Two remarks on $C^\infty$ Anosov diffeomorphisms

TL;DR

This paper investigates $C^ abla$ Anosov diffeomorphisms on closed manifolds, showing conjugacy to hyperbolic automorphisms on the 2-torus and characterizing invariant measures as smooth volume under certain conditions.

Contribution

It establishes conjugacy results for $C^ abla$ Anosov diffeomorphisms on the 2-torus and characterizes absolutely continuous invariant measures as smooth volume, extending understanding of their structure.

Findings

On $T^2$, $f$ is conjugate to a hyperbolic automorphism.

Absolutely continuous invariant measure implies smooth volume.

Proofs rely on well-known results in the field.

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.