On simultaneous linearization of diffeomorphisms of the sphere

Abstract

Let $R_1,R_2,\dots ,R_m$ be rotations generating ${\mathbb{SO}}_{d+1}$, $d\ge 2$, and let $f_1,f_2,\dots ,f_m$ be small smooth perturbations of them. We show that $\{f_{\alpha }\}$ can be linearized simultaneously if and only if the associated random walk has zero Lyapunov exponents. As a consequence, we obtain stable ergodicity of actions of random rotations in even dimensions