Mostly contracting random maps

Abstract

We study the long-term behavior of independent random iterations of Lipschitz transformations on a compact metric space. Such a random map is said to be mostly contracting if all Lyapunov exponents associated with stationary measures are negative. This requires introducing the notion of (maximal) Lyapunov exponent in this general context of Lipschitz transformations on compact metric spaces. We show that this class is open with respect to the appropriate topology and satisfies the strong law of large numbers for non-uniquely ergodic systems, the limit theorem for the law of random iterations, Palis' global conjecture, and that the associated annealed Koopman operator is quasi-compact. This implies many statistical properties such as central limit theorems, large deviations, statistical stability, and the continuity and H\"older continuity of Lyapunov exponents. Examples from this class of random maps include random products of circle diffeomorphisms, interval diffeomorphisms onto their images, and diffeomorphisms of a Cantor set on a line, all considered under the assumption of no common invariant measure. This class also includes projective actions of locally constant linear cocycles under the assumptions of simplicity of the top Lyapunov exponent and a suitable irreducibility condition. Finally, it encompasses the classical theory of finite-state Markov chains, viewed as random walks on a finite set. One of the main tools to prove the above results is the generalization of Kingman's subadditive ergodic theorem and the uniform Kingman's subadditive ergodic theorem for general Markov operators. Another key ingredient is the exponential local contraction theorem, which can be viewed as a non-smooth metric analogue of the contraction mechanism behind stable manifold theory. These results are of independent interest, as they may have broad applications in other contexts.