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Initial powers of Sturmian sequences

Valerie Berthe 1 Charles Holton 2 Luca Q. Zamboni 2
1 ARITH - Arithmétique informatique
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
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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Valerie Berthe, Charles Holton, Luca Q. Zamboni. Initial powers of Sturmian sequences. Acta Arithmetica, Instytut Matematyczny PAN, 2006, 122, pp.315-347. ⟨10.4064/aa122-4-1⟩. ⟨lirmm-00123046⟩

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