Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

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 global stability result, i.e. every partially hyperbolic diffeomorphism as above is 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.