Volume of the set of transitive points of transitive diffeomorphisms

Question

Let $M$ be a compact manifold without boundary, $f:M\to M$ be a diffeomorphism. Then $f$ is said to be (topologically) transitive if $\bigcup_{\mathbb{Z}}f^nU$ is dense for every nonempty open set $U\subset M$.

Assume $f$ is transitive and $\lbrace U_k:k\ge1\rbrace$ is a subbasis of the topology on $M$. Then define the transitive set of $f$ to be $T_f=\bigcap_{k\ge1}(\bigcup_{\mathbb{Z}}f^nU_k)$. Clearly $T_f$ is a dense $G_\delta$ subset of $M$ (so topologically large).

Let $m$ be the Lebesgue measure on $M$. My question is:

• Could $T_f$ be measure-theoretically meager, say $m(T_f)=0$, for some transitive $f$?

Thank you!

Answer 1

Dear Pengfei,

Such examples exist and they can be obtained by the so-called Anosov-Katok method (see, e.g., Theorem 5.1 of the article "Constructions in elliptic dynamics" http://www.ams.org/mathscinet-getitem?mr=2104594 of B. Fayad and A. Katok).

Best,

Matheus