Two remarks on $C^{\infty }$ Anosov diffeomorphisms

Abstract

Let $M$ be a closed oriented $C^{\infty }$ manifold and $f$ a $C^{\infty }$ Anosov diffeomorphism on $M$. We show that if $M$ is the two torus $T^2$, then $f$ is conjugate to a hyperbolic automorphism of $T^2$, either by a $C^{\infty }$ diffeomorphism or by a singular homeomorphism. We also show that for general $M$, if $f$ admits an absolutely continuous invariant measure $\mu$, then $\mu$ is a $C^{\infty }$ volume. The proofs are concatenations of well known results in the field.

3 pages