Substitutions on compact alphabets

Mañibo, Neil; Rust, Dan; Walton, James J.

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.

Citation

Mañibo, N., Rust, D., & Walton, J. J. (2025). Substitutions on compact alphabets. Journal of the London Mathematical Society, 111(3), Article e70123. https://doi.org/10.1112/jlms.70123

• Journal Article Type: Article
• Acceptance Date: Feb 24, 2025
• Online Publication Date: Mar 6, 2025
• Publication Date: 2025-03
• Deposit Date: Feb 26, 2025
• Publicly Available Date: Feb 26, 2025
• Journal: Journal of the London Mathematical Society
• Print ISSN: 0024-6107
• Electronic ISSN: 1469-7750
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 111
• Issue: 3
• Article Number: e70123
• DOI: https://doi.org/10.1112/jlms.70123
• Keywords: substitutions, infinite alphabets, positive operators, quasi-compactness, unique ergodicity
• Public URL: https://nottingham-repository.worktribe.com/output/45856375
• Publisher URL: https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/jlms.70123
• Additional Information: Received: 2023-08-18; Accepted: 2025-02-24; Published: 2025-03-06