arxiv: 1610.00317 · v1 · pith:RJQ3B7CUnew · submitted 2016-10-02 · 🧮 math.DS

Suspension of the Billiard maps in the Lazutkin's coordinate

Jianlu Zhang This is my paper

classification 🧮 math.DS

keywords billiardcdotcoordinatelazutkinsmoothsuspensiontimesunder

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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.

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Suspension of the Billiard maps in the Lazutkin's coordinate

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