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

Abstract

A diffeomorphism f f of a compact manifold M 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 diffeomorphism admits an invariant measure μ \mu that has absolutely continuous conditional measures on unstable manifolds. The measure μ \mu is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, 1 n ∑ i = 0 n − 1 δ f i x \frac {1}{n} \sum _{i=0}^{n-1}\delta _{f^{i}x} tends to either an SBR measure or δ p \delta _{p} for almost every x x with respect to Lebesgue measure. ( δ x \delta _{x} is the Dirac measure at x x .) For each case, we give sufficient conditions by using coefficients of the third order terms in the Taylor expansion of f f at p p .