Backward Inducing and Exponential Decay of Correlations for Partially Hyperbolic Attractors

We study partially hyperbolic attractors of C 2 dieomorphisms on a compact manifold. For a robust (non-empty interior) class of such diffeomorphisms, we construct Sinai-Ruelle-Bowen measures, for which we prove exponential decay of correlations and the central limit theorem, in the space of H\u7folder continuous functions. The techniques we develop (backward inducing, redundancy elimination algorithm) should be useful in the study of the stochastic properties of much more general non-uniformly hyperbolic systems. 1 Introduction In this work, M will always be a compact manifold and H will be the space of H\u7folder continuous functions on M . Moreover, 0 is a physical or SRB (Sinai- Ruelle-Bowen) probability measure for f . That is, 0 gives the time averages 0 = lim n!+1 1 n n 1 X j=0 f j (z) ; p = Dirac measure at p of a set B( 0 ) of points z 2 M with positive Lebesgue measure. This set B( 0 ) will be called the basin of 0 . We say that (f; 0 ) has exponential dec..

Backward Inducing and Exponential Decay of Correlations for Partially Hyperbolic Attractors

We study partially hyperbolic attractors of class C 2 diffeomorphisms in a finite dimensional compact manifold. For a robust (open) class of such diffeomorphisms, we construct Sinai-Ruelle-Bowen measures, for which we prove exponential decay of correlations and the central limit theorem, in the space of Holder continuous functions. For the proof of such results we develop new techniques (backward inducing, redundance elimination algorithm) which should be useful in the study of the stochastic properties of more general systems. 1 Introduction In this work, M will always be a compact manifold, and H will be the space of Holder continuous functions on M . Moreover, in all the cases we treat, ¯ 0 is a physical or SRB (Sinai-Ruelle-Bowen) probability measure for f . That is, ¯ 0 gives the time averages ¯ 0 = lim n!+1 1 n n\Gamma1 X j=0 ffi f j (z) ; ffi p = Dirac measure at p of a set B(¯ 0 ) of points z 2 M with positive Lebesgue measure. This set B(¯ 0 ) will be called the..