Stochastic stability of Anosov diffeomorphisms

Abstract

R. Bowen [1] introduced the notion of pseudo-orbit for a homeomorphism f of a metric space X as follows: A (double) sequence {xi}i∈Z of points Xi in X is called a δ-pseudo-orbit of f iffd(fxi, xi+1) ≤ δfor every i ∈ Z, where d denotes the metric in X. We say f is stochastically stable if for every ε > 0 there exists δ > 0 such that every δ pseudo-orbit {Xi}i∈Z of f is ε-traced by some x ∈ X, i.e.,d(fix, xi) ≤ εfor every i ∈ Z. He proved in [1] that if a compact hyperbolic set Λ for a diffeomorphism f of a compact manifold M has local product structure then the restriction f | Λ of f to Λ is stochastically stable, using stable and unstable manifolds.