Point spectrum of ergodic abelian group actions and the corresponding group-measure factors

Global Aspects of Ergodic Group Actions

The study of dynamical systems has its origins in the classical mechanics of Newton and his successors. That theory concerns the behavior of solutions of certain differential equations on manifolds. The modern theory has flourished in many directions that are best described by focusing on different features of the classical systems. For example, keeping only the topological structure led to the...

Entropy, Smooth Ergodic Theory, and Rigidity of Group Actions

This is a preliminary version of lecture notes based on a 6 hour course given at the workshop Dynamics Beyond Uniform Hyperbolicity held in Provo, Utah June 2017. The goal of this course is two-fold. First we will present a number of tools and results in the smooth ergodic theory of actions of higher-rank abelian groups including Lyapunov exponents, metric entropy, and the relationship between ...

Global Aspects of Ergodic Group Actions and Equivalence Relations

THE AUTOMORPHISM GROUP OF NON-ABELIAN GROUP OF ORDER p^4

Let G be a finite non-abelian group of order p^4 . In this paper we give a structure theorem for the Sylow p-subgroup, Aut_p(G)  , of the automorphism group of G.

On the ratio ergodic theorem for group actions

We study the ratio ergodic theorem (RET) of Hopf for group actions. Under a certain technical condition, if a sequence of sets {Fn} in a group satisfy the RET, then there is a finite set E such that {EFn} satisfies the Besicovitch covering property. Consequently for the abelian group G = ⊕n=1Z there is no sequence Fn ⊆ G along which the RET holds, and in many finitely generated groups, includin...