MIXING PROPERTIES FOR INVERTIBLE MAPS WITH WEAK HYPERBOLIC PRODUCT STRUCTURE

Jin Hatomoto (Japan)

Received March 2, 2008

Abstract

We study an invertible map f which admits a weak hyperbolic product structure region, which is the intersection of two transversal families of weak stable and weak unstable disks, with countably many branches and variable return times. We obtain that f has an invariant probability measure n whose conditional measures on the weak unstable disks are absolutely continuous with respect to the Lebesgue measures on these disks such that the correlation function of for Hölder continuous functions j and y with Hölder exponent h satisfies the following formula: where is a positive increasing function with in a way related to asymptotic mass of the return times on the weak unstable disks, and is the supremum of the diameters of the forward (resp. backward) iterate of the weak stable (resp. unstable) disks. This formula includes exponential, polynomial and stretched exponential decay of correlations.

Keywords and phrases: decay of correlation functions, weak hyperbolic product structure.