The Conley Conjecture and Beyond

As has been pointed out in the introduction, for many closed symplectic manifolds, every Hamiltonian diffeomorphism has infinitely many simple periodic orbits and, in fact, simple periodic orbits of arbitrarily large period whenever the fixed points are isolated. This unconditional existence of infinitely many periodic orbits is often referred to as the Conley conjecture. The conjecture was, indeed, formulated by [Conley1984] for tori, and since then it has been a subject of active research focusing on establishing the existence of infinitely many periodic orbits for broader and broader classes of symplectic manifolds or Hamiltonian diffeomorphisms.

The Conley conjecture was proved for the so-called weakly non-degenerate Hamiltonian diffeomorphisms in [Salamon and Zehnder1992] (see also [Conley and Zehnder1983b]) and for all Hamiltonian diffeomorphisms of surfaces other than $S^{2}$ in [Franks and Handel2003] (see also [Le Calvez2006]). In its original form for the tori, the conjecture was established in [Hingston2009] (see also [Mazzucchelli2013]), and the case of an arbitrary closed, symplectically aspherical manifold was settled in [Ginzburg2010]. The proof was extended to rational, closed symplectic manifolds $M$ with $c_{1}(TM)|_{\pi_{2}(M)}=0$ in [Ginzburg and Gürel2009b], and the rationality requirement was then eliminated in [Hein2012]. In fact, after [Salamon and Zehnder1992], the main difficulty in establishing the Conley conjecture for more and more general manifolds with aspherical first Chern class, overcome in this series of works, lied in proving the conjecture for totally degenerate Hamiltonian diffeomorphisms not covered by [Salamon and Zehnder1992]. (The internal logic in [Franks and Handel2003] and [Le Calvez2006], relying on methods from low-dimensional dynamics, was somewhat different.) Finally, in [Ginzburg and Gürel2012] and [Chance et al.2013], the Conley conjecture was proved for negative monotone symplectic manifolds. (The main new difficulty here was in the non-degenerate case.)

Two other variants of the Hamiltonian Conley conjecture have also been investigated. One of them is the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms with displaceable support; see, e.g., [Frauenfelder and Schlenk2007], [Gürel2008], [Hofer and Zehnder1994], [Schwarz2000] and [Viterbo1992]. Here the form $\omega$ is usually required to be aspherical, but the manifold $M$ is not necessarily closed. The second one is the Lagrangian Conley conjecture or, more generally, the Conley conjecture for Hamiltonians with controlled behavior at infinity on cotangent bundles (see [Hein2011]; [Long2000]; [Lu2009]; [Lu2011]; [Mazzucchelli2011]) or even some twisted cotangent bundles (see [Frauenfelder et al.2012]). In this survey, however, we focus mainly on the case of closed manifolds.

The Conley conjecture looks deceptively similar to the well-known conjecture that every closed simply connected Riemannian manifold (e.g., $S^{n}$) carries infinitely many non-trivial closed geodesics. However, this appears to be a very different problem than the Conley conjecture, for the latter does not distinguish trivial and non-trivial orbits. For instance, the proof of the Lagrangian Conley conjecture for the pure kinetic energy Hamiltonian simply detects the constant geodesics. We will further discuss the connection between the two conjectures in Sects. 4 and 6.

What makes the Conley conjecture difficult and even counterintuitive from a homological perspective is that there seems to be no obvious homological reason for the existence of infinitely many simple periodic orbits. As we have already mentioned, there is no homological growth: the Floer homology of a Hamiltonian diffeomorphism does not change under iterations and remains isomorphic, up to a Novikov ring, to the homology of the manifold. (In that sense, the difficulty is similar to that in proving the existence of infinitely many non-trivial closed geodesics on, say, $S^{n}$ where the rank of the homology of the free loop space remains bounded as a function of the degree.)

Ultimately, one can expect the Conley conjecture to hold for the majority of closed symplectic manifolds. There are, however, notable exceptions. The simplest one is $S^{2}$: an irrational rotation of $S^{2}$ about the $z$ axis has only two periodic orbits, which are also the fixed points; these are the poles. In fact, any manifold that admits a Hamiltonian torus action with isolated fixed points also admits a Hamiltonian diffeomorphism with finitely many periodic orbits. For instance, such a diffeomorphism is generated by a generic element of the torus. In particular, flag manifolds (hence the complex projective spaces and the Grassmannians), and, more generally, most of the coadjoint orbits of compact Lie groups as well as symplectic toric manifolds all admit Hamiltonian diffeomorphisms with finitely many periodic orbits. In dimension two, there are also such examples with interesting dynamics. Namely, there exist area preserving diffeomorphisms of $S^{2}$ with exactly three ergodic measures: two fixed points and the area form; see [Anosov and Katok1970] and, e.g., [Fayad and Katok2004]. These are the so-called pseudo-rotations. By taking direct products of pseudo-rotations, one obtains Hamiltonian diffeomorphisms of the products of several copies of $S^{2}$ with finite number of ergodic measures, and hence with finitely many periodic orbits. It would be extremely interesting to construct a Hamiltonian analog of pseudo-rotations for, say, ${\mathbb{CP}}^{2}$.

In all known examples of Hamiltonian diffeomorphisms with finitely many periodic orbits, all periodic orbits are fixed points, i.e., no new orbits are created by passing to the iterated diffeomorphisms, cf. Sect. 6. Furthermore, all such Hamiltonian diffeomorphisms are non-degenerate, and the number of fixed points is exactly equal to the sum of Betti numbers. Note also that Hamiltonian diffeomorphisms with finitely many periodic orbits are extremely non-generic; see [Ginzburg and Gürel2009c] and Sect. 2.2.

