • •

    the sign of the inflection domain α belongs to.

Chains of consecutive long arcs define patterns formed by the concatenation of the combinatorial symbols of their long arcs.

Chains of consecutive long arcs are given the orientation induced by the topological orientation of MC⁢HT. Therefore, the orientation of a chain coincides with the orientation of the long arc inside positively oriented domains of inflection (and does not coincide with the orientation of the long arc inside negatively oriented domains of inflections).

Remark 5.26.

In particular, at a point z of switch, C1-inflection or C2-inflection type in a chain, the associated ray r⁢(z) and the orientation of the chain points towards the same domain of inflection.

5.8.3. Bounding the number of transverse intersection points between ∂MC⁢HT and the transverse locus of ℑR

Using the fact that an associated ray cannot cross the curve of inflections more than d times (where d=3⁢deg⁡P+deg⁡Q−1), we prove a bound on the number of chains of admissible long arcs that can realize a given pattern.

Lemma 5.27.

In the boundary ∂MC⁢HT of the minimal set, there cannot be 2⁢d+2 disjoint chains of 2⁢d admissible long arcs that realize the same pattern.

Proof.

We assume by contradiction the existence of 2⁢d+2 disjoint chains γ1,…,γ2⁢d+2 of 2⁢d admissible long arcs realizing the same pattern. We refer to the admissible long arc of γi corresponding to the jt⁢h symbol as αi,j.

By definition of the combinatorial symbol, for a given j, the arcs αi,j for 1≤i≤2⁢d+2 belong to the same inflection domain 𝒟j.

We denote by β1,…,β2⁢d+1 the connected components of the transverse locus ℑR∗ at the endpoints of long arcs ordered according to the orientation of the chains.

We are going to prove the existence of a point z of βd+1 such that its associated ray r⁢(z) also intersects d arcs among β1,…,β2⁢d+1, defining a straight line that intersects transversely a real algebraic curve of degree at most d in at least d+1 points, obtaining the desired contradiction.

A first observation is that the chains γ1,…,γ2⁢d+2 cannot cross each other (because the interior of MC⁢HT is connected near endpoints of admissible arcs). Since the complex plane is simply connected, this fact implies that for a given j, the intersection points of the chains γ1,…,γ2⁢d+2 with βj determine a cyclic order that does not depend on j. We assume therefore that the indices of γ1,…,γ2⁢d+2 are elements of ℤ/(2⁢d+2)⁢ℤ and correspond to the previously defined cyclic ordering.

In a given inflection domain 𝒟j, it may happen that the linear orders of these intersection points with βj and βj+1 respectively are different. They may differ by a rotation if βj and βj+1 do not belong to the same connected components of the boundary of ∂𝒟j (if 𝒟j is not simply connected).

It follows that for any 1≤j≤2⁢d, for every k∈ℤ/(2⁢d+2)⁢ℤ except possibly one, there is a quadrilateral in ℂ bounded by αj,k, αj,k+1, one portion of βj and one portion of βj+1. The exception correspond to the case where the linear orderings do not match. Since we have 2⁢d+2 chains, it follows that we can assemble 2⁢d of these strips into a unique long strip 𝒮 bounded by some γk, γk+1 and portions of β0 and β2⁢d+1.

The chains γk and γk+1 are oriented in such a way that for any point z∈∂𝒮∩β0, associated ray r⁢(z) points inside 𝒮. Since the orientation of γk and γk+1 coincides with the topological orientation of MC⁢HT, we can find a point z of ∂𝒮∩βd+1 in the complement of MC⁢HT. Thus, for any such point z, associated ray r⁢(z) cannot cross chains γk and γk+1 and has to leave 𝒮 through β0 or β2⁢d+1. Therefore, r⁢(z) has to cross either β1,…,βd+1 or βd+1,…,β2⁢d+1. This ends the proof. ∎

We deduce a bound on the number of long arcs.

Corollary 5.28.

For any linear differential operator T given by (1.1), the number |𝒜| of long arcs in ∂MC⁢HT satisfies

|𝒜|≤2⁢e16⁢d⁢ln⁡(d)+92⁢d3

where d=3⁢deg⁡P+deg⁡Q−1.

Proof.

Since the number of connected components of the transverse locus ℑR∗ is at most 2⁢d2+6⁢d+2 (see Corollary 3.15), the number of possible combinatorial symbols for an admissible long arc is 2⁢(2⁢d2+6⁢d+2)2 (see Definition 5.25). Therefore, the number of possible patterns for a chain of 2⁢d admissible long arcs is at most 22⁢d⁢(2⁢d2+6⁢d+2)4⁢d. Since d≥3 (see Remark 2.4), we have 2⁢d2+6⁢d+2≤52⁢d2 and we obtain the weaker (but simpler) upper bound (25/2)2⁢d⁢d8⁢d.

Then, using Lemma 5.27, we deduce that the number of disjoint chains of 2⁢d admissible long arcs is at most (25/2)2⁢d⁢(2⁢d+1)⁢d8⁢d.

The number of non-admissible long arcs is bounded by 28⁢d2+52⁢d (Lemma 5.24). It follows that in the worst case, each non-admissible arc is located between two chains of (2⁢d−1)+2⁢d⁢ℕ admissible long arcs. Consequently, the number of long arcs is bounded by

(25/2)2⁢d⁢(2⁢d)⁢(2⁢d+1)⁢d8⁢d+(28⁢d2+52⁢d)+(28⁢d2+52⁢d+1)⁢(2⁢d−1).

Since d≥3, this upper bound can be weakened to 2⁢e16⁢d⁢ln⁡(d)+92⁢d3. ∎

Theorem 1.10 then follows from the fact that |𝒮| and 2⁢|ℐ1|+|ℐ2| are bounded by the number of long arcs (see Lemma 5.24). Corollary 1.11 then follows from the combination of Theorem 1.10 with Lemma 5.21.

6 Global geometry of minimal sets

At present we do not know a general recipe how to describe non-trivial MC⁢HT. Nevertheless we can prove some general statements about their global geometry and provide some illuminating examples.

Recall that MC⁢HT is nontrivial if and only if deg⁡Q−deg⁡P∈{−1,0,1}.

In some cases, description of the convex hull C⁢o⁢n⁢v⁢(MC⁢HT) is easier to obtain. The following has been proved as Corollary 5.16 in [AHN+24].

Proposition 6.1.

Consider a linear differential operator T given by (1.1).Then the intersection of all convex Hutchinson invariant set coincides with the convex hull C⁢o⁢n⁢v⁢(MC⁢HT) of the minimal set MC⁢HT.

The local analysis of boundary points carried on in the previous sections provides interesting partial results towards a characterization of points where ∂MC⁢HT is locally convex.

