Operator ergodic theory for one-parameter decomposable groups

Infinite ergodic theory for Kleinian groups

In this paper we use infinite ergodic theory to study limit sets of essentially free Kleinian groups which may have parabolic elements of arbitrary rank. By adapting a method of Adler, we construct a section map S for the geodesic flow on the associated hyperbolic manifold. We then show that this map has the Markov property and that it is conservative and ergodic with respect to the invariant m...

Ergodic Theory, Groups, and Geometry

Rotation methods in operator ergodic theory

Let E(·) : R → B(X) be the spectral decomposition of a trigonometrically well-bounded operator U acting on the arbitrary Banach space X, and suppose that the bounded function φ : T → C has the property that for each z ∈ T, the spectral integral R [0,2π] φ(e)dEz(t) exists, where Ez(·) denotes the spectral decomposition of the (necessarily) trigonometrically well-bounded operator (zU). We show th...

Algebra, Arithmetic and Multi-parameter Ergodic Theory

While classical ergodic theory deals largely with single ergodic transformations or flows (i.e. with actions of N,Z,R+ or R on measure spaces), many of the lattice models in statistical mechanics (such as dimer models) have multi-dimensional symmetry groups: they carry actions of Z or R with d > 1. However, the transition from Zor R-actions to multi-parameter ergodic theory presents considerabl...

Some Extremely Amenable Groups Related to Operator Algebras and Ergodic Theory

A topological group G is called extremely amenable if every continuous action of G on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact gr...