In any event, the class of manifolds admitting “counterexamples” to the Conley conjecture appears to be very narrow, which leads one to the question of finding further sufficient conditions for the Conley conjecture to hold. There are several hypothetical candidates. One of them, conjectured by the second author of this paper, is that the minimal Chern number $N$ of $M$ is sufficiently large, e.g., $N>\dim M$. (The condition $c_{1}(TM)|_{\pi_{2}(M)}=0$ corresponds to $N=\infty$.) More generally, it might be sufficient to require the Gromov–Witten invariants of $M$ to vanish, as suggested by Michael Chance and Dusa McDuff, or even the quantum product to be undeformed. No results in these directions have been proved to date. Note also that for all known “counterexamples” to the Conley conjecture $\operatorname{H}_{*}(M;{\mathbb{Z}})$ is concentrated in even degrees.

Another feature of Hamiltonian diffeomorphisms with finitely many periodic orbits is that, for many classes of manifolds, the actions or the actions and the mean indices of their simple periodic orbits must satisfy certain resonance relations of Floer homological nature; see [Chance et al.2013], [Ginzburg and Gürel2009b], [Ginzburg and Kerman2010] and [Kerman2012]. (There are also analogs of such resonance relations for Reeb flows, which we will briefly touch upon in Sect. 4.) Although the very existence of homological resonance relations in the Hamiltonian setting is an interesting, new and unexpected phenomenon, and some of the results considered here do make use of these relations, their discussion is outside the scope of this paper.

In this section, we briefly introduce our basic conventions and notation and then state the most up-to-date results on the Conley conjecture and generic existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms.

In this section we recall, very briefly, several definitions and results needed for the proof of the (CY) case of Theorem 2.1 and also some terminology used throughout the paper.

The following proposition settling, in particular, the non-degenerate case of the Conley conjecture for symplectic CY manifolds is a refinement of the main result from [Salamon and Zehnder1992]. It is proved in [Ginzburg2010] and [Ginzburg and Gürel2009b], although the argument given below is somewhat different from the original proof.

The key to the proof is the fact that $\operatorname{HF}_{n}(\varphi^{k})\neq 0$ for all $k$ and that, even when $\omega|_{\pi_{2}(M)}\neq 0$, the condition $c_{1}(TM)|_{\pi_{2}(M)}=0$ guarantees that all recappings of every orbit have the same mean index and the same (graded) local Floer homology.

With Proposition 3.5 established, it remains to deal with the degenerate case of the Conley conjecture, i.e., the case where $\varphi$ has an SDM. We do this in the next section; see Theorem 3.8.

In this section, we show that a Hamiltonian diffeomorphism with an SDM fixed point has infinitely many simple periodic orbits. We assume that $M$ is rational as in [Ginzburg and Gürel2009b]. The case of irrational CY manifolds is treated in [Hein2012].

This theorem is in turn a consequence of the following result.

For instance, to prove case (ii) of Theorem 3.8 when $M$ is CY it suffices to observe that no $k$th iteration of a fixed point can contribute to the Floer homology in degree $n+1$ for any action interval when $k$ is a sufficiently large prime and $\varphi$ has finitely many fixed points. When $\omega|_{\pi_{2}(M)}=0$, the argument is similar, but now the action filtration is used in place of the degree. The proof of case (i) is more involved; see Ginzburg and Gürel ([Ginzburg and Gürel2009b], Sect. 3) where some more general results are also established.

It is worth pointing out that although Theorem 3.9 guarantees the existence of capped simple $k$-periodic orbits $\bar{y}$ with action close to $kc={\mathcal{A}}_{H^{{\natural}k}}(\bar{x}^{k})$, we do not claim that the orbits $y$ are close to $x^{k}$ or even intersect a neighborhood of $x$. In general, essentially nothing is known about the location of these orbits and hypothetically a small neighborhood of $x$ may contain no periodic orbits at all. However, as is proved in [Yan2014], the orbit $x$ is in a certain sense an accumulation point for periodic orbits when, e.g., $M={\mathbb{T}}^{2}$.

The key to the proof is the following geometrical characterization of SDMs:

Returning to the proof of Theorem 3.9, we apply the lemma to the SDM orbit $x$ and observe that the local loop $\varphi_{H}^{t}\circ(\varphi_{K}^{t})^{-1}$ has zero Maslov index and hence is contractible. It is not hard to show that every local contractible loop extends to a global contractible loop; see Ginzburg ([Ginzburg2010], Lemma 2.8). In other words, we can extend the Hamiltonian $K_{t}$ from Lemma 3.10 to a global Hamiltonian such that $\varphi_{K}=\varphi$, not only near $x$ but on the entire manifold $M$.

With this in mind, let us reset the notation. Replacing $H$ by $K$ but retaining the original notation, we can say that for every $\eta>0$ there exists a Hamiltonian $H$ such that

• •

  $\varphi_{H}=\varphi$;

• •

  $x$ is a constant periodic orbit of $H$, and $H_{t}$ has an isolated local maximum at $x$ for all $t$;

• •

  $\|d^{2}H_{t}(x)\|<\eta$ for all $t$.

Furthermore, we can always assume that all such Hamiltonians $H$ are related to each other and to the original Hamiltonian via global loops with zero action and zero Maslov index. Thus, in particular, $c={\mathcal{A}}_{H}(x)=H(x)$ is independent of the choice of $H$ above, and all Hamiltonians have the same filtered Floer homology. Therefore, it is sufficient to prove the theorem for any of these Hamiltonians $H$ with arbitrarily small Hessian $d^{2}H(x)$.

