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 diffeomorphism admits an invariant measure $\mu$ that has absolutely continuous conditional measures on unstable manifolds. The measure $\mu$ is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, $\frac {1}{n} \sum _{i=0}^{n-1}\delta _{f^{i}x}$ tends to either an SBR measure or $\delta _{p}$ for almost every $x$ with respect to Lebesgue measure. ($\delta _{x}$ 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 $p$.