Fluctuations of the nth return time for Axiom A diffeomorphisms

Abstract

We study the time of $n$th return of orbits to some given (union of) rectangle(s) of a Markov partition for an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure. As a by-product, we derive precise large deviation estimates and a central limit theorem for these return times. 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.