Abstract
Let
M
be a closed oriented
C
∞
manifold and
f
a
C
∞
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
∞
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
∞
volume. The proofs are concatenations of well known results in the field.