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April  2017, 37(4): 2227-2242. doi: 10.3934/dcds.2017096

## Suspension of the billiard maps in the Lazutkin's coordinate

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

Received  July 2016 Revised  September 2016 Published  December 2016

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.
Citation: Jianlu Zhang. Suspension of the billiard maps in the Lazutkin's coordinate. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2227-2242. doi: 10.3934/dcds.2017096
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##### References:
The reflective angle keeps equal to the incident angle for every rebound
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