Rates of Convergence in the CLT¶for Two-Dimensional Dispersive Billiards

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We consider billiards in the two-dimensional torus with convex obstacles. Central Limit Theorems have been established for regular functions for the billiard transformation in (2), (1) and (13). We are interested here in the problem of the rate of convergence. In this paper, we establish a rate in O � n − 1 2 +α � (for any α> 0) for the billiard transformation, by adapting the proof of (6, 5, 7). In our proof, we use a strong decorrelation result obtained by the method developped in (13) for the study of general hyperbolic systems. Moreover, we establish a rate of convergence in O � t − 1 6 � in the Central Limit Theorem for the billiard flow.