Suspension of the billiard maps in the Lazutkin's coordinate

Jianlu Zhang

doi:10.3934/dcds.2017096

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2017, Volume 37, Issue 4: 2227-2242. Doi: 10.3934/dcds.2017096

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Suspension of the billiard maps in the Lazutkin's coordinate

Jianlu Zhang

Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, M5S 2E4, Canada

Received: June 30, 2016

Revised: August 31, 2016

Published: May 2017

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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

$H(x,p,t)=p^{3/2}+p^{5/2}V(x,p^{1/2},t),\;\;(x,p,t)∈\mathbb{T}×[0,+∞)×\mathbb{T},$

where $ V(·,·,·) $ 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.

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Mathematics Subject Classification: Primary:37J50;Secondary:70H08.

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Figure 1. The reflective angle keeps equal to the incident angle for every rebound

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