Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

TL;DR

This paper proves that certain partially hyperbolic diffeomorphisms on tori are dynamically coherent and leaf-conjugate to linear models, leading to ergodic and measure-theoretic properties.

Contribution

It establishes global stability and coherence results for partially hyperbolic diffeomorphisms isotopic to Anosov maps on tori, extending understanding of their structure.

Findings

Partially hyperbolic diffeomorphisms are dynamically coherent.

Such diffeomorphisms are leaf-conjugate to linear Anosov maps.

Results imply intrinsic ergodicity and measure equivalence.

Abstract

We show that partially hyperbolic diffeomorphisms of $d$-dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a \textit{global stability result}, i.e. every partially hyperbolic diffeomorphism as above is \textit{leaf-conjugate} to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.