Problem 17

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Symbolic dynamics for billiards



Billiard maps, such as Sinai billiards (also called dispersing billiards) and Bunimovich billiards, have two extra difficulties in comparison with uniformly and non-uniformly hyperbolic diffeomorphisms:
1. Their phase space is a manifold with boundary, so the map and its inverse are not defined everywhere.
2. The table of the billiard is not smooth (it is only smooth by parts), so the billiard map has discontinuities (the "glancing orbits" also create discontinuities).

On top of that, the derivative of the map explodes as we approach the discontinuities. Hence a billiard map has a singular set, formed by the points where the map is not well-defined or it is discontinuous. Bunimovich, Chernov and Sinai constructed countable Markov partitions for two-dimensional dispersing billiard maps with respect to the Lebesgue measure [1]. Recently, Lima and Matheus constructed countable Markov partitions for non-uniformly hyperbolic billiard maps with respect to "adapted" measures (measures s.t. almost every point does not converge to the singular set exponentially fast).

Up to the authors knowledge, this is the first symbolic coding of uniformly and non-uniformly hyperbolic billiard maps for general measures. It applies to Bunimovich billiards (e.g. the pool table with pockets, flower, and stadium) with respect to the Lebesgue measure. A question of interest is to know if Bunimovich billiards have measures of maximal entropy, and if they are "adapted" in the sense described above.


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