Let $M$ be a closed smooth manifold and $m$ be a normalized volume induced by some Riemannian metric on $M$. Let $f\in\mathrm{Diff}^1(M)$ and $R_f$ be the set of recurrent points of $f$ (A point $x$ is recurrent if $x\in\omega(f,x)\cap \alpha(f,x)$, or equally $\liminf_{n\to\pm\infty}d(f^nx,x)=0$). 

The Hopf decomposition of $(M,f)$ is a partition $M=C_f\sqcup D_f$ such that

 - the restriction $f|_{C_f}$ is conservative. That is, if $E\subset {C_f}$ is a measurable subset  with $[f^nE:n\in\mathbb{Z}]$ mutually disjoint, then $m(E)=0$;


 - the restriction $f|_{D_f}$ is dissipative. In fact ${D_f}$ is the disjoint union of $[f^nF:n\in\mathbb{Z}]$ for some measurable subset $F\subset M$.

See [here][1] (Jon Aaronson, An introduction to infinite ergodic theory, Page 15).

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My question is: 

 1. If $m(R_f)>0$, will we have $m(C_f)>0$? 

 2. When will the following be true: $R_f\subset C_f$?


  [1]: http://books.google.com/books?id=3mRo0wTdNv0C&pg=PA53&dq=Aaron+Son&source=gbs_toc_r&cad=4#v=onepage&q&f=false