Two remarks on $C^\infty$ Anosov diffeomorphisms

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 $μ$, then $μ$ is a $C^\infty$ volume. The proofs are concatenations of well known results in the field.

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