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Journal Articles Acta Arithmetica Year : 2006

Initial powers of Sturmian sequences

Valerie Berthe

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.
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Dates and versions

lirmm-00123046 , version 1 (18-03-2009)

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Valerie Berthe, Charles Holton, Luca Q. Zamboni. Initial powers of Sturmian sequences. Acta Arithmetica, 2006, 122, pp.315-347. ⟨10.4064/aa122-4-1⟩. ⟨lirmm-00123046⟩
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