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Pisot Substitution Conjecture and Rauzy Fractals

Kentaro Nakaishi

Abstract

We provide a proof for one version of Pisot conjecture. We make use of the weak mixing property of the subshift of finite type derived from the prefix-suffix automaton to conclude that the substitution dynamical system has pure discrete spectrum.

Pisot Substitution Conjecture and Rauzy Fractals

Abstract

We provide a proof for one version of Pisot conjecture. We make use of the weak mixing property of the subshift of finite type derived from the prefix-suffix automaton to conclude that the substitution dynamical system has pure discrete spectrum.
Paper Structure (15 sections, 28 theorems, 166 equations)

This paper contains 15 sections, 28 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Substitution
  3. Subshift Of Finite Type
  4. Properties Of $A$
  5. Parry Measure And Weak Mixing Subshift
  6. Rauzy Fractals From The Adelic Viewpoint
  7. Multiple Tiling Of $K_{\sigma}$
  8. Perturbation
  9. Special Cylinder
  10. Polynomial-like Behavior
  11. Ergodic part
  12. First Entry To $\mathcal{C}$ Along A Time Series
  13. Non-zero Polynomial Parts Distort The Distribution Of $\tau_{2}$
  14. Consistency Between Classical and Modern Methods
  15. Strong Coincidence Conjecture

Key Result

Lemma 1

$A$ is irreducible and aperiodic ($A^{N+1}>0$).

Theorems & Definitions (67)

  • Definition 1
  • Example 1: Rauzy substitution rauzy1982nombres
  • Remark 3.1
  • Definition 2
  • Remark 3.2
  • Example 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • proof
  • ...and 57 more