Mostly contracting random maps

TL;DR

This paper investigates the long-term dynamics of mostly contracting random Lipschitz maps on compact metric spaces, establishing their statistical properties and stability through advanced ergodic theorems and Lyapunov exponents.

Contribution

It introduces the class of mostly contracting random maps, proves their openness, and establishes key statistical and stability results using generalized ergodic theorems.

Findings

The class of mostly contracting random maps is open and satisfies strong statistical laws.

Lyapunov exponents depend continuously and Hölder continuously on parameters.

The generalized Kingman's subadditive ergodic theorem applies to these systems.

Abstract

We study the long-term behavior of the iteration of a random map consisting of Lipschitz transformations on a compact metric space, independently and randomly selected according to a fixed probability measure. Such a random map is said to be \emph{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, the global Palis' 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 $C^1$ diffeomorphisms, interval $C^1$ diffeomorphisms onto their images, and $C^1$ 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 first Lyapunov exponent and some kind of irreducibility. 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. These results are of independent interest, as they may have broad applications in other contexts.