The coincidence of R\'{e}nyi-Parry measures for $\beta$-transformation

TL;DR

This paper characterizes when two non-integer bases have identical Rényi-Parry measures, proving a specific algebraic condition involving quadratic roots and confirming a conjecture by Bertrand-Mathis.

Contribution

It provides a complete characterization of non-integer bases with identical Rényi-Parry measures, confirming a conjecture and identifying a precise algebraic condition.

Findings

Rényi-Parry measures coincide if and only if bases satisfy a quadratic equation

The bases differ by exactly one, i.e., $eta_2 = eta_1 + 1$

The result confirms Bertrand-Mathis's conjecture

Abstract

We present a complete characterization of two different non-integers with the same R\'{e}nyi-Parry measure. We prove that for two non-integers $\beta_1 ,\beta_2 >1$, the R\'{e}nyi-Parry measures coincide if and only if $\beta_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $\beta_2 = \beta_1 + 1$, which confirms a conjecture of Bertrand-Mathis in \cite[Section III]{Bertrand-1998}.