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 $m(C\_f)>0$)?




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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!



The relations between conservative part and conservativity
Several comments


(1) There is no need to require invariance or finiteness of the measure in order to define the Hopf decomposition - it makes sense for any quasi-invariant measure.


(2) There is no need to evoke metric spaces, homeomorphisms, manifolds, etc. The Hopf decomposition is defined entirely in the measure category.


(3) There is no reason for existence of an equivalent invariant measure on the conservative part. There are ergodic actions (of so-called type III), for which there is no equivalent invariant measure. 


