The coincidence of R\'enyi-Parry measures for β-transformation
Abstract
We present a complete characterization of two different non-integers with the same R\'enyi-Parry measure. We prove that for two non-integers β1 ,β2 >1, the R\'enyi-Parry measures coincide if and only if β1 is the root of equation x2-qx-p=0, where p,q∈N with p≤ q, and β2 = β1 + 1, which confirms a conjecture of Bertrand-Mathis in [Section III]Bertrand-1998.
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