Substitutions on compact alphabets
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.
Keywords
Cite
@article{arxiv.2204.07516,
title = {Substitutions on compact alphabets},
author = {Neil Mañibo and Dan Rust and James J. Walton},
journal= {arXiv preprint arXiv:2204.07516},
year = {2025}
}
Comments
51 pages, 1 figure. Some minor typos fixed and routine proofs removed from S2. To appear (without section on higher dimensions) in Journal of the LMS