Mostly contracting random maps
Pablo G. Barrientos (UFF, Niterói, Brasil)

We study the long-term behavior of the iteration of a random map consisting of Lipschitz transformations on a compact metric space, randomly selected according to a xed probability measure. Such a random map is said to be mostly contracting if all Lyapunov exponents associated with stationary measures are negative. We show in this case that the associated annealed Koopman operator is quasicompact.

This implies many statistical properties, such as the global Palis conjecture on the niteness of physical measures, 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 dieomorphisms, interval dieomorphisms onto their images, and dieomorphisms 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 the simplicity of the rst Lyapunov exponent and a certain type of irreducibility. One of the main tools to prove the above results is the generalization of Kingman’s subadditive ergodic theorem and the uniform Kingmans subadditive ergodic theorem for general Markov operators. These results are of independent interest, as they may have broad applications in other contexts.