Ergodic algebras beyond the weak almost periodic functions

Abstract

Ergodic algebras is a concept introduced in a paper by Zhikov and Krivenko in the 1980’s which generalizes the concept of almost periodic functions and capture the main properties satisfied by generic realizations of a continuous function in a compact topological space endowed with an ergodic dynamical system and an invariant measure. We address the question whether there exist ergodic algebras beyond the weak almost periodic functions, that is, functions whose set of translates is relatively compact in the weak topology of the bounded continuous functions in $${\mathbb {R}}^n$$ . This question is important since, so far, all known ergodic algebras are subalgebras of the weak almost periodic functions. Also, it is known that almost all realizations of a continuous functions in a compact space endowed with a dynamical system and an invariant measure, belongs to an ergodic algebra. In this paper we construct examples of ergodic algebras that are not contained in the weak almost periodic functions. Actually, our examples are not contained in the greater algebra formed by the sum of almost periodic functions with uniformly continuous functions with null mean value in $${\mathbb {R}}^n$$ .