To avoid technical difficulties and illuminate the idea of the proof, let us asume that $d^{2}H(x)=0$ and, of course, that $H_{t}$ has, as above, a strict local maximum at $x$ for all $t$. This case, roughly speaking, corresponds to an SDM $x$ with $d\varphi|_{x}={id}$.

To prove (3.2)‣ for a given $\epsilon>0$, we will use the standard squeezing argument, i.e., we will bound $H$ from above and below by two autonomous Hamiltonians $H_{\pm}$ as in Fig. 1 and calculate the Floer homology of $kH_{\pm}$.

In a Darboux neighborhood $U$ of $x$, the Hamiltonians $H_{\pm}$ are rotationally symmetric. The Hamiltonian $H_{+}$ is constant and equal to $c$ near $x$ on a ball of radius $r$ and then sharply decreases to some $c^{\prime}$ which is close to $c$ and attained on the sphere of radius $R$. Then, after staying constant on a spherical shell, $H_{+}$ increases to some value $c_{+}$, to accommodate $H$, and becomes constant. The radii $r<R$ depend on $\epsilon$; namely, we require that $\pi R^{2}<\epsilon$. The Hamiltonian $H_{-}$ is a bump function decreasing from its maximum $c$ at $x$ to a large negative value $c_{-}$. Thus we have

We require $H_{-}$ to have a strict maximum at $x$ with $d^{2}H_{-}(x)=0$. Then the local Floer homology of $H_{-}^{{\natural}k}$ is equal to ${\mathbb{Z}}_{2}$ and concentrated in degree $n$, i.e., $x$ is also an SDM for $H_{-}$ and all its iterations.

Now, for any $a<b$ outside ${\mathcal{S}}(H^{{\natural}k})$ and ${\mathcal{S}}(H^{{\natural}k}_{\pm})$, we have the maps

induced by monotone homotopies, where $\operatorname{HF}^{(a,\,b)}_{*}\big(H^{{\natural}k}_{\pm}\big)=\operatorname{% HF}^{(a,\,b)}_{*}\big(kH_{\pm}\big)$ since $H_{\pm}$ are autonomous Hamiltonians. Therefore, it is sufficient to prove that the map

is non-zero for some $\delta$ in the range $(0,\,\epsilon)$.

To this end, let us assume first that $H_{\pm}$, as above, are functions on ${\mathbb{R}}^{2n}$ constant outside a neighborhood of $x=0$. The filtered Floer homology of $H_{\pm}$ is still defined for any interval $(a,\,b)$ not containing $c_{\pm}$. Moreover, a decreasing homotopy $H^{s}$ from $H_{+}$ to $H_{-}$ through functions constant outside a compact set induces a map in Floer homology even when the value of $H^{s}$ at infinity passes through $(a,\,b)$. Then we have an isomorphism

when $k$ is sufficiently large and $\delta>0$ is sufficiently small. Namely, $k$ is so large that $k(c-c^{\prime})>\pi R^{2}$. This is the origin of the requirement $k>k_{\epsilon}$. Then the homology of $kH_{\pm}$ is generated by the periodic orbit closest to $x$, and $\delta$ is chosen so that $kc+\delta$ is smaller than the action of $kH_{-}$ on this orbit.

The isomorphism (3.4)‣ is established by a straightforward analysis of periodic orbits and easily follows from the calculation carried out already in [Ginzburg and Gürel2004]. It is based on two facts: that

is an isomorphism when $\delta>0$ is small and that $\operatorname{HF}^{(a,\,b)}_{n+1}\big(kH_{\pm}\big)=0$ for every sufficiently large interval $(a,\,b)$ containing $kc$ and contained in $(kc_{-},\,kc_{+})$.

It remains to transplant this calculation from ${\mathbb{R}}^{2n}$ to a closed manifold $M$. The key to this is the fact that the action interval in question is sufficiently small. This enables one to localize a calculation of filtered Floer homology by essentially turning action localization to spatial localization. A general framework for this process, developed in [Ginzburg and Gürel2009b], is as follows. Let $S$ be a shell in $M$, i.e., a region between two hypersurfaces and bounding a contractible domain $V$ in $M$. (To be more precise, $V$ is bounded in $M$ by a connected component of $\partial S$ and $S\cap V=\emptyset$. The contractibility assumption can be significantly relaxed.) Furthermore, let $F$ be a Hamiltonian which we require to be constant on the shell. For any interval $I=(\alpha,\,\beta)$ not containing $F|_{S}$, consider the subspace of the Floer complex generated by the orbits of $F$ in $V$ with cappings also contained in $V$. If necessary, we perturb $F$ in $V$ to make sure that the orbits with action in $I$ are non-degenerate. Then there exists a constant $\epsilon(S)>0$ such that, when $|I|=\beta-\alpha<\epsilon(S)$, this subspace of the Floer complex is actually a subcomplex and, moreover, a direct summand. [This is an immediate consequence of the fact that a holomorphic curve crossing $S$ must have energy bounded away from zero by some constant $\epsilon(S)$.] Furthermore, continuation maps respect this decomposition as long as the Hamiltonians remain constant on $S$. (However, the value of the Hamiltonians on $S$ can enter the interval $I$ during the homotopy.) Let us denote the resulting Floer homology by $\operatorname{HF}_{*}^{I}(F;V)$.

We apply this construction to $H_{\pm}$ with $S$ being the spherical shell where $H_{+}=c^{\prime}$ and $V$ being the ball of radius $R$ enclosed by this shell. (Hence we also need $H_{-}$ to be constant outside $V$.) As a consequence, (3.4)‣ turns into an isomorphism

entering the map (3.3)‣ as a direct summand. Hence (3.3)‣ is also non-zero.

