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Quantitative weak mixing for typical Salem substitution suspension flows

Juan Marshall-Maldonado, Boris Solomyak

TL;DR

This paper analyzes quantitative weak mixing for suspension flows over Salem-type substitutions, where the Perron–Frobenius eigenvalue lies on the boundary between expanding and neutral behavior. By combining a vector-expansion Erdős–Kahane framework with a detailed return-word/lattice analysis, it proves that for Lebesgue-a.e. roof vector, Lip-cylindrical observables exhibit a spectral decay bound with rate $h_\beta(r)=\exp\left(-\frac{\beta \log\log\log_\alpha (1/2r)}{\log\log\log \log_\alpha (1/2r)}\right)$, hence quantitative weak mixing holds with a sub-polynomial, near-logarithmic rate. The study also shows that no Hölder QWM can occur for any suspension vector, underscoring the exceptional nature of Salem-type borderline behavior; an Appendix further provides a $\log^*$-type bound for a self-similar suspension flow and links the bounds to Fourier decay of Salem Bernoulli convolutions. Collectively, the work extends the understanding of spectral decay and mixing rates in borderline substitution dynamics and connects these rates to broader fractal-measure phenomena.

Abstract

The paper investigates quantitative weak mixing of Salem substitutions flows. We prove that for a substitution whose substitution matrix is irreducible over the rationals and the dominant eigenvalue is a Salem number, for almost every suspension flow with a piecewise constant roof function, quantitative weak mixing holds with a rate that is slightly worse than a power of $\log\log$. We do not know if this is sharp, but we do show that for any suspension flow of this kind, quantitative weak mixing with a polynomial rate is impossible. Results for specific systems are often much weaker than for ``typical'' or ``generic'' ones. In the Appendix we explain how a minor modification of an argument from Bufetov and Solomyak (2014) yields very weak, but nevertheless quantitative weak mixing estimates of $\log^*$ type for the {\em self-similar} suspension flow over a Salem substitution. Simultaneously this provides first quantitative decay rates for the Fourier transform of Salem Bernoulli convolutions.

Quantitative weak mixing for typical Salem substitution suspension flows

TL;DR

This paper analyzes quantitative weak mixing for suspension flows over Salem-type substitutions, where the Perron–Frobenius eigenvalue lies on the boundary between expanding and neutral behavior. By combining a vector-expansion Erdős–Kahane framework with a detailed return-word/lattice analysis, it proves that for Lebesgue-a.e. roof vector, Lip-cylindrical observables exhibit a spectral decay bound with rate , hence quantitative weak mixing holds with a sub-polynomial, near-logarithmic rate. The study also shows that no Hölder QWM can occur for any suspension vector, underscoring the exceptional nature of Salem-type borderline behavior; an Appendix further provides a -type bound for a self-similar suspension flow and links the bounds to Fourier decay of Salem Bernoulli convolutions. Collectively, the work extends the understanding of spectral decay and mixing rates in borderline substitution dynamics and connects these rates to broader fractal-measure phenomena.

Abstract

The paper investigates quantitative weak mixing of Salem substitutions flows. We prove that for a substitution whose substitution matrix is irreducible over the rationals and the dominant eigenvalue is a Salem number, for almost every suspension flow with a piecewise constant roof function, quantitative weak mixing holds with a rate that is slightly worse than a power of . We do not know if this is sharp, but we do show that for any suspension flow of this kind, quantitative weak mixing with a polynomial rate is impossible. Results for specific systems are often much weaker than for ``typical'' or ``generic'' ones. In the Appendix we explain how a minor modification of an argument from Bufetov and Solomyak (2014) yields very weak, but nevertheless quantitative weak mixing estimates of type for the {\em self-similar} suspension flow over a Salem substitution. Simultaneously this provides first quantitative decay rates for the Fourier transform of Salem Bernoulli convolutions.
Paper Structure (17 sections, 15 theorems, 111 equations)

This paper contains 17 sections, 15 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Background and statement of results
  3. Suspension flows.
  4. Spectral theory: general flows
  5. Spectral theory: substitution flows
  6. Statement of results
  7. Proof of the Theorem \ref{['th:main']}
  8. The set-up
  9. The main proposition
  10. Conclusion of the proof: "glueing" near zero and at infinity
  11. Proof of lower bounds (no Hölder QWM for any suspension flow)
  12. Some algebraic number theory (see e.g. Bug04)
  13. Proof of Theorem \ref{['th:noQWM']}
  14. $\log^*$ quantitative weak mixing for the self-similar suspension flow
  15. On the Fourier decay for Salem Bernoulli convolutions
  16. ...and 2 more sections

Key Result

Lemma 2.1

For all $\omega\in{\mathbb R}$ and $r\in (0,\frac{1}{2}]$ we have

Theorems & Definitions (27)

  • Remark 1.1
  • Lemma 2.1
  • Proposition 2.2: BuSo18b
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • ...and 17 more