Substitution invariant sturmian bisequences

Les suites sturmiennes indexées sur $\mathbb{Z}$, de pente $\alpha$ et d’intercept $\rho$, sont laissées fixes par une substitution non triviale si et seulement si $\alpha$ est un nombre de Sturm et $\rho$ appartient à $\mathbb{Q}\left(\alpha \right)$. On remarque aussi que les suites de Beatty permettent de définir des partitions de l’ensemble des entiers relatifs.

We prove that a Sturmian bisequence, with slope $\alpha$ and intercept $\rho$, is fixed by some non-trivial substitution if and only if $\alpha$ is a Sturm number and $\rho$ belongs to $\mathbb{Q}\left(\alpha \right)$. We also detail a complementary system of integers connected with Beatty bisequences.

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