  We study general random dynamical systems of continuous maps on some compact
metric space. Assuming a local contraction condition and uniqueness of the
stationary measure, we establish probabilistic limit laws such as the 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.