6.1 Local convexity of the boundary

Local analysis in terms of correspondences Γ and Δ shows that corner points of MC⁢HT have to satisfy very specific conditions.

Corollary 6.2.

For a linear differential operator T given by (1.1), consider a point α which is a corner point of the boundary ∂MC⁢HT. In other words, there is a neighborhood V of α such that V∩MC⁢HT is contained in a cone with apex α and with the opening strictly smaller than π. Then one of the following statements hold:

  • •

    α is a simple zero of R⁢(z) satisfying ϕα=0 (see 3.1);

  • •

    α is a common root of P⁢(z) and Q⁢(z) of the same multiplicity (i.e. α is neither a zero nor a pole of R⁢(z)).

Besides, if α is a cusp (neighborhoods of α in MC⁢HT can be included in cones of arbitrarily small opening angle), then one of the following statements holds:

  • •

    α is a common root of P⁢(z) and Q⁢(z) of the same multiplicity;

  • •

    MC⁢HT is totally irregular and contained in a half-line.

Proof.

Corollary 3.5 immediately implies that α cannot be a pole or a multiple zero of R⁢(z). Besides, if α is a simple zero, it has to satisfy the condition ϕα=0. Now we assume that α is neither a zero nor a pole of R⁢(z). It remains to prove that α∈𝒵⁢(P⁢Q).

Assume that α∉𝒵⁢(P⁢Q). In this case if some point of the forward trajectory of R⁢(z)⁢∂z starting at α belongs to MC⁢HT, then a germ of the integral curve starting at α is contained in MC⁢HT (see Proposition 2.10) and α cannot be a corner point. We conclude that Γ⁢(α)=∅.

If Δ⁢(α) contains some point y, then a branch of the root trail 𝔱⁢𝔯y containing α belongs to MC⁢HT (see Lemmas 2.11 and 2.16). Thus in this case α cannot be a corner point and we get Δ⁢(α)=0.

Now we take a cone 𝒞 with apex at α, of angle at least π which is locally disjoint from MC⁢HT. If r⁢(α) is contained in 𝒞, but is not one of the two limit rays, we can freely remove a neighborhood of α from MC⁢HT and still get an invariant set. In any other case, we can find an arc contained in a neighborhood of α and the complement of MC⁢HT whose associated rays sweep out a domain containing α. Thus we get a contradiction in this case as well which implies that α has to be in 𝒵⁢(P⁢Q).

Finally if α is a simple zero of R⁢(z) and a cusp, then we have ℒα=𝕊1 (see Definition 3.2). It follows that MC⁢HT has empty interior. All such cases have been completely classified in Section 7 of [AHN+24]. ∎

Further local analysis provides necessary conditions under which boundary points belong to locally convex parts of ∂MC⁢HT.

Proposition 6.3.

For a linear differential operator T given by (1.1), consider a point α∈∂MC⁢HT such that there is a neighborhood V of α with the property that V∩MC⁢HT is contained in a closed half-plane whose boundary contains α.

If α∈𝒵⁢(P⁢Q), then one of the following statements holds:

  • •

    α is a simple pole of R⁢(z);

  • •

    α is a simple zero of R⁢(z) satisfying ϕα=0 (see 3.1);

  • •

    α is a common root of P⁢(z) and Q⁢(z) of the same multiplicity (i.e. α is neither a zero nor a pole of R⁢(z)).

If α∈ℑR∗∖𝒵⁢(P⁢Q), then α is a point of switch type.

If α∉ℑR∪𝒵⁢(P⁢Q), then one of the following statements holds:

  • •

    α is a point of local type;

  • •

    α is a point of global type and for any u∈Δ⁢(α), either Im⁡(f⁢(u,α))=0 or Im⁡(f⁢(u,α)) and Im⁡(R′⁢(α)) have opposite signs (for f defined as in Proposition 2.18).

Proof.

The case α∈𝒵⁢(P⁢Q) follows from Corollary 3.5. If α∈ℑR∗∖𝒵⁢(P⁢Q) and Δ−⁢(α)≠∅, then MC⁢HT contains both the germ of an integral curve of the field −R⁢(z)⁢∂z at α and the germ of the root trail 𝔱⁢𝔯u for some u∈Δ−⁢(α). Proposition 2.19 implies that MC⁢HT cannot be convex at α. Besides, if Γ⁢(α)≠∅, then MC⁢HT cannot be convex in α either because a germ of an integral curve having an inflection point at α is contained in MC⁢HT In the remaining cases, we have Γ⁢(α)∪Δ−⁢(α)=∅. If Δ+⁢(α)≠∅, this characterizes points of switch type (see Theorem 1.9). If Δ−⁢(α)=∅, then we obtain a point of C2-inflection type, α is the starting point of a local arc and Γ⁢(α) is therefore nonempty (see Proposition 5.5).

Now we consider the case α∉ℑR∪𝒵⁢(P⁢Q). If Γ⁢(α)≠∅, then α is a point of local type (α cannot be a point of extruding type because of Proposition 4.30). If Γ⁢(α)=∅, then it follows from Proposition 4.7 that Δ⁢(α)≠∅. Proposition 2.18 then provides the necessary condition. ∎

6.2 Case deg⁡Q−deg⁡P=−1

We have a rational vector field R⁢(z)⁢∂z satisfying R⁢(z)=λz+μz2+o⁢(1/z2) with λ∈ℂ∗ and μ∈ℂ.

6.2.1. Horizontal locus and special line

We define the following loci.

Definition 6.4.

The horizontal locus ℋR is the closure in ℂ of the set formed by points z∉𝒵⁢(P⁢Q), for which σ⁢(z)=arg⁡(λ)±π2.

We also denote by ℒR the special line formed by points z given by the equation I⁢m⁢(z/λ)=I⁢m⁢(μ/λ2).

For the sake of simplicity, the vector field R⁢(z)⁢∂z is normalized by an affine change of variable as R⁢(z)=−1z+o⁢(z−2) (λ=−1 and μ=0). The line ℒR then coincides with the real axis ℝ.

Lemma 6.5.

ℋR is a real plane algebraic curve of degree at most deg⁡P+deg⁡Q. It has two asymptotic infinite branches. The line ℒR is the asymptotic line for both of them.

Proof.

Curve ℋR can be seen as the pull-back of the real axis under the mapping R⁢(z):ℂ⁢ℙ1→ℂ⁢ℙ1. We have R⁢(∞)=0 and ∞ is a simple root of R⁢(z). Therefore, ℋR is smooth near ∞.

It remains to show that the tangent line to ℋR at infinity coincides with the real axis. Actually the tangent line is the line at which the linearization of R⁢(z) at ∞ attains real zeroes. Since this linearization is exactly −1z, the result follows. ∎

Corollary 6.6.

