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 (Jon Aaronson, An introduction to infinite ergodic theory, Page 15).

My question is:

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

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

Let $M$ be a compact 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 (Jon Aaronson, An introduction to infinite ergodic theory, Page 15).

By Poincare Recurrence theorem (Page 17, same book) we have $m(C_f\backslash R_f)=0$.

My question is about the converse direction:

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

2. When will the following be true: $m(R_f\backslash C_f)=0$?