Mostly contracting random maps
Pablo G. Barrientos, Dominique Malicet
TL;DR
This work develops a general theory for mostly contracting Lipschitz random maps on compact metric spaces, extending classical random-product results beyond smooth manifolds. By proving an open-ness result for the mostly contracting regime and establishing uniform Kingman-type theorems for Markov operators, it derives a robust suite of statistical properties, including local and global contraction, quasi-compactness of the annealed Koopman operator, and limit theorems (CLT, LDP) along with statistical stability. The framework further extends to locally constant linear cocycles, where equator/pinnacle decompositions, quasi-irreducibility, and Furstenberg-type representations yield CLTs, Hölder continuity, and localization of maximizing stationary measures; analogous one-dimensional circle diffeomorphism results are obtained under mild hypotheses. These results collectively provide a unifying approach to the long-term behavior of random dynamical systems driven by Lipschitz maps, with broad applicability to random products of circle/interval Cantor diffeomorphisms and linear cocycles. The findings have potential implications for understanding synchronization, fixed-point attractors, and the spectral structure of transfer operators under random perturbations.
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 Hölder 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.