Simultaneous Linearization of Diffeomorphisms of Isotropic Manifolds

TL;DR

This paper extends a linearization result for isometries on isotropic manifolds, showing that smooth perturbations are conjugate to isometries under certain conditions related to Lyapunov exponents, broadening previous work from spheres to projective spaces.

Contribution

It generalizes a linearization theorem from spheres to real, complex, and quaternionic projective spaces and corrects an oversight in prior research.

Findings

Perturbations are conjugate to isometries if Lyapunov exponents are zero.

Extension of linearization results to broader classes of isotropic manifolds.

Identification and correction of an error in earlier work.

Abstract

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 from $S^n$ to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work.