Limit Theorems for Ergodic Group Actions and Random Walks

Abstract: This thesis consists of an introduction, a summary and 7 papers. The papers are devoted to problems in ergodic theory, equidistribution on compact manifolds and random walks on groups.In Papers A and B, we generalize two classical ergodic theorems for actions of abelian groups. The main result is a generalization of Kingman’s subadditive ergodic theorem to ergodic actions of the group Zd.In Papers C,D and E, we consider equidistribution problems on nilmanifolds. In Paper C we study the asymptotic behavior of dilations of probability measures on nilmanifolds, supported on singular sets, and prove, under some technical assumptions, effective convergences to Haar measure. In Paper D, we give a new geometric proof of an old result by Koksma on almost sure equidistribution of expansive sequences. In paper E we give necessary and sufficient conditions on a probability measure on a homogeneous Riemannian manifold to be non–atomic.Papers F and G are concerned with the asymptotic behavior of random walks on groups. In Paper F we consider homogeneous random walks on Gromov hyperbolic groups and establish a central limit theorem for random walks satisfying some technical moment conditions. Paper G is devoted to certain Bernoulli convolutions and the regularity of their value distributions.