Simultaneous linearization of diffeomorphisms of isotropic manifolds

Suppose that M is a closed isotropic Riemannian manifold and that R₁;...; R_m generate the isometry group of M. Let f₁;...; 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.