Arnold Mathematical Journal
Problem Contribution

Received: 16 September 2014 / Revised: 23 September 2014 / Accepted: 14 October 2014

A Baker’s Dozen of Problems

Serge Tabachnikov Supported by NSF grant DMS-1105442 tabachni@math.psu.edu Department of Mathematics Pennsylvania State University, University Park PA 16802 USA ICERM, Brown University, Box 1995 Providence RI 02912 USA

This article is a collection of open problems, with brief historical and bibliographical comments, somewhat in the spirit of the problem with which V. Arnold opened his famous seminar every semester and that were recently collected and published in a book form.

Keywords

Inner and outer billiards, Desargues theorem, Poncelet porism and Cayley criterion, Origami, Projective duality, Bicycle kinematics, Totally skew embedding, Equi-areal dissection

1 Commuting Billiard Ball Maps

Given a convex plane domain, the billiard ball map sends the $\def\R{\mathbb{R}}$ incoming ray (the trajectory of the billiard ball) to the outgoing one: the law of reflection is “the angle of incidence equals the angle of reflection”.

Consider two nested convex domains. Then one has two billiard ball maps, $T_{1}$ and $T_{2}$, acting on the oriented lines that intersect both domains. If the domains are bounded by confocal ellipses, then the respective billiard ball maps commute; see, e.g., Tabachnikov [2005] .

Assume that the two maps commute: $T_{1}\circ T_{2}=T_{2}\circ T_{1}$.

Conjecture 1.

The two domains are bounded by confocal ellipses.

For outer (a.k.a. dual) billiards, an analogous fact is proved in Tabachnikov [1994] . For piece-wise analytic billiards, this conjecture was proved by Glutsyuk [2014] .

Of course, this problem has a multi-dimensional version, open both for inner and outer billiards.

2 Can One-Parameter Families of 2- and 3-Periodic Billiard Trajectories Coexist?

A curve of constant width admits a one-parameter family of 2-periodic (back and forth) billiard trajectories. Likewise, for every $p\geq 3$, there exist billiard tables admitting a one-parameter family of $p$-periodic billiard trajectories; see Baryshnikov and Zharnitsky [2006] for a recent approach using ideas of sub-Riemannian geometry.

Problem 1.

Are there smooth convex curves, other than ellipses, simultaneously admitting one-parameter families of $p$- and $q$-periodic billiard trajectories (for $p\neq q$)?

The simplest case of the question is whether any curve of constant width, other than a circle, admits a one-parameter family of 3-periodic billiard trajectories.

A similar question can be asked about outer billiards.

3 Birkhoff’s Theorem for Lorentz Billiards

The classical Birkhoff theorem states that, for every $n\geq 3$ and $1\leq k\leq n/2$, the billiard system inside a plane oval has at least two $n$-periodic trajectories with the rotation number $k$. Consider the billiard system inside an oval in the Lorentz plane with the pseudo-Euclidean metric $ds^{2}=dx^{2}-dy^{2}$. Is there an analog of Birkhoff’s theorem in this set-up?

Billiard trajectories in pseudo-Euclidean space can be of three types: space-like, time-like, and light-like, see Khesin and Tabachnikov [2009] for Lorentz billiards. One would expect separate existence statements for space-like and time-like trajectories .

A convex body in $\R^{n}$ has at least $n$ diameters (2-periodic billiard trajectories). If the ambient space is pseudo-Euclidean, $\R^{p,q}$, then there are at least $p$ space- and at least $q$ time-like diameters ( Khesin and Tabachnikov [2009] ). A lower bound on the number of periodic billiard trajectories in multi-dimensional Euclidean space is obtained in Farber and Tabachnikov [2002] . What happens with multi-dimensional pseudo-Euclidean billiards?

4 Polygonal Outer Billiards in the Hyperbolic Plane

The outer billiard about a convex polygon $P$ in the plane $\R^{2}$ is a piece-wise isometry, $T$, of the exterior of $P$ defined as follows: given a point $x$ outside of $P$, find the support line to $P$ through $x$ having $P$ on the left, and define $T(x)$ to be the reflection of $x$ in the support vertex (Fig. 1 ). See Dogru and Tabachnikov [2005] , Tabachnikov [2005] .

Figure 1: The outer billiard map in the plane.
The orbit structure for a regular dodecagon

C. Culter proved that every polygon in the affine plane admits periodic outer billiard orbits, see Tabachnikov [2007] . Outer billiard can be defined on the sphere and in the hyperbolic plane. On the sphere, there exist polygons without