I am just addressing the existence of a smooth invariant measure by a diffeomorphism.

Let $\Omega$ be the standard volume on your Riemannian manifold,
and $\phi$ a smooth function on M. A quick computation shows that
$e^\phi \Omega$ is invariant by f if and only if the following cohomological equation is satisfied:
$$ \phi(f^{-1}(x))-\phi(x)=log\ Jf(x)$$
where Jf is the jacobian of f. This implies for example that $Jf^n(x)=1$ for all $x\in Fix(f^n)$. 

This later condition is in fact sufficient for C2 transitive Anosov diffeomorphisms (see e.g. Katok-Hasselblatt th 19.2.7). For these diffeos, this is also equivalent to saying that the SRB measure for f and the SRB measure for the inverse of f are equal (this is interesting because transitive Anosov diffeos always admit SRB measures, but of course not always smooth invariant measures).