Simultaneous linearization of diffeomorphisms of isotropic manifolds

Suppose that M is a closed isotropic Riemannian manifold and that R 1 , ... , R m generate the isometry group of M . Let f 1 , ... , 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.