The general case where we only have $\|d^{2}H(x)\|<\eta$ is handled in a similar way, but the construction of $H_{\pm}$ is considerably more involved and the choice of the modified Hamiltonians $H$ requires more attention; see [Ginzburg2010] and [Ginzburg and Gürel2009b]. $\square$

Interestingly, no other proof of the Conley conjecture is known for general symplectic manifolds. A more conceptual or just plain different argument may shed new light on the nature of the phenomena considered here and is likely to have other applications. [For the torus, a different proof is given in the original work [Hingston2009] and then in [Mazzucchelli2013]. However, it is not clear to us how to translate that proof to symplectic topological language.]

The collection of all closed symplectic manifolds breaks down into two classes: those for which the Conley conjecture holds and those for which the Conley conjecture fails. Of course, the non-trivial assertion is then that, as we have seen, the former class is non-empty and even quite large. The situation with closed contact manifolds is more involved even if we leave aside such fundamental questions as the Weinstein conjecture and furthermore focus exclusively on the contact homological properties of the manifold.

First of all, there is a class of contact manifolds for which every Reeb flow has infinitely many simple closed orbits because the rank of the contact or symplectic homology grows as a function of the index or of some other parameter related to the order of iteration. This phenomenon is studied in, e.g., [Colin and Honda2013], [Hryniewicz and Macarini2015] and [McLean2012] and the results generalize and are inspired by a theorem from [Gromoll and Meyer1969], establishing the existence of infinitely many closed geodesics for manifolds whose free loop space homology grows. (A technical but important fact closely related to Theorem 3.2 and underpinning the proof is that the iterates of a given orbit can make only bounded contributions to the homology; see [Ginzburg and Gürel2010], [Gromoll and Meyer1969], [Hryniewicz and Macarini2015] and [McLean2012] for various incarnations of this result.) By [Vigué-Poirrier and Sullivan1976], [Abbondandolo and Schwarz2006], [Salamon and Weber2006] and [Viterbo1999], among contact manifolds in this class are the unit cotangent bundles $ST^{*}M$ whenever $\pi_{1}(M)=0$ and the algebra $\operatorname{H}^{*}(M;{\mathbb{Q}})$ is not generated by one element, and some others; [Colin and Honda2013], [Hryniewicz and Macarini2015] and [McLean2012]. As is already pointed out in Sect. 2.1, this homologically forced existence of infinitely many Reeb orbits has very different nature from the Hamiltonian Conley conjecture where there is no homological growth.

Then there are contact manifolds admitting Reeb flows with finitely many closed orbits. Among these are, of course, the standard contact spheres and, more generally, the pre-quantization circle bundles over symplectic manifolds admitting torus actions with isolated fixed points; see Gürel ([Gürel2015], Example 1.13). Note that the class of such pre-quantization circle bundles includes the Katok–Ziller flows, i.e., Finsler metrics with finitely many closed geodesics on $S^{n}$ and on some other manifolds; see [Katok1973] for the original construction and also [Ziller1983]. Another important group of examples also containing the standard contact spheres arises from contact toric manifolds; see [Abreu and Macarini2012]. These two classes (pre-quantization circle bundles and contact toric manifolds) overlap, but do not entirely coincide. Although this is not obvious, Reeb flows with finitely many periodic orbits may have non-trivial dynamics, e.g., be ergodic; see [Katok1973].

Finally, as is shown in [Ginzburg et al.2014], there is a non-empty class of contact manifolds for which every Reeb flow (meeting certain natural index conditions) has infinitely many simple closed orbits, although there is no obvious homological growth—the rank of the relevant contact homology remains bounded. One can expect this class to be quite large, but at this point such unconditional existence of infinitely many closed Reeb orbits has only been proved for the pre-quantization circle bundles of certain aspherical manifolds; see Theorem 4.1. The proof of this theorem is quite similar to its Hamiltonian counterpart.

This picture is, of course, oversimplified and not even close to covering the entire range of possibilities, even on the homological level. For instance, hypothetically, the Reeb flows for overtwisted contact structures have infinitely many simple closed orbits, but where should one place such contact structures in our “classification”? [See [Eliashberg1998] and [Yau2006] for a proof of the existence of one closed orbit in this case.]

One application of Theorem 4.1 is the existence of infinitely many simple periodic orbits for all low energy levels of twisted geodesic flows on surfaces of genus $g\geq 2$ with non-vanishing magnetic field; see Sect. 5.

Just as in the Hamiltonian setting, the mean indices or the actions and the mean indices of simple periodic orbits of Reeb flows must, in many instances, satisfy certain resonance relations when the number of closed orbits is finite. The mean index resonance relations for the standard contact sphere were discovered by Viterbo in [Viterbo1989], and the Morse–Bott case for geodesic flows was considered in [Rademacher1989]. Viterbo’s resonance relations were generalized to non-degenerate Reeb flows on a broad class of contact manifolds in [Ginzburg and Kerman2010]. These resonance relations resemble the equality between two expressions for the Euler characteristic of a closed manifold: the homological one and the one using indices of zeroes of a vector field. The role of the homological expression is now taken by the mean Euler characteristic of the contact homology of the manifold, introduced in [van Koert2005], and the sum of the indices is replaced by the sum of certain local invariants of simple closed orbits. The degenerate case of the generalized Viterbo resonance relations was studied in [Ginzburg and Gören2015], [Hryniewicz and Macarini2015] and [Long et al.2014] and the Morse–Bott case in [Espina2014]. There are also variants of resonance relations involving both the actions and the mean indices; see [Gürel2015] and also [Ekeland1984] and [Ekeland and Hofer1987].

