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arXiv · 1505.01418

Convex billiards on convex spheres

Pengfei Zhang
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Abstract

In this paper we study the dynamical billiards on a convex 2D sphere. We investigate some generic properties of the convex billiards on a general convex sphere. We prove that $C^\infty$ generically, every periodic point is either hyperbolic or elliptic with irrational rotation number. Moreover, every hyperbolic periodic point admits some transverse homoclinic intersections. A new ingredient in our approach is that we use Herman's result on Diophantine invariant curves to prove the nonlinear stability of elliptic periodic points for a dense subset of convex billiards.

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Pengfei Zhang. 2015-05-06. Convex billiards on convex spheres. https://doi.org/10.1016/j.anihpc.2016.07.001

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