Uniform families of ergodic operator nets

Abstract

We study mean ergodicity in amenable operator semigroups and establish the connection to the convergence of strong and weak ergodic nets. We then use these results in order to show the convergence of uniform families of ergodic nets that appear in topological Wiener-Wintner theorems.

Compact operator semigroups applied to dynamical systems

“…Applying Theorem 4.3 to the σ(X ′ , X)-topology immediately gives a characterization of weak* mean ergodicity. Next we characterize weak and strong mean ergodicity (see also Theorem 1.7 in [Nag73]) extending results of M. Schreiber for operator semigroups on Banach spaces to barrelled locally convex spaces (see Theorem 1.7 in [Sch13b], see also Corollary 1 of [Sat78] for a similar result). For a familiy T of operators on a locally convex space X we use the notation…”

The primitive spectrum of a semigroup of Markov operators

“…for each S ∈ S with respect to the strong operator topology. We note that there always are right ergodic operator nets for S (see Corollary 1.5 of [Sch13]). We give some examples (see Examples 1.2 of [Sch13]).…”

“…For the rest of this section we make the following assumption (cf. Examples 1.2 (e) of [Sch13]). with T s f (x) := f (sx) for f ∈ C(K), s ∈ S and x ∈ K, which is strongly continuous by Theorem 4.17 of [EFHN15].…”

“…by Theorem 1.7 of [Sch13]. Choose c ∈ (c 1 , c 2 ) and set U 1 := f −1 ((−∞, c)) and U 2 := f −1 ((c, ∞)).…”

“…Now S consists of Markov operators and therefore fix(S) separates fix(S ′ ). Thus S is mean ergodic by Theorem 1.7 of [Sch13]. (ii) If K = T then there is a homeomorphism ϕ : T −→ T such that S is not uniquely ergodic, but minimal (see Theorem 5.8 of [Par81]).…”