Center foliation rigidity for partially hyperbolic toral diffeomorphisms

TL;DR

This paper investigates the rigidity of center foliations in partially hyperbolic toral automorphisms, proving smooth conjugacy results under small perturbations and establishing conditions for smooth conjugacy in specific cases.

Contribution

It provides new conditions under which perturbations of certain toral automorphisms are smoothly conjugate, extending rigidity results to broader classes.

Findings

Smooth leaf conjugacy exists for small smooth perturbations of the automorphism.

Bi-Holder conjugacy implies smoothness under certain conditions.

Symplectic perturbations enforce smooth conjugacy for the automorphism.

Abstract

We study perturbations of a partially hyperbolic toral automorphism L which is diagonalizable over C and has a dense center foliation. For a small perturbation of L with a smooth center foliation we establish existence of a smooth leaf conjugacy to L. We also show that if a small perturbation of an ergodic irreducible L has smooth center foliation and is bi-Holder conjugate to L, then the conjugacy is smooth. As a corollary, we show that for any symplectic perturbation of such an L any bi-Holder conjugacy must be smooth. For a totally irreducible L with two-dimensional center, we establish a number of equivalent conditions on the perturbation that ensure smooth conjugacy to L.