Topological Wiener–Wintner ergodic theorems via non-abelian Lie group extensions

Abstract

Abstract. We generalize a series of topological Wiener-Wintner ergodic theorems due to Walters to the context of group extensions of measure-preserving transformations where the group is a non-abelian compact Lie group. Applications to random ergodic theorems for a shift map are given.

Topological Wiener–Wintner theorems for amenable operator semigroups

“…. , n−1} the following corollary has been proved by Santos and Walkden [23,Corollary 4.4] generalizing a previous result of Walters [28,Theorem 5], who considered the case N = 1. Proof.…”

“…Denote by U (N ) the group of unitary operators on C N and take a continuous cocycle γ ∈ Γ(G× K, U (N )). Motivated by papers of Walters [28] and Santos and Walkden [23] we study the mean ergodicity of the semigroup γS := {γ(g, ·)S g : g ∈ G} on C(K, C N ), where γ(g, ·)S g f (x) = γ(g, x)S g f (x) for f ∈ C(K, C N ) and x ∈ K.…”

“…1 n n j=1 λ j S j f for a continuous function f ∈ C(K) on a compact space K, the Koopman operator S : f → f • ϕ of a continuous transformation ϕ : K → K and λ in the unit circle T. Subsequently, Robinson's result has been generalized in various ways by Walters [28], Santos and Walkden [23] and Lenz [18,19].…”

“…The problem when averages of the form (⋆) even converge uniformly in λ ∈ T has been studied independently by Assani [1] and Robinson [21] with their results subsequently generalized by Walters [28], Santos and Walkden [23] and Lenz [19]. In [27] we have developed the concept of a uniform family of ergodic nets that allows us to treat this question also in our more general setting.…”

Uniform families of ergodic operator nets

“…In the second part we develop the adequate framework for investigating uniform convergence in so-called topological Wiener-Wintner theorems. In the simplest situation these theorems deal with the convergence of averages 1 N N −1 n=0 λ n S n for some operator S on spaces C(K) and λ in the unit circle T. Assani [2] and Robinson [16] asked when this convergence is uniform in λ ∈ T. Subsequently, their results have been generalised in different ways by Walters [19], Santos and Walkden [17] and Lenz [11,12]. We propose and study uniform families of ergodic nets as an appropriate concept for unifying and generalizing these and other results.…”