# The relations between conservative part and conservativity

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!

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For $C^{1+\alpha}$ Anosov diffeomorphism, Gurevic and Oseledec proved the following dichotomy that either $f$ is completely dissipative, or there exists a smooth invariant measure. A recent preprint of Z. Kosloff (arxiv.org/abs/1410.7707) proved that, there exists a $C^1$ Anosov diffeomorphism which is conservative and ergodic (in the sense of abstract ergodic theory), but does not even admit a $\sigma$-finite invariant measure being absolutely continuous with respect to Lebesgue. So the answer to my question should be 'yes sometimes', but 'no in general'. – Pengfei Dec 1 '14 at 16:23

Hi! In fact your third comment is the reason why require much stronger regularity in my question: do there exist the examples of type III in the category of smooth diffeomorphisms on closed manifolds? That is, in the class of special $\mathbb{Z}$-actions on special spaces. – Pengfei Apr 3 '13 at 13:52
I see. I wouldn't immediately know any type III examples for smooth $\mathbb Z$ actions. The ones I know are either $\mathbb Z$ actions on pretty abstract spaces (e.g., shift on product spaces) or smooth actions of much bigger groups (e.g., boundary actions of Fuchsian or Kleinian groups). – R W Apr 3 '13 at 17:01