The closure MC⁢HT¯ of the minimal set in the extended plane contains asymptotic directions 0 and π. Besides, the curve ℋR is contained in the minimal set MC⁢HT.

Proof.

Looking at separatrices of the vector field R⁢(z)⁢∂z and using Proposition 2.10 we get that the closure MC⁢HT¯ in the extended plane contains asymptotic directions 0 and π. The associated rays of points of ℋR are thus asymptotically tangent to MC⁢HT and ℋR is contained in the minimal set. ∎

Proposition 6.7.

Consider a linear differential operator T given by (1.1) such that deg⁡Q−deg⁡P=−1. Then the minimal convex Hutchinson invariant set C⁢o⁢n⁢v⁢(MC⁢HT) is a bi-infinite strip (domain bounded by two parallel lines).

More precisely, C⁢o⁢n⁢v⁢(MC⁢HT) is the smallest strip containing ℋR∪𝒵⁢(P⁢Q).

Proof.

The minimal convex Hutchinson invariant set C⁢o⁢n⁢v⁢(MC⁢HT) is the complement of the union of every open half-plane disjoint from MC⁢HT. Since ℋR is contained in MC⁢HT (Corollary 6.6), these open half-planes have to be disjoint from ℋR. Conversely, any open half-plane H disjoint from ℋR is such that I⁢m⁢(R⁢(z)) is either positive or negative for every z∈H. Therefore, provided H does not contain any zero or pole of R⁢(z), one can conclude that it can be removed from any Hutchinson invariant set. In other words, C⁢o⁢n⁢v⁢(MC⁢HT) is the complement to the union of all half-planes disjoint from ℋR∪𝒵⁢(P⁢Q). Since ℋR has asymptotically horizontal infinite branches, the boundary line of every half-plane disjoint from ℋR has to be horizontal.

It remains to prove that such half-planes exist. It follows from the asymptotic description of ℋR in Lemma 6.5 that |I⁢m⁢(z)| is bounded on ℋR. Therefore we can find two (disjoint) open half-planes that are also disjoint from ℋR. These half-planes contain half-planes which, in addition, are disjoint from 𝒵⁢(P⁢Q). ∎

6.2.2. Asymptotic geometry of the minimal set

Following Proposition 6.7, C⁢o⁢n⁢v⁢(MC⁢HT) is the smallest horizontal strip containing the curve ℋR∪𝒵⁢(P⁢Q). The closure of the projection of C⁢o⁢n⁢v⁢(MC⁢HT) on the vertical axis is an interval [y−,y+] where y−≤0≤y+.

Lemma 6.8.

For 0<y<y0, denote by Mt the intersection point between the associated ray r⁢(t+i⁢y) and the horizontal line I⁢m⁢(z)=y0. Then the following statements hold:

  • •

    for t⟶+∞, Re⁡(Mt)⟶−∞ if y<y02;

  • •

    for t⟶+∞, Re⁡(Mt)⟶+∞ if y02<y<y0.

Analogous statements hold for t⟶−∞ or y0<y<0.

Proof.

For large values of t, we have Re⁡(R⁢(z))=−1t+o⁢(t−1) and Im⁡(R⁢(z))=yt2+o⁢(t−2). Provided t is large enough, Im⁡(R⁢(z)) is positive and the associated ray r⁢(z) intersects the line I⁢m⁢(z)=y0. Then the real part of the intersection point equals t−(y0−y)⁢ty+o⁢(t). After simplification, we obtain (2⁢y−y0)⁢ty+o⁢(t). The sign of the main term is then determined by the sign of 2⁢y−y0. ∎

Proposition 6.9.

If deg⁡Q−deg⁡P=−1, the minimal set MC⁢HT is connected in ℂ.

Proof.

Following Proposition 2.24, the complement (MC⁢HT)c of MC⁢HT in ℂ has exactly two connected components and it has been proved in Proposition 6.7 that each of them contains a half-plane. We refer to the domain containing an upper half-plane as 𝒟+ and to the domain containing a lower half-plane as 𝒟−. Since MC⁢HT contains ℋR, we deduce that Im⁡(R⁢(z)) is positive on 𝒟+ and negative on 𝒟−.

Proving that MC⁢HT is connected in ℂ amounts to showing that 𝒟+ and 𝒟− have only one topological end. We will prove this statement for 𝒟+ (the proof for 𝒟− is identical). We assume by contradiction that 𝒟+ has a topological end κ distinct from the end of the upper half-plane contained in 𝒟+ (we will refer to this end as the main end of 𝒟+).

For any sequence {zn} of points in (MC⁢HT)c approaching κ, we have (up to taking a subsequence) the sequence {arg⁡(zn)} converging either to 0 or to π (since otherwise, κ would not be distinct from the main end). Let’s assume without loss of generality that it is 0. Again, we can assume that {I⁢m⁢(zn)} converges to some value ye∈[0,y+].

If ye>0, then Lemma 6.8, shows that for any horizontal line Lf with yf∈]ye,2ye[, the associated rays of the points in MC⁢HTc converging to the end κ sweep out points of Lf whose real part is arbitrarily close to +∞. Assuming that ye is the maximal possible limit value, we deduce that no infinite component of MC⁢HT can separate κ from the upper main end containing asymptotic directions of ]0,π[.

Hence, for a sequence {zn} of points in (MC⁢HT)c approaching κ, the only accumulation value of {I⁢m⁢(zn)} is 0. In this case, the associated rays r⁢(zn) accumulate onto the ℝ-axis which is therefore contained in the closure of 𝒟+.

Now we prove that the open upper half-plane defined by Im⁡(z)>0 is disjoint from MC⁢HT. We assume by contradiction the existence of a point z0 such that y0=Im⁡(z0) is positive and z0∈MC⁢HT. We denote by Ly0 the horizontal line formed by points satisfying Im⁡(z)=y0. Since there is a family of associated rays accumulating onto the ℝ-axis, there exists a path (t+i⁢f⁢(t))t∈ℝ such that for any t, f(t)∈]0,y04[ and t+i⁢f⁢(t)∈𝒟+. Applying Lemma 6.8 to the intersection between Ly0 and the family of associated rays starting from t+i⁢f⁢(t), a continuity argument proves that z0 belongs to some associated ray of the family (as t moves from −∞ to +∞, the intersection of the associated rays with Ly0 moves from the right end to the left end of this horizontal line). Therefore, z0 cannot belong to MC⁢HT and the open upper half-plane defined by Im⁡(z)>0 is disjoint from MC⁢HT.

Then, there are interior points of connected a component X of MC⁢HT located above κ whose imaginary value is negative. It follows that the associated rays of points of 𝒟+ approaching κ intersect the interior of X (these associated rays accumulate on the ℝ-axis). Therefore, there is no such end κ and MC⁢HT is connected. ∎

Proposition 6.10.

