Simultaneous linearization of diffeomorphisms of isotropic manifolds

Abstract

Suppose that $M$ is a closed isotropic Riemannian manifold and that $R_1,\dots ,R_m$ generate the isometry group of $M$. Let $f_1,\dots ,f_m$ be smooth perturbations of these isometries. We show that the $f_i$ are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian [Duke Math. J. {136}, 475–505 (2007)] from $S^n$ to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work.