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Billiards in generic convex bodies have positive topological entropy

Mário Bessa, Gianluigi Del Magno, João Lopes Dias, José Pedro Gaivão, Maria Joana Torres

Abstract

We show that there exists a $C^2$ open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and exponential growth of the number of periodic orbits.

Billiards in generic convex bodies have positive topological entropy

Abstract

We show that there exists a open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and exponential growth of the number of periodic orbits.
Paper Structure (27 sections, 33 theorems, 165 equations)

This paper contains 27 sections, 33 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Billiard map
  3. Smooth convex bodies
  4. $C^r$-topology
  5. Perturbing the body
  6. The billiard map
  7. The derivative of the billiard map
  8. Projections
  9. Jacobi coordinates
  10. Derivative of the billiard map
  11. Generic properties concerning periodic orbits
  12. Proof of Theorem \ref{['main thm']}
  13. Perturbing the $k$-jets: Klingenberg-Takens theorem for billiards
  14. Perturbing the $k$-jet of the billiard map
  15. Proof of Theorem \ref{['thm: K-T first']}
  16. ...and 12 more sections

Key Result

Theorem 1.1

There is a $C^2$-open and dense set of smooth convex bodies whose billiard maps have a nontrivial hyperbolic basic set.

Theorems & Definitions (55)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 45 more