Birkhoff averages of Poincare cycles for Axiom-A diffeomorphisms

Details Birkhoff averages of Poincare cycles for Axiom-A diffeomorphisms Birkhoff averages of Poincare cycles for Axiom-A diffeomorphisms

2003-12-15

arxiv.org/abs/math/0312291v1

Mathematics

Dynamical Systems

17 pages; submitted

Scientific paper

We study the time of $n$th return of orbits to some given (union of) rectangle(s) of a Markov partition of an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure (associated to a Holderian potential). As a by-product, we derive precise large deviation estimates and a central limit theorem for Birkhoff averages of Poincare cycles. We emphasize that we look at the limiting behavior in term of number of visits (the size of the visited set is kept fixed). Our approach relies on the spectral properties of a one-parameter family of induced transfer operators on unstable leaves crossing the visited set.

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