The Coincidence of Rényi–Parry Measures for β-Transformation

Abstract: We present a complete characterization of two 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 [A. Bertrand-Mathis, Acta Math. Hungar. 78, no. 1-2 (1998):71–78].

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

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