Séminaire de Théorie Analytique des Nombres et Problèmes Diophantiens

Le 1er février 2001

Teturo Kamae


Sequence entropy and *Sturmian words.


 
 Résumé : For an infinite word $\alpha=\alpha_0\alpha_1\alpha_2\cdots$ over a finite alphabet A we define the *complexity by

\begin{displaymath}C_\alpha^*(k)=\sup_\tau\sharp\{\alpha_{n+\tau(0)} \alpha_{n+\tau(1)}\cdots\alpha_{n+\tau(k-1)};~n=0,1,2,\cdots\} \end{displaymath}

where the ``sup'' is taken over all subsequences $0=\tau(0)<\tau(1)<\cdots<\tau(k-1)$ of integers. We prove that $\alpha$ is ultimately periodic if $C_\alpha^*(k)\le 2k-1$ for some k. Infinite words $\alpha$ with $C_\alpha^*(k)=2k$for any k are called a *Sturmian words and are discussed.
 
 

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