Conditions for the Existence of SBR Measures for “Almost Anosov” Diffeomorphisms

Abstract

A diffeomorphism f of a compact manifold M is called "almost Anosov" if it is uniformly hyperbolic away from a finite set of points. We show that under some nondegeneracy condition, every almost Anosov dif-feomorphism admits an invariant measure µ that has absolutely continuous conditional measures on unstable manifolds. The measure µ is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, 1 n n−1 i=0 δ f i x tends to either an SBR measure or δp for almost every x with respect to Lebesgue measure. (δx is the Dirac measure at x.) For each case, we give sufficient conditions by using coefficients of the third order terms in the Taylor expansion of f at p.