There is a compact set K and a positive constant B>0 such that the intersection MC⁢HT∩Kc is contained in the closure of the domain bounded by the hyperbolas given by

y=y+2(1+Bx),y=y−2(1+Bx):x>0 (6.1)
y=y−2(1−Bx),y=y+2(1−Bx):x<0. (6.2)
Proof.

By Lemma 6.5 for any y in J=[y−,y−2⁢[∪]⁢y+2,y+], there is a positive constant A>0 such that the union of the two semi-infinite horizontal strips characterized by Im⁡(z)∈J and |Re⁡(z)|>A is disjoint from ℋR.

Consider some positive number B>A and introduce the domain DB characterized by the inequalities:

  • •

    Im⁡(z)>y+ if Re⁡(z)∈[−B,B];

  • •

    Im⁡(z)>g⁢(t) where g⁢(t)=y+2⁢|t|+B|t| if t=Re⁡(z)∉[−B,B].

For any point z such that Im⁡(z)>y+, the associated ray r⁢(z) remains in DB. Now we assume that z=t+i⁢y satisfies the conditions

|t|>Bandy+2⁢|t|+B|t|<|y|≤y+.

Without loss of generality, we assume that t<−B.

In order to prove that the associated ray r⁢(z) remains in DB, we have to show that for any t<−B and any s∈[t,−B], we have

Im⁡(R⁢(z))Re⁡(R⁢(z))>g⁢(s)−g⁢(t)s−t.

Since g⁢(s)−g⁢(t)s−t≤B⁢y+2⁢s⁢t≤−y+2⁢t, we just have to prove that

Im⁡(R⁢(z))Re⁡(R⁢(z))>−y+2⁢t.

In our case Re⁡(R⁢(z))=−1t+o⁢(t−2) and Im⁡(R⁢(z))=yt2+o⁢(t−3) imply that

Im⁡(R⁢(z))Re⁡(R⁢(z))=−yt+o⁢(t−2).

Since y−y+2>y+⁢B2⁢|t|>0, the inequality holds provided B is large enough.

By replacing y+ by y−, we get an analogous result for the lower part of the complement to MC⁢HT. ∎

6.2.3. Examples

Consider a family of operators of the form Tα=Q⁢(z)⁢dd⁢z+P⁢(z) where Q⁢(z)=(z−α)k and P⁢(z)=z⁢(α−z)k with the common root α∈ℂ of degree k∈ℕ∗.

The family Tα provides a rich assortment of examples. We have R⁢(z)=−1z. The special line is the real axis ℝ which coincides with the horizontal locus ℋR. Besides, the integral curves of R⁢(z)⁢∂z are hyperbolas (level sets of x⁢y).

Proposition 6.11.

If α∈ℝ, then the minimal set MC⁢HT of operator Tα coincides with the real axis ℝ.

Proof.

This follows immediately from Proposition 6.7 and 6.9. ∎

If α does not belong to the real axis, we get different pictures depending on whether or not α belongs to the imaginary axis. Without loss of generality, we will assume that I⁢m⁢(α)>0.

Proposition 6.12.

If α is of the form y0⁢i with y0>0, then the minimal set MC⁢HT is the union of the segment [y02⁢i,y0⁢i] with the horizontal strip formed by points z satisfying 0≤I⁢m⁢(z)≤y02.

Proof.

From Proposition 6.7 it follows immediately that the convex hull of MC⁢HT is contained in the strip bounded by ℝ and the horizontal line I⁢m⁢(z)=I⁢m⁢(y0). For any point of segment [0,y0⁢i], the associated ray contains α so [0,y0⁢i]⊂MC⁢HT.

For any point of the horizontal strip given by the inequalities 0≤I⁢m⁢(z)≤y02, a simple computation proves that its associated ray intersects the segment [0,y0⁢i].

Finally, for any point z such that I⁢m⁢(z)>y02 and R⁢e⁢(z)≠0, the associated ray is disjoint from the segment [0,y0⁢i]. This completely characterizes the minimal set. ∎

The latter case provides an example of a partially irregular minimal set whose irregularity locus is contained in a R-invariant line (the imaginary axis in this case).

In the general case, the boundary of MC⁢HT is more complicated. Up to conjugation, we can restrict us to the case when Re⁡(α),Im⁡(α)>0.

Proposition 6.13.

If α is of the form x0+y0⁢i with x0,y0>0, then the minimal set MC⁢HT of Tα is bounded by the following arcs:

  • •

    the real ℝ-axis ;

  • •

    global arc (t,f1⁢(t)) where f1⁢(t)=y0⁢t2⁢t−x0 for t∈[x0,+∞[;

  • •

    local arc (t,f2⁢(t)) where f2⁢(t)=x0⁢y0t for t∈[x0,xe];

  • •

    global arc (t,f3(t) where f3⁢(t)=x0⁢y0⁢t(2⁢x0⁢t+x0)2 for t∈[0,xe];

  • •

    global arc (t,f4⁢(t)) where f4⁢(t)=y0⁢t2⁢t−x0 for t∈]−∞,0].

Here, (xe,ye) is a point of extruding type. Its coordinates are xe=(3+2⁢2)⁢x0 and ye=y03+2⁢2.

Proof.

The convex hull of MC⁢HT is contained in the strip bounded by ℝ and the horizontal line I⁢m⁢(z)=I⁢m⁢(y0), see Proposition 6.7. The arcs (t,f1⁢(t)) and (t,f4⁢(t)) are characterized by the fact that the associated rays starting from their points contain x0+i⁢y0 (this can be checked by a direct computation). In particular, they belong to two distinct branches of the same hyperbola. Besides, the domain 𝒟 between ℝ− and arc (t,f4⁢(t)) is automatically contained in MC⁢HT.

Following Proposition 2.10, the backward trajectory of the vector field R⁢(z)⁢∂z starting at x0+y0⁢i is contained in MC⁢HT. The domain between this portion of the integral curve and the arc (t,f1⁢(t)) is also contained in MC⁢HT.

We denote by 𝒟′ the domain in the open right upper quadrant where the associated ray intersects the domain 𝒟. At each point (t,γ⁢(t)) of the upper boundary of 𝒟′, the associated ray is tangent to the branch of hyperbola (s,f4⁢(s)) for some s≤0. Since R⁢(z)=−1z, the argument of t+i⁢γ⁢(t) equals the negative of the slope of (s,f4⁢(s)) at s. Since d⁢f4d⁢s⁢(s)=−x0⁢y0(2⁢s−x0)2, we get

γ⁢(t)t=x0⁢y0(2⁢s−x0)2.

Since the tangent line has to intersect the imaginary axis at 2⁢γ⁢(t)⁢i, we obtain the following equation:

f4⁢(s)−2⁢γ⁢(t)s=−x0⁢y0(2⁢s−x0)2.

