Substitutions on compact alphabets

TL;DR

This paper develops a framework for continuous substitutions on compact Hausdorff alphabets, linking topological dynamics and ergodic theory through spectral analysis of substitution operators, and provides criteria for unique ergodicity.

Contribution

It extends Perron--Frobenius theory to Banach lattices and offers computable conditions for quasi-compactness and unique ergodicity of substitution operators.

Findings

Primitive, constant length substitutions on isolated point alphabets are uniquely ergodic.

Spectral properties of substitution operators relate to ergodic properties.

Extension of Perron--Frobenius theory to Banach lattices.

Abstract

We develop a systematic approach to continuous substitutions on compact Hausdorff alphabets. Focussing on implications of irreducibility and primitivity, we highlight important features of the topological dynamics of their (generalised) subshifts. We then reframe questions from ergodic theory in terms of spectral properties of a corresponding substitution operator. This requires an extension of standard Perron--Frobenius theory to the setting of Banach lattices. As an application, we identify computable criteria that guarantee quasi-compactness of the substitution operator. This allows unique ergodicity to be verified for several classes of examples. For instance, it follows that every primitive and constant length substitution on an alphabet with an isolated point is uniquely ergodic, a result which fails when there are no isolated points.