Convex billiards on convex spheres

Zhang, Pengfei

doi:10.1016/j.anihpc.2016.07.001

Zhang, Pengfei

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 793-816.

Résumé

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 Herman's result on Diophantine invariant curves that we use to prove the nonlinear stability of elliptic periodic points for a dense subset of convex billiards.

MR Zbl

DOI : 10.1016/j.anihpc.2016.07.001

Classification : 37D40, 37D50, 37C20, 37E40
Mots clés : Convex billiards, Generic properties, Homoclinic intersections, Diophantine number, Nonlinearly stable

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     title = {Convex billiards on convex spheres},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {793--816},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.07.001/}
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Zhang, Pengfei. Convex billiards on convex spheres. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 793-816. doi : 10.1016/j.anihpc.2016.07.001. http://www.numdam.org/articles/10.1016/j.anihpc.2016.07.001/

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