A generalization of Sturmian sequences: Combinatorial structure and transcendence
Rebecca Risley, Luca Quardo Zamboni · Acta Arithmetica · 2000
In this paper 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 give two combinatorial algorithms for constructing characteristic Arnoux-Rauzy sequences. The first method, which is the central idea of the paper, involves a simple combinatorial algorithm for constructing all bispecial words. This description is new even in the Sturmian case. The second is a S-adic description of the characteristic sequence similar to that given by Arnoux and Rauzy for k = 2, 3. Arnoux-Rauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every Arnoux-Rauzy sequence contains arbitrarily large subwords of the form V 2+ɛ and in the Sturmian case arbitrarily large subwords of the form V 3+ɛ. Combined with a recent combinatorial version 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, is transcendental. This yields a class of transcendental numbers of arbitrarily large linear complexity. I