A Generalization of Sturmian Sequences: Combinatorial Structure and Transcendence

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We investigate a class of minimal sequences on a finite alphabet Ak = {1,2,...,k} having (k - 1)n + 1 distinct subwords of length n. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of binary Sturmian sequences. We describe two simple combinatorial algorithms for constructing characteristic Arnoux-Rauzy sequences (one of which is new even in the Sturmian case). 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+ε and, in the Sturmian case, arbitrarily ... continued below

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iii, 45 leaves

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Risley, Rebecca N. August 1998.

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  • Risley, Rebecca N.

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We investigate a class of minimal sequences on a finite alphabet Ak = {1,2,...,k} having (k - 1)n + 1 distinct subwords of length n. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of binary Sturmian sequences. We describe two simple combinatorial algorithms for constructing characteristic Arnoux-Rauzy sequences (one of which is new even in the Sturmian case). 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+ε and, in the Sturmian case, arbitrarily large subwords of the form V^3+ε. Finally, we prove that an irrational number whose base b-digit expansion is an Arnoux-Rauzy sequence is transcendental.

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iii, 45 leaves

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  • August 1998

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  • March 24, 2014, 8:07 p.m.

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  • May 29, 2015, 6:21 p.m.

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Risley, Rebecca N. A Generalization of Sturmian Sequences: Combinatorial Structure and Transcendence, thesis, August 1998; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc278440/: accessed November 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .