Suspension of the Billiard maps in the Lazutkin's coordinate

TL;DR

This paper demonstrates that billiard maps in Lazutkin's coordinates can be smoothly interpolated by a Hamiltonian flow, enabling the construction of transitive trajectories between caustics using variational methods.

Contribution

It introduces a formal Hamiltonian interpolation of billiard maps in Lazutkin's coordinates, facilitating new dynamical constructions.

Findings

Billiard maps can be interpolated by a Hamiltonian flow in Lazutkin's coordinates.

The Hamiltonian has a specific formal expression involving powers of momentum.

This interpolation allows for the construction of transitive trajectories between caustics.

Abstract

In this paper we proved that under the Lazutkin's coordinate, the billiard map can be interpolated by a time-1 flow of a Hamiltonian $H(x,p,t)$ which can be formally expressed by \[ H(x,p,t)=p^{3/2}+p^{5/2}V(x,p^{1/2},t),\quad(x,p,t)\in\T\times[0,+\infty)\times\T, \] where $V(\cdot,\cdot,\cdot)$ is $C^{r-5}$ smooth if the convex billiard boundary is $C^r$ smooth. Benefit from this suspension we can construct transitive trajectories between two adjacent caustics under a variational framework.