Center foliation rigidity for partially hyperbolic toral diffeomorphisms

We study perturbations of a partially hyperbolic toral automorphism L which is diagonalizable over C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb C$$\end{document} 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-Hölder conjugate to L, then the conjugacy is smooth. As a corollary, we show that for any symplectic perturbation of such an L any bi-Hölder 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.