Transitive partially hyperbolic diffeomorphisms in dimension three

TL;DR

This paper proves that transitive conservative partially hyperbolic diffeomorphisms in 3-manifolds are ergodic and provides a classification of accessibility classes, advancing understanding of dynamical behavior in three-dimensional topology.

Contribution

It establishes the equivalence of transitivity and ergodicity for a broad class of 3D partially hyperbolic diffeomorphisms and characterizes their accessibility classes.

Findings

Transitivity implies ergodicity in the studied class.

Complete classification of accessibility classes for transitive 3D systems.

Extension of ergodic theory results to manifolds with virtually solvable fundamental groups.

Abstract

We prove that any $C^{1+\alpha}$ transitive conservative partially hyperbolic diffeomorphism of a closed 3-manifold with virtually solvable fundamental group is ergodic. Consequently, in light of \cite{FP-classify}, this establishes the equivalence between transitivity and ergodicity for $C^{1+\alpha}$ conservative partially hyperbolic diffeomorphisms in \emph{any} closed 3-manifold. Moreover, we provide a characterization of compact accessibility classes under transitivity, thereby giving a precise classification of all accessibility classes for transitive 3-dimensional partially hyperbolic diffeomorphisms.