Mostly Contracting Random Maps

This volume studies the long-term behavior of independent random iterations of Lipschitz transformations on a compact metric space. 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 setting. It is shown that this class is open and satisfies the strong law of large numbers for non-uniquely ergodic systems, a limit theorem for random iterations, the Palis’ global conjecture, and quasi-compactness of the associated annealed Koopman operator. These results yield central limit theorems, large deviations, statistical stability, and continuity and Hölder continuity of Lyapunov exponents. The class includes random products of C¹ diffeomorphisms of the circle, projective actions of locally constant linear cocycles, and finite-state Markov chains. Key tools include generalizations of Kingman’s subadditive ergodic theorem and an exponential local contraction theorem.