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Local spectral estimates and quantitative weak mixing for substitution $\mathbb{Z}$-actions

Alexander I. Bufetov, Juan Marshall-Maldonado, Boris Solomyak

TL;DR

The paper develops local spectral estimates for substitution $\mathbb{Z}$-actions by linking spectral-measure regularity to twisted Birkhoff sums and a quantitative Veech criterion. It proves a dichotomy: for primitive aperiodic substitutions with irreducible characteristic polynomials and no unit-circle eigenvalues, weak mixing implies a uniform log-Hölder bound on spectral measures, yielding power-log rates for quantitative weak mixing; for Salem-type substitutions, Hölder bounds hold at algebraic spectral parameters $\omega\in\mathbb{Q}(\alpha)$ but cannot be uniform across all such parameters. A key novelty is the approximation of toral automorphism orbits by lattice points, enabling a

Abstract

The paper investigates Hölder and log-Hölder regularity of spectral measures for weakly mixing substitutions and the related question of quantitative weak mixing. It is assumed that the substitution is primitive, aperiodic, and its substitution matrix is irreducible over the rationals. In the case when there are no eigenvalues of the substitution matrix on the unit circle, our main theorem says that a weakly mixing substitution $\mathbb{Z}$-action has uniformly log-Hölder regular spectral measures, and hence admits power-logarithmic bounds for the rate of weak mixing. In the more delicate Salem substitution case, our second main result says that Hölder regularity holds for algebraic spectral parameters, but the Hölder exponent cannot be chosen uniformly.

Local spectral estimates and quantitative weak mixing for substitution $\mathbb{Z}$-actions

TL;DR

The paper develops local spectral estimates for substitution -actions by linking spectral-measure regularity to twisted Birkhoff sums and a quantitative Veech criterion. It proves a dichotomy: for primitive aperiodic substitutions with irreducible characteristic polynomials and no unit-circle eigenvalues, weak mixing implies a uniform log-Hölder bound on spectral measures, yielding power-log rates for quantitative weak mixing; for Salem-type substitutions, Hölder bounds hold at algebraic spectral parameters but cannot be uniform across all such parameters. A key novelty is the approximation of toral automorphism orbits by lattice points, enabling a

Abstract

The paper investigates Hölder and log-Hölder regularity of spectral measures for weakly mixing substitutions and the related question of quantitative weak mixing. It is assumed that the substitution is primitive, aperiodic, and its substitution matrix is irreducible over the rationals. In the case when there are no eigenvalues of the substitution matrix on the unit circle, our main theorem says that a weakly mixing substitution -action has uniformly log-Hölder regular spectral measures, and hence admits power-logarithmic bounds for the rate of weak mixing. In the more delicate Salem substitution case, our second main result says that Hölder regularity holds for algebraic spectral parameters, but the Hölder exponent cannot be chosen uniformly.
Paper Structure (13 sections, 20 theorems, 134 equations)

This paper contains 13 sections, 20 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Background
  4. Log-Hölder regularity and quantitative weak mixing
  5. Substitutions of Salem type
  6. Comparison with the results on suspension flows.
  7. Scheme of the proofs.
  8. Approximation of toral automorphisms orbits by lattice points
  9. Proof of Theorem \ref{['th-main1']}
  10. The Salem case
  11. Dynamics of Salem automorphisms
  12. Application to Salem substitutions
  13. Non-uniformity of the Hölder exponent and the proof of Theorem \ref{['th:lower']}

Key Result

Theorem 2.1

Let $\zeta$ be a primitive aperiodic substitution with the incidence matrix ${\sf S}$. Then $e^{2\pi i \omega},\ \omega\in (0,1)$, is an eigenvalue for the substitution ${\mathbb Z}$-action $(X_\zeta,T,\mu)$ if and only if for all return words $v$,

Theorems & Definitions (36)

  • Theorem 2.1: Host Host, see also CS1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Remark 2.9
  • Lemma 2.10
  • ...and 26 more