Replacing γ⁢(t) by x0⁢y0⁢t(2⁢s−z0)2, we get t=s2x0.

Since s is the negative square root of x0⁢t, we deduce that γ⁢(t)=x0⁢y0⁢t(2⁢x0⁢t+x0)2. In particular, for s=−x0, we get t=x0 and γ⁢(x0)=y09.

The arc γ and the backward trajectory starting at x0+i⁢y0 (which is a branch of hyperbola) intersect each other at some point xe+i⁢ye. From a computation, we obtain xe=(3+2⁢2)⁢x0 and therefore ye=y03+2⁢2.

It is then geometrically clear that for any point z above the curve formed by arcs defined by functions f1,f2,f3,f4, the associated ray cannot intersect any of these arcs. ∎

The latter example provides an illustration of a point of extruding type. Since the boundary arcs are explicit algebraic curves, we can obtain the exact picture shown in Figure 11.

Refer to caption
Figure 11. The case when α=1+0.8⁢i.

6.3 Case deg⁡Q−deg⁡P=0

For the sake of simplicity, we normalize the vector field R⁢(z)⁢∂z by an affine change of variable so that R⁢(z)=1+μzκ+o⁢(z−κ−1) for some μ∈ℂ∗ and κ≥1. The case of a constant vector field is already treated in Section 2.3 of [AHN+24].

Under the assumptions Im⁡(μ)≠0 and κ=1, we are going to prove that the minimal set MC⁢HT is connected. Firstly we show that MC⁢HT is regular and disjoint from the curve of inflections ℑR outside a compact set.

Lemma 6.14.

Assuming that Im⁡(μ)⁢(−1)κ>0, there is a cone 𝒞 and a compact set K such that:

  • •

    for any z∈𝒞, Im⁡(R⁢(z))>0 and Im⁡(R′⁢(z))>0;

  • •

    MC⁢HT⊂𝒞∪K.

Besides, MC⁢HT is a regular subset of ℂ.

Proof.

Computing R⁢(t) and R′⁢(t) for a negative real number t, we obtain that R⁢(t)=1+μtκ+o⁢(t−κ−1) and thus Im⁡(R⁢(t))∼Im⁡(μ)⁢t−κ. The sign of the former is thus the sign of Im⁡(μ)⁢(−1)κ. Similarly we obtain that it is also the sign of Im⁡(R′⁢(t)) for t close enough to infinity and negative.

It follows from Proposition 2.25 that MC⁢HT is contained in an infinite cone 𝒞0 whose asymptotic directions are ]π−ϵ,π+ϵ[ for some ϵ∈]0,π2[. Besides, since the asymptotic directions of infinite branches of the algebraic curves defined by equations Im⁡(R)=0 and Im⁡(R′)=0 are not horizontal, 𝒞0 and thus MC⁢HT are covered by the union of a cone 𝒞 and a compact set K such that for any z∈𝒞, Im⁡(R⁢(z))>0 and Im⁡(R′⁢(z))>0.

Any R-invariant line (see Definition 2.5) has to be horizontal and therefore it intersects the cone 𝒞. Thus some points of any R-invariant line Λ have the associated rays that are not contained in Λ. Therefore, there are no R-invariant lines for such a vector field R⁢(z)⁢∂z. The minimal set MC⁢HT has hence no tails and Theorem 2.23 guarantees that MC⁢HT is regular. ∎

Corollary 6.15.

Assuming that Im⁡(μ)⁢(−1)κ>0, consider a sequence (αn)n∈ℕ of points of ∂MC⁢HT such that |αn|→+∞ and Δ⁢(αn)≠∅ for any n∈ℕ. Then there exist a subsequence (αf⁢(n))n∈ℕ and a line ℒy0 given by Im⁡(z)=y0 such that:

  • •

    the line ℒy0 is disjoint from the interior of MC⁢HT;

  • •

    the line ℒy0 contains a point of ∂MC⁢HT;

  • •

    Re⁡(αf⁢(n))→−∞;

  • •

    Im⁡(αf⁢(n))≤y0 for any n∈ℕ.

Proof.

Up to taking a subsequence, we can also assume that every an belongs to the cone 𝒞 defined in Lemma 6.14. Lemma 4.17 implies that for any n, points of Δ⁢(αf⁢(n)) belong to ℑ−, ℑR or 𝒵⁢(P⁢Q). Therefore, following Lemma 6.14, points of Δ⁢(αn) accumulate in a compact set as n→∞. We denote by z0 one of their accumulation points and by Ly0 the horizontal line containing z0 (here y0=Im⁡(z0)).

Thus, up to taking a subsequence of α, we get a sequence (yn)n∈ℕ such that yn→y0 and yn∈Δ⁢(αn) for any n∈ℕ. As αn goes to infinity while Δ⁢(αn) remains in a compact set, the associated rays r⁢(αn) accumulate on Ly0. Thus the line Ly0 is disjoint from the interior of MC⁢HT. Besides, since αn∈𝒞 for any n∈ℕ, we have I⁢m⁢(R⁢(αn))>0 and therefore Im⁡(αn)≤y0. ∎

Lemma 6.16.

Provided that Im⁡(μ)≠0 and κ=1, no integral curve has a horizontal asymptotic line at infinity.

Proof.

Since R⁢(z)=1+μz+o⁢(z−2), the integral curve γ⁢(t) satisfies Re⁡(γ⁢(t))∼t as t→±∞. Then I⁢m⁢(γ′⁢(t))=I⁢m⁢(μ)t+o⁢(t−1). We obtain that I⁢m⁢(γ⁢(t)) has logarithmic growth as t→±∞ and therefore the integral curve has no asymptotic lines at infinity. ∎

Corollary 6.17.

Provided that Im⁡(μ)≠0 and κ=1, the minimal set MC⁢HT is connected in ℂ. Besides, ∂MC⁢HT has exactly two infinite arcs: one is a local arc starting at infinity while the other is a global arc ending at infinity.

Proof.

Without loss of generality, we can assume that Im⁡(μ)>0. Proposition 2.20 shows that there are finitely many connected components of MC⁢HT. Moreover they are attached to the point π∈𝕊1 at infinity in some linear order. We refer to these components of MC⁢HT as X1,…,Xk where X1 is the lowest component while Xk is the highest component. Besides the boundary ∂Xi of any component Xi has exactly two topological ends. We call them the lower end ∂Xi− and the upper end ∂Xi+.

Since ∂MC⁢HT∩ℑR is contained in a compact set (see Lemma 6.14), points of ∂X that are close enough to infinity are either of local, global or of extruding types. Proposition 4.12 proves that every local arc has an endpoint in ℑR∪𝒵⁢(P⁢Q). Thus the ends of ∂X are represented either by a local arc starting at infinity or by a global arc. Since Im⁡(μ)>0, these points belong to ℑ− so the orientation constraint shows that only the upper end can be represented by a local arc. Otherwise, the local arc would have the point at infinity as its endpoint. Equivalently, any lower end ∂Xi− has to be represented by an infinite global arc ending at infinity (see Lemma 4.23).

