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The coincidence of Rényi-Parry measures for $β$-transformation

Yan Huang, Zhiqiang Wang

TL;DR

The paper characterizes all pairs of non-integer $β>1$ for which the Rényi-Parry measures of the corresponding $β$-transformations coincide. It analyzes the density function $h_{β}$ of the Rényi-Parry measure and the orbit structure $\mathcal{O}_{β}$ of $1$ under $T_{β}$ to derive necessary and sufficient conditions. The main result shows that $ν_{β_1} = ν_{β_2}$ with $β_1\neq β_2$ occurs if and only if $β_1$ is the root of $x^2 - qx - p = 0$ with $p\leq q$ in $\mathbb{N}$ and $β_2 = β_1 + 1$, in which case $β_1$ and $β_2$ are Pisot numbers of degree $2$. The proof combines explicit density computations for sufficiency with a detailed orbit-structure-based necessity argument, thereby confirming Bertrand-Mathis's conjecture and illuminating measure rigidity phenomena for β-transformations.

Abstract

We present a complete characterization of two different non-integers with the same Rényi-Parry measure. We prove that for two non-integers $β_1 ,β_2 >1$, the Rényi-Parry measures coincide if and only if $β_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $β_2 = β_1 + 1$, which confirms a conjecture of Bertrand-Mathis in \cite[Section III]{Bertrand-1998}.

The coincidence of Rényi-Parry measures for $β$-transformation

TL;DR

The paper characterizes all pairs of non-integer for which the Rényi-Parry measures of the corresponding -transformations coincide. It analyzes the density function of the Rényi-Parry measure and the orbit structure of under to derive necessary and sufficient conditions. The main result shows that with occurs if and only if is the root of with in and , in which case and are Pisot numbers of degree . The proof combines explicit density computations for sufficiency with a detailed orbit-structure-based necessity argument, thereby confirming Bertrand-Mathis's conjecture and illuminating measure rigidity phenomena for β-transformations.

Abstract

We present a complete characterization of two different non-integers with the same Rényi-Parry measure. We prove that for two non-integers , the Rényi-Parry measures coincide if and only if is the root of equation , where with , and , which confirms a conjecture of Bertrand-Mathis in \cite[Section III]{Bertrand-1998}.
Paper Structure (2 sections, 10 theorems, 42 equations, 1 figure)

This paper contains 2 sections, 10 theorems, 42 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main-result']}

Key Result

Theorem 1

Hochman-Shmerkin-2015 Let $\beta_1, \beta_2 > 1$ with $\beta_1\nsim \beta_2$ and $\beta_1$ a Pisot number. If $\mu$ is jointly invariant under $T_{\beta_1},T_{\beta_2}$, and if all ergodic components of $\mu$ under $T_{\beta_2}$ have positive entropy, then $\mu$ is the common Rényi-Parry measure for

Figures (1)

  • Figure 1: The $\beta$-transformation $T_{\beta}(x)$ and density function $\widetilde{h}_\beta(x)$ for $\beta=\frac{1+\sqrt 5}{2}$.

Theorems & Definitions (17)

  • Theorem
  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • ...and 7 more