I revised the question. In smooth ergodic theory, a diffeomorphism is said to be conservative (I), if it preserves the Lebesgue measure. So for some of us, conservativity is just short for measure-preserving.

On the other hand, we can define the conservative part $C_f$ for general measure-class preserving maps (see below). We can also say $f$ is conservative (II) if $C_f=M$.

My question is:

given a smooth map $f:M\to M$, when could we upgrade from conservative (II) to conservativity (I) (up to a change of Riemannian metric, or to some measure $\mu\sim m$)?

More generally, when could the restriction $f|_{C_f}$ be conservative (I) (assuming $m(C_f)>0$)?

Let $(X,\mu)$ be a standard measure space and $f:X\to X$ be an isomorphism under which $\mu$ is quasi-invariant. That is, $f^\ast\mu\ll \mu$ and $\mu\ll f^\ast\mu$. A measurable set $E$ is said to be ** wandering** if all $f^nE$, $n\in\mathbb{Z}$ are mutually disjoint.

(We may call it *topologically* wandering if $E$ is an open subset. So we generalize the classical definition of wandering.)

It has been proved that there exists a *maximal* wandering set $W$ (up to a $\mu$-null set). Then the *dissipative part* $D_f$ of $(X,f,\mu)$ is $D_f=\bigsqcup_{\mathbb{Z}}f^nW$. Then $C_f=X\backslash D_f$ is called the conservative part of $(X,f,\mu)$. The induced partition $\lbrace C_f,D_f\rbrace$ is called Hopf decomposition (named by Halmos?)

For example, the dissipative part is trivial if $\mu$ is probability and preserved by $f$ **simultaneously**.

Observation: by introducing an artificial measure $\nu=\sum_{\mathbb{Z}}f^n(\mu|_W)$, the map can be made $\nu$-preserving on the dissipative part $D_f$.

What about the conservative part? Could we make it measure-preserving with respect to some measure?

More specifically, let $M$ be a closed manifold, $f:M\to M$ be a smooth diffeomorphism (say $C^\infty$ if necessary), and $m$ be the normalized Lebesgue measure (automatically quasi-invariant). Assume $m(C_f)>0$. When could we find some $\mu\sim m|_{C_f}$ that is preserved by $f$?

Thank you!