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Substitutions on compact alphabets

Neil Mañibo, Dan Rust, James J. Walton

TL;DR

Substitutions on compact alphabets extends the classical finite-alphabet framework to compact Hausdorff alphabets by introducing and analyzing the substitution operator $M$ on $E=C(\mathcal{A})$ and its normalised form $T=M/r$. The authors prove that natural length functions exist and are essentially unique under irreducibility, enabling a geometric tiling interpretation via suspension flows and invariant measures; they also provide computable quasi-compactness criteria that guarantee unique ergodicity for broad classes, including primitive and constant-length substitutions with isolated points. The work builds bridges between subshifts and tilings through fusion tilings, establishing a correspondence between invariant measures and eigenmeasures and extending results to higher dimensions such as the pinwheel tiling; it further derives discrepancy bounds from a spectral gap. Collectively, these results deliver practical criteria for certifying unique ergodicity and yield a robust operator-theoretic toolkit for studying infinite-alphabet substitutions and higher-dimensional ILC tilings.

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.

Substitutions on compact alphabets

TL;DR

Substitutions on compact alphabets extends the classical finite-alphabet framework to compact Hausdorff alphabets by introducing and analyzing the substitution operator on and its normalised form . The authors prove that natural length functions exist and are essentially unique under irreducibility, enabling a geometric tiling interpretation via suspension flows and invariant measures; they also provide computable quasi-compactness criteria that guarantee unique ergodicity for broad classes, including primitive and constant-length substitutions with isolated points. The work builds bridges between subshifts and tilings through fusion tilings, establishing a correspondence between invariant measures and eigenmeasures and extending results to higher dimensions such as the pinwheel tiling; it further derives discrepancy bounds from a spectral gap. Collectively, these results deliver practical criteria for certifying unique ergodicity and yield a robust operator-theoretic toolkit for studying infinite-alphabet substitutions and higher-dimensional ILC tilings.

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.
Paper Structure (20 sections, 70 theorems, 110 equations, 1 figure)

This paper contains 20 sections, 70 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Topology and dynamics
  3. Substitutions
  4. Primitivity
  5. Realising set substitutions with compact alphabets
  6. The substitution operator and natural length functions
  7. The substitution operator
  8. The spectrum of
  9. Uniqueness and consequences of a natural length function
  10. Existence of natural length functions and unique ergodicity
  11. Operator convergence, length functions, and eigenmeasures
  12. Fusion tilings, invariant measures, and power convergence
  13. Invariant measures and suspension flows
  14. Applications
  15. Quasi-compact substitutions and unique ergodicity
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\varrho$ be a primitive substitution on a compact Hausdorff alphabet with corresponding subshift $X_\varrho$ and substitution operator $M$. Let $r$ be the spectral radius of $M$. If $T = M/r$ is strongly power convergent, then $X_\varrho$ is uniquely ergodic.

Figures (1)

  • Figure 1: The pinwheel substitution

Theorems & Definitions (177)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Remark 2.9
  • ...and 167 more