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Yan Huang, Zhiqiang Wang, The Coincidence of Rényi–Parry Measures for β-Transformation, International Mathematics Research Notices, Volume 2025, Issue 8, April 2025, rnaf089, https://doi.org/10.1093/imrn/rnaf089
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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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