A characterization of subshifts with bounded powers

TL;DR

This paper characterizes subshifts with bounded powers using a metric approach, constructing graph approximations and relating Lipschitz equivalence to the property, while also exploring zeta-functions and complexity exponents.

Contribution

It introduces a novel metric characterization of bounded powers in minimal, aperiodic subshifts via graph-based metrics and connects this to zeta-functions and complexity measures.

Findings

Bounded powers characterized by Lipschitz equivalence of metrics.

Construction of graph families approximating subshifts.

Relation between zeta-function abscissa and complexity exponents.

Abstract

We consider minimal, aperiodic symbolic subshifts and show how to characterize the combinatorial property of bounded powers by means of a metric property. For this purpose we construct a family of graphs which all approximate the subshift space, and define a metric on each graph which extends to a metric on the subshift space. The characterization of bounded powers is then given by the Lipschitz equivalence of a suitably defined infimum metric with the corresponding supremum metric. We also introduce zeta-functions and relate their abscissa of convergence to various exponents of complexity of the subshift.