## A generalization of Sturmian sequences: Combinatorial structure and transcendence.(English)Zbl 0953.11007

Summary: We study dynamical properties of a class of uniformly recurrent sequences on a $$k$$-letter alphabet with complexity $$p(n)=(k-1)n+1.$$ These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of the (binary) Sturmian sequences of Morse and Hedlund. We describe two combinatorial methods for constructing characteristic Arnoux-Rauzy sequences. The first, which is the central idea of the paper, is a simple combinatorial algorithm for generating the bispecial words. This description seems new even in the Sturmian case. The second is a ‘dual’ reformulation of an algorithm due to Arnoux and Rauzy; we use it to obtain a characterization of primitive morphic Arnoux-Rauzy sequences.
As an application of the first combinatorial description, we show that every Arnoux-Rauzy sequence contains arbitrarily large subwords of the form $$V^{2+\epsilon }$$ and in the Sturmian case arbitrarily large subwords of the form $$V^{\frac {5+\sqrt 5}{2}}.$$
Using a recent combinatorial interpretation of Ridout’s Theorem due to S. Ferenczi and C. Mauduit, we prove that an irrational number whose base $$b$$-digit expansion is an Arnoux-Rauzy sequence on $$k\geq 2$$ letters, is transcendental. This result is an extension of a theorem of Ferenczi and Mauduit for $$k\in \{2,3\},$$ and yields a class of transcendental numbers of complexity $$p(n)=(k-1)n+1$$.

### MSC:

 11B85 Automata sequences 68R15 Combinatorics on words 11J91 Transcendence theory of other special functions
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