A volume preserving nonuniformly hyperbolic diffeomorphism with arbitrary number of ergodic components and close to the identity

Abstract

We prove that for any $\ell\in\NN\cup\{\infty\}$ and any $r\in \NN$, every compact smooth Riemannian manifold $\cM$ of $\dim \cM\ge 5$ carries a $C^\infty$ volume preserving nonuniformly hyperbolic diffeomorphism, which has exactly $\ell$ ergodic components (in fact, Bernoulli components) and is $C^r$ close to the identity.