Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symplectic billiards for pairs of polygons

Peter Albers, Fabian Lander, Jannik M. Westermann

Abstract

We introduce symplectic billiards for pairs of possibly non-convex polygons. After establishing basic properties, we give several criteria on pairs of polygons for the symplectic billiard map to be fully periodic, i.e. $\textit{every}$ orbit is periodic. First fully periodic examples were discovered by Albers-Tabachnikov [AT18] and Albers-Banhatti-Sadlo-Schwartz-Tabachnikov in [ABS+19]. Our criteria allow us to construct a plethora of new examples. Moreover, we provide an example of a pair of polygons where the symplectic billiard map is fully periodic while having orbits of arbitrarily large period. After giving a class of examples which provably have isolated periodic orbits (and are thus not fully periodic) we exhibit the first example without any periodic orbits at all. It is open whether being fully periodic with unbounded period or having no periodic orbits at all is possible in the single polygon setting. Finally, we prove that if one replaces polygons by smooth strictly convex curves then there are always infinitely many periodic orbits.

Symplectic billiards for pairs of polygons

Abstract

We introduce symplectic billiards for pairs of possibly non-convex polygons. After establishing basic properties, we give several criteria on pairs of polygons for the symplectic billiard map to be fully periodic, i.e. orbit is periodic. First fully periodic examples were discovered by Albers-Tabachnikov [AT18] and Albers-Banhatti-Sadlo-Schwartz-Tabachnikov in [ABS+19]. Our criteria allow us to construct a plethora of new examples. Moreover, we provide an example of a pair of polygons where the symplectic billiard map is fully periodic while having orbits of arbitrarily large period. After giving a class of examples which provably have isolated periodic orbits (and are thus not fully periodic) we exhibit the first example without any periodic orbits at all. It is open whether being fully periodic with unbounded period or having no periodic orbits at all is possible in the single polygon setting. Finally, we prove that if one replaces polygons by smooth strictly convex curves then there are always infinitely many periodic orbits.
Paper Structure (6 sections, 16 theorems, 48 equations, 41 figures)

This paper contains 6 sections, 16 theorems, 48 equations, 41 figures.

Table of Contents

  1. Introduction
  2. The symplectic billiard map on pairs of polygons
  3. Criteria for periodicity
  4. (Crooked) kites have isolated periodic orbits
  5. The necktie has no periodic orbits
  6. Pairs of strictly convex smooth tables

Key Result

Lemma 2.8

For any $(x,y) \in \mathcal{P}'$ there exists a unique $z \in P_\pm \cap (x + T_y P_\mp \setminus \{0\})$ such that $xz \subset \mathrm{int}(P_\pm)$. Hence $\mathcal{P}'$ is contained in $\mathcal{P}_\mathrm{max}$. Moreover, if $z$ is not a vertex then $\det(\nu_y,\nu_z) \neq 0$, i.e. $(y,z) \in \m

Figures (41)

  • Figure 1: The symplectic billiard reflection in a curve and in two polygons.
  • Figure 2: The necktie -- a pair of polygons for which the symplectic billiard map has no periodic orbits at all, see Section \ref{['section:necktie']}.
  • Figure 3: The symplectic billiard map on a convex polygon.
  • Figure 4: The symplectic billiard map on a non-convex polygon.
  • Figure 5: The symplectic billiard map on two polygons.
  • ...and 36 more figures

Theorems & Definitions (55)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 45 more