Abstract

We present a complete characterization of two different non-integers with the same Rényi–Parry measure. We prove that for two non-integers |$\beta _{1},\beta _{2}>1$|⁠, 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].

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