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Modulus of continuity for spectral measures of suspension flows over Salem type substitutions

Juan Marshall-Maldonado

Abstract

We study the spectrum of the self-similar suspension flows of sub-shifts arising from primitive substitutions. We focus on the case where the substitution matrix has a Salem number α as dominant eigenvalue. We obtain a Hölder exponent for the spectral measures for points away from zero and belonging to the field Q(α). This exponent depends only on three parameters of each of these points: its absolute value, the absolute value of its real conjugate and its denominator.

Modulus of continuity for spectral measures of suspension flows over Salem type substitutions

Abstract

We study the spectrum of the self-similar suspension flows of sub-shifts arising from primitive substitutions. We focus on the case where the substitution matrix has a Salem number α as dominant eigenvalue. We obtain a Hölder exponent for the spectral measures for points away from zero and belonging to the field Q(α). This exponent depends only on three parameters of each of these points: its absolute value, the absolute value of its real conjugate and its denominator.

Paper Structure

This paper contains 14 sections, 22 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Dynamical systems arising from substitutions
  4. Spectral theory
  5. Salem numbers
  6. Polynomial approximation of functions
  7. Bessel functions
  8. Outline of the proof
  9. Proof of Theorem \ref{['theorem1']} and Theorem \ref{['theorem2']}
  10. Preliminary results
  11. Conclusion
  12. Proof of Proposition \ref{['lattice']}
  13. Dependence of $r_0$
  14. Appendix

Key Result

Theorem 1.1

Let $\zeta$ be a Salem type, aperiodic and primitive substitution on $\mathcal{A}$, $\alpha$ its Perron-Frobenius eigenvalue and $\vec{p}$ the positive (left-)eigenvector of the substitution matrix. Let $\mathfrak{X}^{\vec{p}}_{\zeta}$ be the corresponding self-similar suspension flow and for any $a for all $0< r < r_0$ and $a\in\mathcal{A}$.

Figures (1)

  • Figure 1: Graph of the function $F(x) = \mathds{1}_{[\delta,1-\delta]}(-4\cos(2\pi x))$ for $\delta = 0.2$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Example 2.1: see holton1998geometric
  • Example 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Proposition 2.6: bufetov2014modulus, Lemma 4.3
  • ...and 31 more