Topological Dynamics indexed by words

TL;DR

This paper extends classical topological dynamics to systems indexed by words over infinite alphabets, establishing recurrence results and broadening the scope of dynamical systems theory.

Contribution

It introduces a new framework for topological dynamics indexed by words, generalizing classical results and applying to semigroup-indexed systems.

Findings

Proves recurrence results for word-indexed dynamical systems.

Extends classical Furstenberg-Weiss theory to new indexing sets.

Provides recurrence results for semigroup-indexed systems.

Abstract

Starting with a combinatorial partition theorem for words over an infinite alphabet dominated by a fixed sequence, established recently by the authors, we prove recurrence results for topological dynamical systems indexed by such words. In this way we extend the classical theory developed by Furstenberg and Weiss of dynamical systems indexed by the natural numbers to systems indexed by words. Moreover, applying this theory to topological systems indexed by semigroups that can be represented as words we get analogous recurrence results for such systems.