## A simple characterization of Sturm numbers. (Une caractérisation simple des nombres de Sturm.)(French)Zbl 0930.11051

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 B. Parvaix [J. Théor. Nombres Bordx. 9, 351-369 (1997; Zbl 0904.11008)], and first studied by D. Crisp, W. Moran, A. Pollington and 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].

### MSC:

 11J70 Continued fractions and generalizations 37B10 Symbolic dynamics 11B85 Automata sequences 68R15 Combinatorics on words

### Citations:

Zbl 0904.11008; Zbl 0786.11041
Full Text:

### References:

 [1] Berstel, J., Recent results in sturmian words. Developments in Language Theory II, pp. 13-24, World Sci. Publishing, River Edge, NJ, 1996. · Zbl 1096.68689 [2] Berthé, V., Fréquences des facteurs des suites sturmiennes. Theoret. Comput. Sci.165 (1996), 295-309. · Zbl 0872.11018 [3] Crisp, D., Moran, W., Pollington, A., Shuie, P., Substitution invariant cutting sequences. J. Théor. Nombres Bordeaux5 (1993), 123-137. · Zbl 0786.11041 [4] Parvaix, B., Propriétés d’invariance des mots sturmiens. J. Théor. Nombres Bordeaux9 (1997), 351-369. · Zbl 0904.11008 [5] Redmond, D., Number Theory: An Introduction, chapter 4. pp. 210-235. Monographs and Textbooks in Pure and Applied Mathematics, 201. Marcel Dekker, Inc., New York, 1996. · Zbl 0847.11001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.