A simple characterization of Sturm numbers
A real number is called a Sturm number if it is the slope of a Sturmian sequence (binary coding of a billiard sequence on the square), which is a fixed point of a morphism. These numbers were called ``nombres de Sturm'' by \textit{B. Parvaix} [J. Théor. Nombres Bordx. 9, 351-369 (1997; Zbl 0904.11008)], and first studied by \textit{D. Crisp}, \textit{W. Moran}, \textit{A. Pollington} and \textit{P. Shiue} [J. Théor. Nombres Bordx. 5, 123-137 (1993; Zbl 0786.11041)], who gave the following characterization for the characteristic Sturmian sequences (billiard sequences starting from the origin) that are fixed point of a morphism: they are exactly the sequences with slope \(x:[\alpha_0,\alpha_1,\alpha_2\dots]\) where \[ \begin{alignedat}{3} \text{if }&x>1 &\quad &x=[a_0,\overline{a_1,\dots,a_n}]&\quad &\text{with }1\leq a_0\leq a_n\\ \text{if }&0<x<1 &\quad &x=[0,a_0,\overline{a_1,\dots,a_n}]&\quad &\text{with }a_0\leq a_n\\ \text{or} \text{if }&\tfrac 12<x<1 &\quad &x=[0,1,a_0,\overline{a_1,\dots,a_n}]&\quad &\text{with }a_0\leq a_n\\ \text{if }&0<x<\tfrac 12 &\quad &x=[0,1,a_0,\overline{a_1,\dots,a_n}]&\quad &\text{with }1\leq a_0\leq a_n.\end{alignedat} \] In the paper under review a simple arithmetic condition is given: either \(x\) is quadratic with a negative conjugate, or \(x\) is quadratic in \(]0,1[\) with a conjugate outside \(]0,1[\). Note that ``Shiue'' is misspelled in Ref. [3].
scientific article; zbMATH DE number 1307383
scientific article