Weak almost periodicity and the strong ergodic limit theorem for periodic evolution systems

SEMIGROUP ACTIONS , WEAK ALMOST PERIODICITY, AND INVARIANT MEANS

Let S be a topological semigroup acting on a topological space X. We develop the theory of (weakly) almost periodic functions on X, with respect to S, and form the (weakly) almost periodic compactifications of X and S, with respect to each other. We then consider the notion of an action of Son a Banach space, and on its dual, and after defining S-invariant means for such a space, we give a...

The Entangled Ergodic Theorem in the Almost Periodic Case

An entangled ergodic theorem was introduced in [1] in connection with the quantum central limit theorem, and clearly formulated in [6]. Namely, let U be a unitary operator on the Hilbert space H, and for m ≥ k, α : {1, . . . , m} 7→ {1, . . . , k} a partition of the set {1, . . . , m} in k parts. The entangled ergodic theorem concerns the convergence in the strong, or merely weak (s–limit, or w...

Compactifications, Hartman functions and (weak) almost periodicity

In this paper we investigate Hartman functions on a topological group G. Recall that (ι, C) is a group compactification of G if C is a compact group, ι : G → C is a continuous group homomorphism and ι(G) ⊆ C is dense. A bounded function f : G 7→ C is a Hartman function if there exists a group compactification (ι, C) and F : C → C such that f = F ◦ ι and F is Riemann integrable, i.e. the set of ...

Weak-coupling limit for ergodic environments

The main aim of this work is to establish an averaging principle for a wide class of interacting particle systems of birth-and-death type in the continuum. This principle is an important step in the analysis of Markov evolutions and is usually applied for the associated semigroups related to backward Kolmogorov equations, c.f. [Kur73]. Our approach is based on the study of forward Kolmogorov eq...

Periodicity and Almost-periodicity