Billiards in generic convex bodies have positive topological entropy

We show that there exists a C2 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.

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2024-04, Vol.442, p.109592, Article 109592
Main Authors: Bessa, Mário, Del Magno, Gianluigi, Lopes Dias, João, Gaivão, José Pedro, Torres, Maria Joana
Format: Article
Language:English
Subjects:
Multidimensional billiards
Topological entropy
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Summary:We show that there exists a C2 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.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2024.109592