Leaving aside the exact form of the resonance relations, we only mention here some of their applications. The first one, in dynamics, is a contact analog of Theorem 2.2: the generic existence of infinitely many simple closed orbits for a large class of Reeb flows; see [Ginzburg and Gürel2009c] and also [Ekeland1984], [Rademacher1994] and [Hingston1984] for related earlier results. Another application, also in dynamics, is to the proof of the existence of at least two simple closed Reeb orbits on the standard contact $S^{3}$. This result is further discussed in the next section; see Theorem 4.3. (We refer the reader to [Gürel2015] for some other applications in dynamics.) Finally, on the topological side, the resonance relations can be used to calculate the mean Euler characteristic, [Espina2014].

Consider a closed symplectic manifold $(M,\omega)$ such that the form $\omega$ or, to be more precise, its cohomology class $[\omega]$ is integral, i.e., $[\omega]\in\operatorname{H}^{2}(M;{\mathbb{Z}})/\operatorname{Tor}$. Let $\pi\colon P\to M$ be an $S^{1}$-bundle over $M$ with first Chern class $-[\omega]$. The bundle $P$ admits an $S^{1}$-invariant 1-form $\alpha_{0}$ such that $d\alpha_{0}=\pi^{*}\omega$ and $\alpha_{0}(R_{0})=1$, where $R_{0}$ is the vector field generating the $S^{1}$-action on $P$. In other words, when we set $S^{1}={\mathbb{R}}/{\mathbb{Z}}$ and identify the Lie algebra of $S^{1}$ with ${\mathbb{R}}$, the form $\alpha_{0}$ is a connection form on $P$ with curvature $\omega$. (Note our sign convention.)

Clearly, $\alpha_{0}$ is a contact form with Reeb vector field $R_{0}$, and the connection distribution $\xi=\ker\alpha_{0}$ is a contact structure on $P$. Up to a gauge transformation, $\xi$ is independent of the choice of $\alpha_{0}$. The circle bundle $P$ equipped with this contact structure is usually referred to as a pre-quantization circle bundle or a Boothby–Wang bundle. Also, recall that a degree two (real) cohomology class on $P$ is said to be atoroidal if its integral over any smooth map ${\mathbb{T}}^{2}\to P$ is zero. (Such a class is necessarily aspherical.) Finally, in what follows, we will denote by ${\mathfrak{f}}$ the free homotopy class of the fiber of $\pi$.

The main tool used in the proof of Theorem 4.1 stated below is the cylindrical contact homology. As is well known, to have this homology defined for a contact form $\alpha$ on any closed contact manifold $P$ one has to impose certain additional requirements on the closed Reeb orbits of $\alpha$. (See [Bourgeois2009]; [Eliashberg et al.2000] and references therein for the definition and a detailed discussion of contact homology.) Namely, following [Ginzburg et al.2014], we say that a non-degenerate contact form $\alpha$ is index–admissible if its Reeb flow has no contractible closed orbits with Conley–Zehnder index $2-n$ or $2-n\pm 1$, where $\dim P=2n+1$. In general, $\alpha$ or its Reeb flow is index–admissible when there exists a sequence of non-degenerate index–admissible forms $C^{1}$-converging to $\alpha$.

This requirement is usually satisfied when $(P,\alpha)$ has some geometrical convexity properties. For instance, the Reeb flow on a strictly convex hypersurface in ${\mathbb{R}}^{2m}$ is index–admissible, [Hofer et al.1998]. Likewise, as is observed in [Benedetti2014], the twisted geodesic flow on a low energy level for a symplectic magnetic field on a surface of genus $g\geq 2$ is index–admissible; see Sect. 5 for more details. Finally, let us call a closed Reeb orbit $x$ non-degenerate (or weakly non-degenerate, SDM, etc.) if its Poincaré return map is non-degenerate (or, respectively, weakly non-degenerate, SDM, etc.), cf. Sect. 2.2.1. (The Poincaré return map is the map, or rather the germ of a map, $\Sigma\to\Sigma$ defined on a small cross section at $x(0)$ and sending a point $z\in\Sigma$ to the first intersection of the Reeb orbit through $z$ with $\Sigma$.)

It follows from Theorem 4.1 and the discussion below that, when the Reeb flow of $\alpha$ is weakly non-degenerate, the number of simple periodic orbits of the Reeb flow of $\alpha$ with period (or equivalently action) less than $a>0$ is bounded from below by $C_{0}\cdot a/\ln a-C_{1}$, where $C_{0}=\inf\alpha(R_{0})$ and $C_{1}$ depends only on $\alpha$. As mentioned in Sect. 2.2, this is a typical lower growth bound in the Conley conjecture type results. Note also that the weak non-degeneracy requirement here plays a technical role and probably can be eliminated.

The key to the proof of Theorem 4.1 is the observation that, as a consequence of (i), all free homotopy classes ${\mathfrak{f}}^{k}$, $k\in{\mathbb{N}}$, are distinct and hence give rise to an ${\mathbb{N}}$-grading of the cylindrical contact homology of $(P,\alpha)$. (In fact, it would be sufficient to assume that $[\omega]$ is aspherical and $\pi_{1}(M)$ is torsion free; both of these requirements follow from (i).) This grading plays essentially the same role as the order of iteration in the Hamiltonian Conley conjecture. With this observation in mind, the proof of the weakly non-degenerate case is quite similar to its Hamiltonian counterpart. [Condition (ii) is purely technical and most likely can be dropped.]

To complete the proof, one then has to deal with the case where the Reeb flow of $\alpha$ has a simple SDM orbit, i.e., a simple isolated orbit with an SDM Poincaré return map. This is also done similarly to the Hamiltonian case, but there are some nuances.

