Abstract : In this paper we investigate powers of prefixes of Sturmian sequences. We give an explicit formula for ice(ω), the initial critical exponent of a Sturmian sequence ω, defined as the supremum of all real numbers p > 0 for which there exist arbitrary long prefixes of ω of the form up, in terms of its S-adic representation. This formula is based on Ostrowski's numeration system. Furthermore we characterize those irrational slopes α of which there exists a Sturmian sequence ω beginning in only finitely many powers of 2 + ε, that is for which ice(ω) = 2. In the process we recover the known results for the index (or critical exponent) of a Sturmian sequence. We also focus on the Fibonacci Sturmian shift and prove that the set of Sturmian sequences with ice strictly smaller than its everywhere value has Hausdorff dimension 1.
Type de document :
Article dans une revue
Acta Arithmetica, Instytut Matematyczny PAN, 2006, 122, pp.315-347
https://hal-lirmm.ccsd.cnrs.fr/lirmm-00123046
Contributeur : Valerie Berthe
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Soumis le : mercredi 18 mars 2009 - 11:30:43
Dernière modification le : jeudi 11 janvier 2018 - 06:26:07
Document(s) archivé(s) le : mercredi 7 avril 2010 - 01:54:19
