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

TL;DR

This paper constructs smooth, volume-preserving, nonuniformly hyperbolic diffeomorphisms on high-dimensional manifolds that are arbitrarily close to the identity and have a specified number of ergodic components.

Contribution

It demonstrates the existence of such diffeomorphisms with any number of ergodic components on manifolds of dimension at least 5, close to the identity in the smooth topology.

Findings

Existence of volume-preserving nonuniformly hyperbolic diffeomorphisms with arbitrary ergodic components.

Construction of such diffeomorphisms close to the identity.

Applicable to all compact smooth manifolds of dimension ≥ 5.

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.