Consider a closed contact manifold $(P^{2n+1},\ker\alpha)$ with a strong symplectic filling $(W,\omega)$, i.e., $W$ is a compact symplectic manifold such that $P=\partial W$ with $\omega|_{P}=d\alpha$ and a natural orientation compatibility condition is satisfied. Let ${\mathfrak{c}}$ be a free homotopy class of loops in $W$.

This result is a contact analog of Theorem 3.8. Theorem 4.1 readily follows from the first case of Theorem 4.2 where we take the pre-quantization disk bundle over $M$ as $W$. [Here we only point out that $\pi_{2}(W)=\pi_{2}(M)=0$ since $M$ is aspherical and refer the reader to [Ginzburg et al.2014] for more details.]

The proof of Theorem 4.2 uses the filtered linearized contact homology. To be more specific, denote by $\operatorname{HC}_{*}^{(a,\,b)}(\alpha;W,{\mathfrak{c}}^{k})$ the linearized contact homology of $(P,\alpha)$ with respect to the filling $(W,\omega)$ for the action interval $(a,\,b)$ and the free homotopy class ${\mathfrak{c}}^{k}$, graded by the Conley–Zehnder index. Set $\Delta=\Delta(x)$ and $c={\mathcal{A}}(x)$ where $x$ is the SDM orbit from the theorem. Similarly to the Hamiltonian case (cf. Theorem 3.9), one first shows that, under the hypotheses of the theorem, for any $\epsilon>0$ there exists $k_{\epsilon}\in{\mathbb{N}}$ such that

Theorem 4.2 is a consequence of (4.1)‣ (see [Ginzburg et al.2014]), although the argument is less obvious than its Hamiltonian counterpart for, say, symplectic CY manifolds.

The proof of (4.1)‣ given in [Ginzburg et al.2013] follows the same path as the proof of Theorem 3.9. Namely, we squeeze the form $\alpha$ between two contact forms $\alpha_{\pm}$ constructed using the Hamiltonians $H_{\pm}$ near the SDM orbit, calculate the relevant contact homology for $\alpha_{\pm}$ (or rather a direct summand in it), and show that the map in contact homology induced by the cobordism from $\alpha_{+}$ to $\alpha_{-}$ is non-zero. This map factors through $\operatorname{HC}_{k\Delta+n+1}^{(kc+\delta_{k},\,kc+\epsilon)}(\alpha;W,{% \mathfrak{c}}^{k})$, and hence this group is also non-trivial.

Note that Theorem 4.2 as stated, without further assumptions on ${\mathfrak{c}}$, affords no control on the free homotopy classes of the simple orbits or their growth rate. A related point is that, at the time of this writing, there seems to be no satisfactory version of Theorem 4.2 which would not rely on the existence of the filling $W$. The difficulty is that without a filling one is forced to work with cylindrical contact homology to prove a variant of (4.1)‣, but then it is not clear if the forms $\alpha_{\pm}$ can be made index–admissible without additional assumptions on $\alpha$ along the lines of index–positivity. Such a filling–free version of the theorem would, for instance, enable one to eliminate the weak non-degeneracy assumption in the growth assertion in Theorem 4.1. Another serious limitation of Theorem 4.2 is that the SDM orbit is required to be simple. This condition, which is quite restrictive but probably purely technical, is used in the proof in a crucial way to construct the forms $\alpha_{\pm}$.

Another application of Theorem 4.2 considered in [Ginzburg et al.2013] (and also in [Ginzburg and Gören2015]; [Liu and Long2013]) is the following result originally proved in [Cristofaro-Gardiner and Hutchings2012].

In fact, a much stronger result holds. Namely, every Reeb flow on a closed three-manifold has at least two simple closed Reeb orbits. This fact is proved in [Cristofaro-Gardiner and Hutchings2012] using the machinery of embedded contact homology and is outside the scope of this survey. The idea of the proof from [Ginzburg et al.2013] is that if a Reeb flow on the standard contact $S^{3}$ had only one simple closed orbit $x$, this orbit would be an SDM, and, by Theorem 4.2, the flow would have infinitely many periodic orbits. Showing that $x$ is indeed an SDM requires a rather straightforward index analysis with one non-trivial ingredient used to rule out a certain index pattern. In [Ginzburg et al.2013], this ingredient is strictly three-dimensional and comes from the theory of finite energy foliations (see [Hofer et al.1995]; [Hofer et al.1996]). The argument in [Ginzburg and Gören2015]; [Liu and Long2013] uses a variant of the resonance relation for degenerate Reeb flows proved in [Ginzburg and Gören2015] and [Long et al.2014]. Theorem 4.2 can also be applied to give a simple proof, based on the same idea, of the result from [Bangert and Long2010] that any Finsler geodesic flow on $S^{2}$ has at least two closed geodesics; see [Ginzburg and Gören2015]. [Of course, this fact also immediately follows from [Cristofaro-Gardiner and Hutchings2012].]

Interestingly, no multiplicity results along the lines of Theorem 4.3 have been proved in higher dimensions without restrictive additional assumptions on the contact form. Conjecturally, every Reeb flow on the standard contact sphere $S^{2n-1}$ has at least $n$ simple closed Reeb orbits. This conjecture has been proved when the contact form comes from a strictly convex hypersurface in ${\mathbb{R}}^{2n}$ and the flow is non-degenerate or $2n\leq 8$; see [Long and Zhu2002], [Long2002] and [Wang2013] and references therein. In the degenerate strictly convex case, the lower bound is $\lfloor n/2\rfloor+1$. Without any form of a convexity assumption, it is not even known if a Reeb flow on the standard contact $S^{5}$ must have at least two simple closed orbits. It is easy to see, however, that a non-degenerate Reeb flow on the standard $S^{2n-1}$ has at least two simple closed orbits; see, e.g., Gürel ([Gürel2015], Rmk. 3.3).

