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!



Volume of the set of transitive points of transitive diffeomorphisms
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


