Limit theorems in the stadium billiard

Details Limit theorems in the stadium billiard Limit theorems in the stadium billiard

2005-02-22

arxiv.org/abs/math/0502453v1

Mathematics

Dynamical Systems

46 pages

Scientific paper

10.1007/s00220-005-1511-6

We prove that the Birkhoff sums for ``almost every'' relevant observable in the stadium billiard obey a non-standard limit law. More precisely, the usual central limit theorem holds for an observable if and only if its integral along a one-codimensional invariant set vanishes, otherwise a $\sqrt{n\log n}$ normalization is needed. As one of the two key steps in the argument, we obtain a limit theorem that holds in Young towers with exponential return time statistics in general, an abstract result that seems to be applicable to many other situations.

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