The Coincidence of Rényi–Parry Measures for <i>β</i>-Transformation

Abstract We present a complete characterization of two different non-integers with the same Rényi–Parry measure. We prove that for two non-integers $\beta _{1},\beta _{2}&amp;gt;1$, the Rényi–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 [1, Section III].