The relations between conservative part and conservativity

Question

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!

Answer 1

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.

Answer 2

Regarding the modified question I first guess that you want your measure to be absolutely continuous with respect to Lebesgue measure (otherwise you can use existence of invariant measures if your manifold is compact and so on but they might be singular).

In certain situations, for example Anosov diffeomorphisms, if your map is $C^2$ then a result of Gurevic and Oseledec shows that the two notions of conservativity agree (that is a recurrent $C^2$ Anosov diffeomorphism has a volume absolutely continuous invariant measure with smooth density). The same proof is true so long as you have absolute continuity of the foliations but then the density might be less regular. There are type $III$, $C^1$ Anosov diffeomorphisms of $\mathbb{T}^2$ so the absolute continuity of folliations is essential in some way.

If you ask about a more general class of diffeomorphisms then Michel Hermann's monograph (in French) has examples of type $III$ diffeomorphisms. See also the paper "On $C^2$-diffeomorphisms of the circle which are of type $III_1$." by Jane Hawkins and Klaus Schmidt and other papers of Hawkins with a similar name. Again if the diffeomorphism is $C^3$ then the situation is different.