Partially hyperbolic diffeomorphisms homotopic to the identity in dimension three

TL;DR

This paper proves that conservative partially hyperbolic diffeomorphisms homotopic to the identity are ergodic unless the 3-manifold's fundamental group is virtually solvable, confirming a conjecture in this setting.

Contribution

It establishes ergodicity for a broad class of diffeomorphisms in three dimensions, confirming the Hertz-Hertz-Ures Ergodicity Conjecture for those homotopic to the identity.

Findings

Such diffeomorphisms are ergodic in most cases.

Ergodicity holds unless the fundamental group is virtually solvable.

The result applies to conservative systems in three-dimensional manifolds.

Abstract

We show that any conservative partially hyperbolic diffeomorphism homotopic to the identity is accessible unless the fundamental group of its ambient 3-manifold is virtually solvable. As a consequence, such diffeomorphisms are ergodic, giving an affirmative answer to the Hertz-Hertz-Ures Ergodicity Conjecture in the homotopy class of identity.