Synchronization rates and limit laws for random dynamical systems

We study general random dynamical systems of continuous maps on some compact metric space. Assuming a local contraction condition and proximality, we establish probabilistic limit laws such as the (functional) central limit theorem, the strong law of large numbers, and the law of the iterated logarithm. Moreover, we study exponential synchronization and synchronization on average. In the particular case of iterated function systems on \({\mathbb {S}}^1\), we analyze synchronization rates and describe their large deviations. In the case of \(C^{1+\beta }\)-diffeomorphisms, these deviations on random orbits are obtained from the large deviations of the expected Lyapunov exponent.