Ergodic Theorems on Amenable Groups

Pointwise Theorems for Amenable Groups

In this paper we describe proofs of the pointwise ergodic theorem and Shannon-McMillan-Breiman theorem for discrete amenable groups, along Følner sequences that obey some restrictions. These restrictions are mild enough so that such sequences exist for all amenable groups.

Ergodic Theorems on Amenable Groups

In 1931, Birkhoff gave a general and rigorous description of the ergodic hypothesis from statistical meachanics. This concept can be generalized by group actions of a large class of amenable groups on σ-finite measure spaces. The expansion of this theory culminated in Lindenstrauss’ celebrated proof of the general pointwise ergodic theorem in 2001. The talk is devoted to the introduction of abs...

Ergodic Theoretic Characterization of Left Amenable Lau Algebras

Non-linear ergodic theorems in complete non-positive curvature metric spaces

Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...

Sharp ergodic theorems for group actions and strong ergodicity

Let μ be a probability measure on a locally compact group G, and suppose G acts measurably on a probability measure space (X,m), preserving the measure m. We study ergodic theoretic properties of the action along μ-i.i.d. random walks on G. It is shown that under a (necessary) spectral assumption on the μ-averaging operator on L2(X,m), almost surely the mean and the pointwise (Kakutani’s) rando...