The coincidence of Rényi-Parry measures for $\beta$-transformation

We present a complete characterization of two different non-integers with the same Rényi-Parry measure. We prove that for two non-integers ${\beta }_1,{\beta }_2>1$, 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 $p\le q$, and ${\beta }_2={\beta }_1+1$, which confirms a conjecture of Bertrand-Mathis in \cite[Section III]{Bertrand-1998}.