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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).

In general one cannot expect that

As pointed out by Deane, any volume form can be realised as the volume associated to a smooth f-invariant measure Riemannian metric. Embed your manifold M in R^n, extend your volume form $f dvol_{eucl}$ to be given by a riemannian neighborhood of the manifold, then take the restriction of the metric $f^{2/m}g_{eucl}$ to M . Yet In fact, it is even possible to find a smooth conjuguate of f that preserves any given riemannian volume: this is the Moser trick.

Let M be a compact riemannian manifold, $\Omega_0$ and $\Omega_1$ two volume forms with the same volume. Then there is a diffeo g such that $g^*\Omega_0=\Omega_1$.

The Moser theorem can also used to build a diffeo that does not preserve a Riemannian volume. Take an ergodic diffeo that preserves such a volume, and send that volume on a volume form that does not come from a riemannian metric.

2 added a paragraph on Moser theorem

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).

In general one cannot expect that a smooth f-invariant measure to be given by a riemannian metric. Yet it is possible to find a smooth conjuguate of f that preserves any given riemannian volume: this is the Moser trick.

Let M be a compact riemannian manifold, $\Omega_0$ and $\Omega_1$ two volume forms with the same volume. Then there is a diffeo g such that $g^*\Omega_0=\Omega_1$.

The Moser theorem can also used to build a diffeo that does not preserve a Riemannian volume. Take an ergodic diffeo that preserves such a volume, and send that volume on a volume form that does not come from a riemannian metric.

1

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).