Multiplicity of topological systems

TL;DR

This paper introduces the concept of topological multiplicity for invertible systems, exploring its properties, examples, and its application to subshifts with linear complexity growth.

Contribution

It defines topological multiplicity, analyzes its properties, and applies it to subshifts, providing new insights into the complexity of topological systems.

Findings

Finite multiplicity systems have specific structural properties.

Examples illustrate the concept and its implications.

Subshifts with linear growth complexity have bounded multiplicity.

Abstract

We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number $k$ of real continuous functions $f_1,\cdots, f_k$ such that the functions $f_i\circ T^n$, $n\in\mathbb Z$, $1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on $X$ endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.