A hyperbolic diffeomorphism with countably many ergodic components near identity

TL;DR

This paper constructs a smooth, volume-preserving hyperbolic diffeomorphism on a 4D manifold that is arbitrarily close to the identity and exhibits infinitely many ergodic components, illustrating complex dynamical behavior near identity.

Contribution

It introduces a novel example of a hyperbolic diffeomorphism with countably many ergodic components close to the identity map.

Findings

Existence of a hyperbolic diffeomorphism with infinitely many ergodic components

The diffeomorphism can be made arbitrarily close to the identity

The construction demonstrates complex dynamics near the identity map

Abstract

We construct a smooth hyperbolic volume preserving diffeomorphism on a four dimensional compact Riemannian manifold which has countably many ergodic components and is arbitrarily close to the identity map.