| Results are divided into three chapters. Each chapter studies several classes of topological dynamical systems and derives results relating chaoticity to the existence of invariant measures, and equidistribution behavior. Chapter 1 is an introduction. In Chapter 2, we prove that if a countable group G acts transitively on a compact metric space, preserving a probability measure of full support, then the system is either equicontinuous, or has sensitive dependence on initial conditions. Assuming ergodicity, we get the same conclusion without countability. Additionally, we prove that when a finitely generated, solvable group, acts transitively and certain cyclic sub-actions have dense sets of minimal points, then the system has sensitive dependence on initial conditions.;In Chapter 3, we prove that for a minimal translation T on a compact 2-step nilmanifold X and any Borel probability measure mu on X, the push-forward Tn⋆ mu of mu under Tn tends toward Haar measure if and only if mu projects to Haar measure on the maximal torus factor. For an arbitrary nilmanifold we get the same result along a sequence of uniform density 1. Using different methods, we derive the same result for a large class of iterated skew products. Additionally we prove a multiplicative ergodic theorem for functions taking values in the upper unipotent group. Finally, we characterize limits of Tn⋆ mu for some skew product transformations with expansive fibers. All results are presented in terms of twisting and weak twisting, properties which strengthen unique ergodicity in a way analogous to how mixing and weak mixing strengthen ergodicity for measure preserving systems.;In Chapter 4, we study hyperbolic automorphisms of tori. It is known that Haar measure is the unique invariant measure of maximal entropy. We make this statement effective by showing that an invariant measure having large entropy must integrate Lipschitz functions similarly to Haar measure. On the 2-torus we prove a stronger statement which gives meaningful results even for zero entropy measures. Finally, we prove an analogous result for x2 on the circle (the map x mapsto 2x on R/Z ) and give an application to equidistribution of finite invariant sets. |