For any component Xi, the lower end ∂Xi− of its boundary is approached by a sequence of points of global type. Applying Corollary 6.15 to such a sequence we prove the existence of a horizontal line Li lying below the component Xi and disjoint from the interior of MC⁢HT. Thus no component of MC⁢HT lying below the line Li can contain an infinite local arc because the latter has no asymptotic line at infinity (see Lemma 6.16). Consequently, among the ends of ∂MC⁢HT, only ∂Xk+ can be represented by a local arc.

It remains to prove that MC⁢HT has only one connected component. Assuming that k>1, we consider the upper end ∂X1+. We already know that it can be approached by points (αn)n∈ℕ for which Δ⁢(an)≠∅. Applying Corollary 6.15, we prove the existence of a line L such that:

  • •

    L is disjoint from the interior of MC⁢HT;

  • •

    there is some point z0∈L∩∂MC⁢HT;

  • •

    points of (αn)n∈ℕ lie below the line L.

We deduce from the first and the third bullet points that the interior of component X1 lies below line L.

Besides, since for each n∈ℕ, αn belongs to ℑ−, Δ⁢(αn) is a direct support point of MC⁢HT for the associated ray r⁢(αn) (see Lemma 4.4). Then, z0 is also a direct support point of MC⁢HT for (oriented) line L. It follows that the interior of the component of MC⁢HT containing z0 lies above line L. In other words, z0 belongs to some component Xi such that i>1.

For any point z of ∂Xi close enough to ∂Xi−, the associated ray r⁢(z) has to cross the portion of the line L formed by points whose real part is smaller than Re⁡(z0) (otherwise, the associated ray would have to cross the interior of Xi). This is impossible since z lies on or above L and I⁢m⁢(R⁢(z))>0. This is a contradiction. There is no such component Xi and MC⁢HT is connected. A neighborhood of its upper end is contained in a local arc while a neighborhood of its lower end is contained in a global arc. ∎

6.4 Connected components of minimal sets

Putting together partial results for the different values of deg⁡Q−deg⁡P, we are able to state a bound on the number of connected components of MC⁢HT in ℂ. It is already known that the closure of MC⁢HT in the extended plane ℂ∪𝕊1 is always connected.

Proof of Theorem 1.12.

For any operator T satisfying |deg⁡Q−deg⁡P|>1, it has been proved in Theorem 1.11 of [AHN+24] that MC⁢HT=ℂ. Besides, when deg⁡Q−deg⁡P=1, Section 6.3 and Corollary 5.20 of the same paper proves that MC⁢HT is connected and contractible. For deg⁡Q−deg⁡P=−1, it follows from Proposition 6.9.

The only case where there could be several connected components is deg⁡Q−deg⁡P=0. If R⁢(z) is constant, then there are two situations. If P,Q are both constant, then there is no meaningful notion of minimal set (see Section 2.3.1 in [AHN+24]). Otherwise, MC⁢HT is formed by parallel half-lines starting at points of 𝒵⁢(P⁢Q). Since every point of 𝒵⁢(P⁢Q) is a common root of P and Q (otherwise R⁢(z) would not be constant) we get that there are at most 12⁢deg⁡P+12⁢deg⁡Q such half-lines.

If R⁢(z) is not constant, then we have R⁢(z)=λ+μzκ+o⁢(z−κ) for some λ,μ∈ℂ∗ and κ∈ℕ∗. If κ=1 and Im⁡(μ/λ)≠0, then Corollaries 6.17 proves that MC⁢HT is connected. Otherwise, Proposition 2.20 provides an upper bound 12⁢deg⁡P+12⁢deg⁡Q. ∎

6.5 Case deg⁡Q−deg⁡P=1

In [AHN+24] we found that, outside a rather trivial case222When deg⁡P=0 and deg⁡Q=1, MC⁢HT coincides with the unique root of Q⁢(z) when λ∉ℝ<0 and coincides with ℂ otherwise., a necessary and sufficient condition for the compactness of MC⁢HT in case deg⁡Q−deg⁡P=1 is Re⁡(λ)≥0. Moreover in case Re⁡(λ)<0, we get MC⁢HT=ℂ.

We will describe MC⁢HT for Re⁡(λ)=0. Unfortunately, in the most interesting situation Re⁡(λ)>0, we do not have a general description of MC⁢HT, but we provide a number of partial results, observations and examples.

6.5.1. Re⁡(λ)=0

In this case a complete characterization of ∂MC⁢HT can be carried out.

Theorem 6.18.

Consider a linear differential operator T given by (1.1) such that deg⁡Q−deg⁡P=1 and Re⁡(λ)=0. In this case, the neighborhood of infinity is foliated by a family 𝒞 of closed integral curves of the vector field R⁢(z)⁢∂z.

The boundary ∂MC⁢HT of the minimal set of T is described as the first closed leaf (according to the natural ordering starting at infinity) of the family 𝒞 containing a point of 𝒵⁢(P⁢Q)∪ℑR.

If the latter leaf γ contains a point of the curve of inflections ℑR, then the latter point is a tangency point between γ and ℑR. Moreover it is the first leaf that is non-strictly convex (the curvature at the tangency point vanishes).

In particular, ∂MC⁢HT is formed by finitely many local arcs. It is real-analytic and convex (but can fail to be strictly convex). It contains neither zeros nor poles of R⁢(z)⁢∂z.

Proof.

It follows from λ∈ℂ∗ and Re⁡(λ)=0 that Im⁡(λ)≠0. The curve of inflections ℑR is therefore compact. The neighborhood of infinity is foliated by a family 𝒞 of integral curves of vector field R⁢(z)⁢∂z. The orientation of these integral curves depends on the sign of Im⁡(λ). By compactness of ℑR, another neighborhood 𝒞′ of infinity is foliated by strictly convex integral curves (the curvature of integral curves vanishes precisely on ℑR).

We first consider the case when some point α of 𝒵⁢(P⁢Q) belongs to 𝒞′. Denoting by γ the periodic leaf α belongs to, we deduce from Proposition 2.10 that γ belongs to MC⁢HT and bounds a strictly convex domain 𝒟. Provided the complement of 𝒟 does not contain any other point of 𝒵⁢(P⁢Q), we obtain that 𝒟 coincides with MC⁢HT. Since α is disjoint from ℑR, it follows from Corollary 3.12 that it cannot be a zero or a pole of R⁢(z) (α is a root of both P and Q of the same multiplicity).

