Estimates for correlations in certain Bunimovich billiards
by
Zhang Hongkun
North China Electric Power Universty
Coauthors: Jinguo Lian

Ergodic and mixing chaotic billiards may have quite different statistical properties depending on the rate of mixing (the rate of the decay of correlations). In this paper we study certain type of Bunimovich billiards which is much harder to investigate; the main reason is a weak (non-uniform) hyperbolicity of the collision map. Indeed, whenever the moving particle gets into the focussing boundaries, it experiences a large number of collisions that do not contribute much to the expansion or contraction of tangent vectors. We first localize spots in the phase space where expansion (contraction) of tangent vectors slows down. Then verify that the return map F is uniformly hyperbolic and prove F enjoys exponential decay of correlations. Finally, we obtain an upper bound for the slow decay of correlations of the original system based upon the tail bound on the return time function.

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Date received: February 25, 2007