We conclude this section by pointing out that the machinery of contact homology which the proof of Theorem 4.1 relies on is yet to be fully put on a rigorous basis.

Even when the Conley conjecture fails, the existence of infinitely many simple periodic orbits is, as we have already seen, a generic feature of Hamiltonian diffeomorphisms (and Reeb flows) for a broad class of manifolds. There is, however, a different and more interesting, from our perspective, phenomenon responsible for the existence of infinitely many periodic orbits. The starting point here is a celebrated theorem of Franks.

In fact, the theorem, already in its original form, was proved for area preserving homeomorphisms. This aspect of the problem is outside the scope of the paper, and here we focus entirely on smooth maps. Furthermore, in the setting of the theorem, there are also strong growth results; see [Franks and Handel2003], [Le Calvez2006] and [Kerman2012] for this and other refinements of Theorem 6.1. The proof of the theorem given in [Franks1992]; [Franks1996] utilized methods from low–dimensional dynamics. Recently, a purely symplectic topological proof of the theorem was obtained in [Collier et al.2012]; see also [Bramham and Hofer2012] for a different approach.

In the former case, we glue together two copies of $S^{2}$ punctured at $y$ and $z$ by inserting narrow cylinders at the seams as in Arnold ([Arnold1989], App. 9). As a result, we obtain a torus ${\mathbb{T}}^{2}$, and the Hamiltonian diffeomorphism $\varphi$ gives rise to an area preserving map $\psi\colon{\mathbb{T}}^{2}\to{\mathbb{T}}^{2}$. This map is not necessarily a Hamiltonian diffeomorphism, but it is symplectically isotopic to ${id}$ and its Floer homology $\operatorname{HF}_{*}(\psi)$ is defined. Hence, either $\operatorname{HF}_{*}(\psi)=0$ or $\operatorname{HF}_{*}(\psi)\cong\operatorname{H}_{*+1}({\mathbb{T}}^{2})$ when $\psi$ is Hamiltonian. Now one shows that, roughly speaking, any of the points $x^{\pm}\in{\mathbb{T}}^{2}$ corresponding to $x$ represents a non-trivial homology class of degree different from 0 and $\pm 1$, which is impossible.

When $z$ is a hyperbolic point, we use the points $x$ and $y$ to produce the torus, and again a simple Floer homological argument leads to a contradiction. Indeed, for a sufficiently large iteration of $\psi$, each elliptic point has large Conley–Zehnder index, since Theorem 3.8 rules out SDM points, and each hyperbolic point has even index. Moreover, hyperbolic points give rise to non-trivial homology classes (cf. [Ginzburg and Gürel2009c], Thm. 1.7). Thus $\operatorname{HF}_{*}(\psi)\neq 0$ but $\operatorname{HF}_{1}(\psi)=0$, which is again impossible. (Alternatively, one can just apply Theorem 6.2 below to deal with this case.) $\square$

Even though all proofs of Franks’ theorem are purely low-dimensional, it is tempting to think of the result as a particular case of a more general phenomenon. For instance, one hypothetical generalization of Franks’ theorem would be that a Hamiltonian diffeomorphism with more than necessary non-degenerate (or just homologically non-trivial in the sense of Sect. 3.1.2) fixed points has infinitely many periodic orbits. Here more than necessary is usually interpreted as a lower bound arising from some version of the Arnold conjecture. For ${\mathbb{CP}}^{n}$, the expected threshold is $n+1$ and, in particular, it is 2 for $S^{2}$ as in Franks’ theorem, cf. Hofer and Zehnder ([Hofer and Zehnder1994], p. 263).

However, this conjectural generalization of Franks’ theorem seems to be too restrictive, and from the authors’ perspective it is fruitful to put the conjecture in a broader context. Namely, it appears that the presence of a fixed point that is unnecessary from a homological or geometrical perspective is already sufficient to force the existence of infinitely many simple periodic orbits. Let us now state some recent results in this direction.

Here, clearly, the hyperbolic periodic orbit is unnecessary from every perspective. In contrast with Franks’ theorem and the Conley conjecture type results, at the time of writing, there are no growth results in this setting. The theorem actually holds for a broader class of manifolds $M$, and the requirements on $M$ can be stated purely in terms of the quantum homology of $M$; see Ginzburg and Gürel ([Ginzburg and Gürel2014], Thm. 1.1). Among the manifolds meeting these requirements are, in addition to ${\mathbb{CP}}^{n}$, the complex Grassmannians $\operatorname{Gr}(2;N)$, $\operatorname{Gr}(3;6)$ and $\operatorname{Gr}(3;7)$; the products ${\mathbb{CP}}^{m}\times P^{2d}$ and $\operatorname{Gr}(2;N)\times P^{2d}$, where $P$ is symplectically aspherical and $d\leq m$ in the former case and $d\leq 2$ in the latter; and the monotone products ${\mathbb{CP}}^{m}\times{\mathbb{CP}}^{m}$. There is also a variant of the theorem for non-contractible hyperbolic orbits, which is applicable to, for example, the product ${\mathbb{CP}}^{m}\times P^{2d}$. Note also that the generalization of Franks’ theorem to ${\mathbb{CP}}^{n}$, at least for non-degenerate Hamiltonian diffeomorphisms, would follow if one could replace in Theorem 6.2 a hyperbolic fixed point by a non-elliptic one.

Another result fitting into this context is the following.