In the remaining cases, we can assume that 𝒵⁢(P⁢Q) is disjoint from 𝒞′. The cylinder 𝒞 is bounded by a singular curve formed by separatrices (integral curves connecting singularities of R⁢(z)⁢∂z). We denote by Σ the union of these separatrices and by 𝒮 the smallest simply connected subset containing Σ. By Proposition 2.10, Σ and 𝒮 are contained in MC⁢HT. For the same reason, a point z of cylinder 𝒞 is contained in MC⁢HT if and only if the periodic integral curve containing z belongs entirely to MC⁢HT. Therefore, the boundary of MC⁢HT coincides with some periodic integral curve of the cylinder 𝒞.

Since the associated rays cannot cross the interior of MC⁢HT, its boundary ∂MC⁢HT (which is a periodic integral curve) has to be convex. Therefore, it is contained in the domain of inflection of infinity (or in its boundary). Since the domain 𝒞′ does not belong to the interior of MC⁢HT (its complement is clearly a TC⁢H-invariant set), these conditions characterize the boundary γ of 𝒞′ as the boundary of MC⁢HT.

The curve γ cannot cross the curve of inflections because it is convex. If it did not intersect ℐR there would be a strictly smaller invariant set whose boundary is an integral curve between ℐR and γ. Thus γ has a tangency point with ℐR. At this point, the curvature of γ vanishes.

The boundary ∂MC⁢HT is formed by local arcs joining points of 𝒵⁢(P⁢Q) (with the same multiplicity of P and Q) and some points of the tangency locus. By Proposition 4.7, there arcs are strictly convex and real-analytic. ∎

6.5.2. Re⁡(λ)>0

As we mentioned above, we do not have a general description of MC⁢HT, but only a number of interesting examples. Observe that in this case ∞ is a sink of R⁢(z)⁢∂z).

A qualitative description of the convex hull C⁢o⁢n⁢v⁢(MC⁢HT) is the best that we can obtain with our current knowledge.

Proposition 6.19.

Consider a linear differential operator T given by (1.1) with deg⁡Q−deg⁡P=1. The boundary ∂C⁢o⁢n⁢v⁢(MC⁢HT) of the convex hull C⁢o⁢n⁢v⁢(MC⁢HT) of the minimal set is formed by:

  • •

    finitely many straight segments;

  • •

    finitely many portions of integral curves of vector field R⁢(z)⁢∂z.

In particular, the latter are strictly convex and belong to local arcs of ∂MC⁢HT. In particular, ∂C⁢o⁢n⁢v⁢(MC⁢HT) is piecewise-analytic.

Proof.

We denote by 𝒮 the set of points where the boundary ∂C⁢o⁢n⁢v⁢(MC⁢HT) is strictly convex. They also belong to ∂MC⁢HT (these points belong to the support of the hull). It follows from Theorem 1.9 that outside finitely many points, 𝒮 is formed by either local or global arcs of ∂MC⁢HT. If such a point z belongs to a global arc, then the line containing the associated ray r⁢(z) is a support line of C⁢o⁢n⁢v⁢(MC⁢HT) at z and every point of Δ⁢(z). It follows that [z,Δm⁢a⁢x⁢(z)] is a straight segment contained in ∂C⁢o⁢n⁢v⁢(MC⁢HT). Consequently any arc of 𝒮 has to be a portion of local arc.

We know that ∂C⁢o⁢n⁢v⁢(MC⁢HT) is formed by straight segments and portions of local arcs. It remains to prove that there are finitely many of them. We consider an arc α of ∂C⁢o⁢n⁢v⁢(MC⁢HT) contained in a local arc γ of ∂MC⁢HT. The endpoint of α (with the orientation defined by R⁢(z)⁢∂z) has at the same time to be the endpoint of γ (since otherwise the associated rays starting at points of α would intersect MC⁢HT). Therefore, the endpoint of every such arc α in ∂C⁢o⁢n⁢v⁢(MC⁢HT) belongs to 𝒵⁢(P⁢Q)∪ℑR (see Proposition 4.12). Since there are finitely many such points in 𝒮, there are finitely many such arcs in ∂C⁢o⁢n⁢v⁢(MC⁢HT).

If the boundary of the convex hull is not formed by finitely many straight segments and portions of integral curves, then there are infinitely many corner points of angle smaller than π between the pairs of consecutive straight segments of the boundary. It follows from Corollary 6.2 that these points belong to 𝒵⁢(P⁢Q). Therefore, we have finitely many corner points and finitely many straight segments. ∎

In the examples below (including a very interesting family of operators in which Q⁢(z) has simple roots and P⁢(z)=Q′⁢(z)), C⁢o⁢n⁢v⁢(MC⁢HT) is a polygon.

Proposition 6.20.

Consider a linear differential operator T given by (1.1), such that every root α of Q⁢(z) is simple and satisfies P⁢(α)≠0 and ϕα=0.

Then, C⁢o⁢n⁢v⁢(MC⁢HT) coincides with the convex hull of 𝒵⁢(Q).

Proof.

The argument is similar to the one used in the proof of the classical Gauss–Lucas theorem (see [Mor]). If the differential form P⁢(z)⁢d⁢zQ⁢(z) has all positive residues, then the roots of P⁢(z) are contained in the convex hull of 𝒵⁢(Q).

The proof is based on consideration of the electrostatic force F created by the system of point charges placed at the poles of P⁢(z)⁢d⁢zQ⁢(z) where the value of each charge equals the residue at the corresponding pole. This electrostatic force F equals the conjugate of P⁢(z)⁢d⁢zQ⁢(z) and one can show that if we take any line L not intersecting the convex hull of 𝒵⁢(Q) then at any point p∈L, F points inside the half-plane of ℂ∖L not containing 𝒵⁢(Q). Now recall that the associated ray has the same direction as the conjugate of P/Q. Thus, the associated ray r⁢(p) does not intersect the convex hull of 𝒵⁢(Q). ∎

6.5.3. The first family of examples

Consider a family of operators of the form Tλ=Q⁢(z)⁢dd⁢z+P⁢(z) where Q⁢(z)=λ⁢(z−1)k⁢z and P⁢(z)=(z−1)k for some principal coefficient λ∈ℂ∗ and some degree k∈ℕ∗.

Integral curves of the vector field R⁢(z)⁢∂z are logarithmic spirals parametrized by γ⁢(t)=γ⁢(0)⁢eλ⁢t. In particular, they are concentric circles for R⁢e⁢(λ)=0.

Depending on the value of λ, the shape of the minimal set MC⁢HT can change drastically. Namely,

  • •

    if R⁢e⁢(λ)<0, then MC⁢HT=ℂ (see Theorem 1.11 of [AHN+24]);

  • •

    if R⁢e⁢(λ)=0, then MC⁢HT is the closed unit disk (see Theorem 6.18);

  • •

    if R⁢e⁢(λ)>0 and I⁢m⁢(λ)=0, then MC⁢HT is segment [0,1].

