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arXiv · 2501.08116

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

Yan Huang·Zhiqiang Wang
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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}.

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Yan Huang, Zhiqiang Wang. 2025-01-14. The coincidence of R\'{e}nyi-Parry measures for $\beta$-transformation. https://arxiv.org/abs/2501.08116

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