As a consequence, regardless of whether the fixed-point set is finite or not, $\varphi$ has infinitely many simple periodic orbits. In this theorem the condition that the eigenvalues of $Q$ are real can be slightly relaxed. Conjecturally, it should be enough to require $Q$ to be non-degenerate and $x$ to have mean index different from the mean index of the origin for $Q$. However, hyperbolicity of $Q$ is used in an essential way in the proof of the theorem. Also, interestingly, in contrast with Franks’ theorem, the requirement that $x$ is homologically non-trivial is essential and cannot be omitted, even in dimension two. As an easy consequence of Theorem 6.3, we obtain

In the two-dimensional case, a stronger result is proved in Abbondandolo ([Abbondandolo2001], Thm. 5.1.9). In the setting of Theorem 6.3 and Theorem 6.4, one can be more precise about which simple periods occur. Namely, for a certain integer $m>0$, starting with a sufficiently large prime number, among every $m$ consecutive primes, there exists at least one prime which is the period of a simple periodic orbit. Thus, as in many other results of this type, we have the lower growth bound ${const}\cdot k/\ln k$.

Theorem 6.2 and, with some extra work, Theorem 6.3 imply the case of Franks’ theorem where $\varphi$ is assumed to have a hyperbolic periodic orbit, e.g., when $\varphi$ is non-degenerate. Furthermore, it is conceivable that one could prove Franks’ theorem as a consequence of Theorem 6.2. Such a proof would certainly be of interest, but it would most likely be much more involved than the argument in [Collier et al.2012].

Let us now turn to non-contractible orbits. Recall that for a (time-dependent) Hamiltonian flow $\varphi_{H}^{t}$ generated by a Hamiltonian $H\colon S^{1}\times M\to{\mathbb{R}}$ there is a one-to-one correspondence between the one-periodic orbits of $\varphi_{H}^{t}$ and the fixed points of $\varphi=\varphi_{H}$. Furthermore, as is easy to see from the proof of the Arnold conjecture, the free homotopy class of an orbit $x$ is independent of the Hamiltonian generating the time-one map $\varphi$. Thus the notion of a contractible one-periodic orbit (or even a “contractible fixed point”) of $\varphi$ is well-defined. Of course, the same applies to $k$-periodic orbits.

On a closed symplectic manifold $M$ a Hamiltonian diffeomorphism need not have non-contractible one-periodic orbits. Indeed, the Hamiltonian Floer homology vanishes for any non-trivial free homotopy class when $M$ is compact, since all one-periodic orbits of a $C^{2}$-small autonomous Hamiltonian are its critical points (hence contractible). Thus, from a homological perspective, non-contractible periodic orbits are totally unnecessary.

To state our next result, recall that a symplectic form $\omega$ on $M$ is said to be atoroidal if for every map $v\colon{\mathbb{T}}^{2}\to M$, the integral of $\omega$ over $v$ vanishes. We have

Thus the number of simple non-contractible periodic orbits with period less than or equal to $k$, or the number of distinct homology classes represented by such orbits, is bounded from below by ${const}\cdot k/\ln k$. An immediate consequence of the theorem is that $\varphi$ has infinitely many simple periodic orbits with homology classes in ${\mathbb{N}}[x]$ whether or not the set of one-periodic orbits (in the class $[x]$) is finite. Moreover, in this theorem, as in Theorem 6.3, the non-degeneracy condition can be relaxed and replaced by the much weaker requirement that $x$ is isolated and homologically non-trivial. Finally, in both theorems, the orbit $x$ need not be one-periodic; the theorems (with obvious modifications) still hold when $x$ is just a periodic orbit.

Among the manifolds meeting the requirements of Theorem 6.5 are, for instance, closed Kähler manifolds with negative sectional curvature and, more generally, any closed symplectic manifold with $[\omega]|_{\pi_{2}(M)}=0$ and hyperbolic $\pi_{1}(M)$. Furthermore, Hamiltonian diffeomorphisms having a periodic orbit in a non-trivial homology class exist in abundance. It is plausible that a $C^{1}$-generic, or even $C^{\infty}$-generic, Hamiltonian diffeomorphism has an orbit in a non-trivial homology class when the fundamental group (or the first homology group) of $M$ is large enough; see [Tal2013] for some possibly relevant results for surfaces. However, as is easy to see, already for $M={\mathbb{T}}^{2}$, a fixed Hamiltonian diffeomorphism need not have non-contractible periodic orbits (e.g., $\varphi_{H}$ for a small bump function $H$), and even $C^{\infty}$-generically one cannot prescribe the homology class of an orbit in advance; [Ginzburg and Gürel2015] and [Gürel2013].

Hypothetically, one can expect an analog of the theorem to hold when the condition that $\omega$ is atoroidal is omitted or relaxed, e.g., replaced by the requirement that $(M,\omega)$ is toroidally monotone. We refer the reader to [Ginzburg and Gürel2015] for some further results in this direction.

The proofs of all these theorems are based on the same idea that an unnecessary periodic orbit is a seed creating infinitely many periodic orbits. In Theorem 6.3 and Theorem 6.5 the argument is that, roughly speaking, the change in filtered Floer homology, for a carefully chosen action range (and/or degree), between different iterations of $\varphi$ requires new simple periodic orbits to be created. The proof of Theorem 6.2 relies on a result, perhaps of independent interest, asserting that the energy needed for a Floer connecting trajectory of an iterated Hamiltonian to approach a hyperbolic orbit and cross its fixed neighborhood cannot become arbitrarily small: it is bounded away from zero by a constant independent of the order of iteration. Then the product structure in quantum homology is used to show that there must be Floer connecting trajectories with energy converging to zero for some sequence of iterations unless new periodic orbits are created.

The conjectures discussed in Sect. 6.2 have obvious analogs for Reeb flows and symplectomorphisms.