When R⁢e⁢(λ)>0 and I⁢m⁢(λ)≠0, MC⁢HT has a more complicated shape we describe below in terms of local and global arcs. Up to conjugation, we will assume that Im⁡(λ)>0.

Proposition 6.21.

If λ satisfies Re⁡(λ),Im⁡(λ)>0, then the minimal set MC⁢HT of operator Tλ is bounded by the following arcs:

  • •

    local arc γ where γ⁢(t)=e−λ⁢t and t∈]0,t0[;

  • •

    global arc α where α⁢(t)=11+λ⁢t and t∈]0,t1[.

These two arcs intersect at 1 and the point γ⁢(t0)=α⁢(t1) of extruding type characterized as the first intersection point between α and γ defined on ℝ>0.

Proof.

The backward trajectory of the vector field R⁢(z)⁢∂z starting at 1 is parametrized by γ⁢(t)=e−λ⁢t and t∈[0,∞). 2.10 shows that this arc is entirely contained in MC⁢HT.

Points z for which the associated ray contains 1 are characterized by the condition 1−zλ⁢z∈ℝ>0. They form an arc parametrized by α⁢(t)=11+λ⁢t for t∈[0,+∞[. This arc is also contained in MC⁢HT.

Since R⁢(z)=λ⁢z, it is geometrically clear that these two arcs bound MC⁢HT. The boundary ∂MC⁢HT is formed by a portion of each of them with two singular points at 1 (when t=0) and the first intersection point in the parametrization. There are different ways to see that such an intersection occurs. One of them is to note that limt→∞α⁢(t)=0 and limt→∞arg⁡(α′⁢(t))=limt→∞=arg⁡(−λ(1+λ⁢t)2) exists. Since the vector field R⁢(z) has residue with positive real and imaginary parts, it follows that α⁢(t) and γ⁢(t) intersect infinitely many times. The endpoint distinct from 1 common to α and γ is the first intersection point between the two parametrized arcs defined on ℝ>0. ∎

Refer to caption
Figure 12. Illustration of the boundary of the minimal set when λ=1+6⁢i in Prop. 6.21.

6.5.4. A second family of examples

Consider the family T=z⁢(zk−1)⁢dd⁢z+(zk+1), where k is a positive integer. We are going to prove that for any k, the minimal set MC⁢HT is the unit disk.

Lemma 6.22.

Set f⁢(z)=z+t⁢z⁢(zk−1)zk+1, with t>0. Then |f⁢(z)|>1 whenever |z|>1.

Proof.

We substitute z=r1k⁢ei⁢θ with r>1. After some algebraic manipulations, we find that

|f⁢(z)|2|z|2=|f⁢(r1k⁢ei⁢θ)|2r2/k=1+t⁢2⁢r2−2+r2⁢t−2⁢cos⁡(θ⁢k)⁢r⁢t+tr2+2⁢cos⁡(θ⁢k)⁢r+1. (6.3)

Setting c≔cos⁡k⁢θ and rewriting further, we get

|f⁢(r1k⁢ei⁢θ)|2r2/k=1+t⁢2⁢(r2−1)+t⁢((r−c)2+(1−c2))(r+c)2+(1−c2). (6.4)

Since −1≤c≤1, it follows that

|f⁢(r1k⁢ei⁢θ)|2r2/k>1+t⁢r2−1(r+1)2>1.

Consequently, |f⁢(z)|>|z| whenever |z|>1 and the statement follows. ∎

Lemma 6.23.

The separatrices of the vector field R⁢(z)⁢∂z=z⁢(zk−1)zk+1⁢∂z are the arcs of the unit circle, connecting roots of P⁢(z) with roots of Q⁢(z).

Proof.

Assuming that z is not a root of Q, we have that

∫zk+1z⁢(zk−1)⁢𝑑z=k−1⁢log⁡((1−zk)2zk).

Now for z=ei⁢θ, we find that

Im⁡log⁡((1−zk)2/zk)=arg⁡((1−zk)2/zk)=arg⁡(−2+ei⁢k⁢θ+e−i⁢k⁢θ)=π.

As the integral trajectories are level curves of Im⁢∫d⁢zR⁢(z) away from zeros or poles of R⁢(z), it follows that the unit circle consists of the integral trajectories of R⁢(z)⁢∂z. Since the roots of P lie on the unit circle and the zeros of Q on the unit circle have positive residues, it follows that these integral trajectories must be separatrices that are contained in MC⁢HT. ∎

Corollary 6.24.

For T=z⁢(zk−1)⁢∂z+(zk+1), the minimal set MC⁢HT coincides with the unit disk.

Proof.

By Lemma 6.22, we have that all the associated rays for points lying outside the unit disk never intersect the unit disk. Therefore, MC⁢HT is contained in the unit disk. Since the unit circle consists of separatrices of −R⁢(z)⁢∂z (see Lemma 6.23), it follows that MC⁢HT contains the unit circle. The associated ray of any point (distinct from 0) of the open unit disk intersects the unit circle so MC⁢HT coincides with the unit disk. ∎

References

  • [ABS] P. Alexandersson, P. Bränden, B. Shapiro, An inverse problem in Pólya-Schur theory. I. Non-degenerate and degenerate operators, submitted.
  • [AHN+24] P. Alexandersson, N. Hemmingsson, D. Novikov, B. Shapiro, G. Tahar, Linear first order differential operators and complex dynamics, Journal of Differential Equations, Vol 391, pp 265-320, 2024.
  • [Bea] A. F. Beardon. Iteration of rational functions, volume 132 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. Complex analytic dynamical systems.
  • [BB] J. Borcea, P. Brändén, Pólya-Schur master theorems for circular domains and their boundaries. Ann. of Math. (2) 170 (2009), no. 1, 465–492.
  • [BCh] P. Bränden, M. Chasse, Classification theorems for operators preserving zeros in a strip. J. Anal. Math. 132 (2017), 177–215.
  • [CsCr] G. Csordas, T. Craven, Composition theorems, multiplier sequences and complex zero decreasing sequences. in Value distribution theory and related topics, 131–166, Adv. Complex Anal. Appl., 3, Kluwer Acad. Publ., Boston, MA, 2004.
  • [Dia] K. Dias, A. Garijo. On the separatrix graph of a rational vector field on the Riemann sphere. Journal of Differential Equations, Vol 282: 541-565, 2021.
  • [Hut] J. E. Hutchinson. Fractals and self-similarity. Indiana Univ. Math. J., 30(5):713–-747, 1981.
  • [Mor] M. Marden. Geometry of polynomials 2nd edition, Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I. 1966.
  • [Will] S. Willard. General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts. 1970.