Simultaneous linearization of diffeomorphisms of isotropic manifolds (Q6566457)

A basic problem in dynamics is determining whether two discrete dynamical systems, originated from the iteration of two diffeomorphism, are equivalent in some, continuous, smooth, or analytic category.\N\NLet \(g_1, \ldots , g_m\) be a finite number \(m \geq 2\) of diffeormorphisms of a smooth compact manifold \(M\). The author considers the random dynamical system on the manifold, obtained by applying, at each step, one of these diffeomorphisms \(g_j\) chosen randomly. In addition, he assumes that \(M\) is a closed isotropic Riemannian manifold (different of the circle \(\mathbb{S}^1\)), and that \(R_1, \ldots, R_m\) generate the isometry group of \(M\).\N\NLet \(f_1, \ldots, f_m\) be smooth perturbations of these isometries \(\{ R_j\}\). The main result shows that the \(f_i\) are simultaneously smooth conjugate to isometries of \(M\) if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero.\N\NThis extends a simultaneous linearization result of \textit{D. Dolgopyat} et al. [Duke Math. J. 136, 475--505 (2007; Zbl 1113.37038)] from \(M=\mathbb{S}^n\) to real, complex, quaternionic projective spaces, and the Cayley projective plane.\N\NThe article is very well written. Even if the proof of the main theorem follows the general arguments in [loc. cit.] in several instances, the author describes in full detail the differences and contributions of his work. In particular, an explanation for the case of \(M=\mathbb{S}^1\) is provided.