On simultaneous linearization of diffeomorphisms of the sphere (Q869987)

Small smooth random perturbations \(f_1,\dots, f_m\) of rotations \(R_1,\dots ,R_m\) generating SO\(_{d+1}\), \(d\geq 2\), are considered. It is shown that a necessary and sufficient condition for the zero Lyapunov exponent of the associated random walk is that \(\{f_{\alpha}\}\) can be linearized simultaneously. As a consequence it follows that if \(d\) is even and \(f_\alpha\) are sufficiently close to \(R_\alpha\) and preserve volume, then \(\{f_\alpha\}\) is ergodic.