SRB measures for pointwise hyperbolic systems on open regions

Abstract

A diffeomorphism f : M → M is pointwise partially hyperbolic on an open invariant subset N ⊂ M if there is an invariant decomposition TNM = Eu ⊕ Ec ⊕ Es such that Dxf is strictly expanding on E and contracting on E at each x ∈ N. We show that under certain conditions f has unstable and stable manifolds, and admits a finite or an infinite u-Gibbs measure μ. If f is pointwise hyperbolic on N, then μ is a Sinai-Ruelle-Bowen (SRB) measure or an infinite SRB measure. As applications, we show that some almost Anosov diffeomorphisms and gentle perturbations of Katok’s map have the properties.