Outer symplectic billiard map at infinity

TL;DR

This paper demonstrates that the second iteration of the outer symplectic billiard map can be approximated by a specific Hamiltonian flow at infinity, revealing asymptotic behavior of orbits and periodic points.

Contribution

It introduces an explicit Hamiltonian approximation for the outer symplectic billiard map at infinity and analyzes the growth of orbits and periodic points.

Findings

$T^2$ approximates a Hamiltonian flow for points far from $M$

Orbit distance from the origin grows at most as $ oot{k}$ for escaping orbits

Periodic orbits are close to $M$ in terms of their period $k$

Abstract

We show that the second iteration $T^2$ of the outer symplectic billiard map with respect to a convex domain $M$ in a symplectic vector space is approximated by an explicit Hamiltonian flow for points far away from $M$. More precisely, denote by $N$ the symplectic polar dual of the symmetrization $M\ominus M$ of $M$. If we write $N$ as the unit level set of a 1-homogeneous function $H$, then the difference between $T^2$ and the time-2-Hamiltonian flow of $H$ applied to a point $x$ is smaller than $c/|x|$ for some constant $c$ depending only on $M$. Moreover, we show that if an orbit escapes to infinity, then its distance to the origin grows not faster than $\sqrt{k}$ in the number of iterations. Finally, we prove that a $k$-periodic orbit needs to be close, in terms of $k$, to $M$.