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Homoclinic points for convex billiards

Zhihong Xia, Pengfei Zhang

Abstract

In this paper we investigate some generic properties of a billiard system on a convex table. We show that generically, every hyperbolic periodic point admits some homoclinic orbit.

Homoclinic points for convex billiards

Abstract

In this paper we investigate some generic properties of a billiard system on a convex table. We show that generically, every hyperbolic periodic point admits some homoclinic orbit.

Paper Structure

This paper contains 9 sections, 13 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Topology on the set of tables
  4. Limit sets of invariant manifolds
  5. Some known properties
  6. Prime-end compactification and extension
  7. Recurrence and Saddle connections
  8. Homoclinic points on generic convex billiard systems
  9. Recurrence and homoclinic orbits

Key Result

Theorem 1.1

Let $r\ge3$. For a $C^r$-generic convex billiard system $Q$, we have that

Figures (1)

  • Figure 1: Existence of homoclinic points from adjacent pair of branches.

Theorems & Definitions (23)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 2.1: Oli1
  • Remark 1
  • Proposition 2.2
  • Remark 2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 13 more