UC Irvine

This thesis is dedicated to studying the quantitative regularity of measures produced by actions of random dynamical systems and to applying these geometric estimates to products of random matrices. First, we consider smooth random dynamical systems defined by a distribution with a finite moment of the norm of the differential, and prove that under suitable non-degeneracy conditions any stationary measure must be Hölder continuous. This result is a vast generalization of the classical statement on Hölder continuity of stationary measures of random walks on linear groups. Second, we extend this framework to rougher settings by considering Lipschitz and Hölder continuous random dynamical systems defined by a distribution with a finite logarithmic moment, proving that under suitable non-degeneracy conditions every stationary measure must be log-Hölder continuous. Finally, we leverage these regularization properties to prove a Central Limit Theorem for non-stationary random products of SL(2, R) matrices, generalizing the classical results by Benoist and Quint that was obtained in the case of iid random matrix products.The content of this thesis is based on three papers, two of which ([24, 23]) are joint work with Anton Gorodetski and Victor Kleptsyn, and the last of which ([34]) was prepared with their continuous support and advice.