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Initial powers of Sturmian sequences. (English) Zbl 1117.37005

This paper deals with powers of factors occurring at the beginning of Sturmian sequences. Indeed, it is known that any Sturmian sequence begins in infinitely many squares. However, the characteristic Sturmian sequence with slope the golden ratio begins in no \((3 + \sqrt5)/2\) powers at all, while every sequence outside the shift orbit of this sequence begins in arbitrary long words repeated at least three times. Then, the authors define the initial critical exponent of an infinite sequence \(w\) (denoted by ice\((w)\)) as the supremum \(p\) of all real numbers for which there exist arbitrarily long prefixes \(u\) of \(w\) such that \(u^p\) is also a prefix of \(w\). The authors exhibit an explicit computation for ice\((w)\) (Proposition 3.3). This formula involves the digits of the sequence into an expansion related to the Ostrowski numeration system. All along the paper, the authors investigate and make use of the relation between ice\((w)\) and the already studied \(\text{ind}^{*}(w)\), that is, the limit superior of powers of longer and longer words appearing in a sequence. This allows to find explicit lower bounds for ice\((w)\) for some specific families of Sturmian sequences. Finally, the authors give a characterization of a sequence with the smallest ice, that is, ice\((w)=2\) and they exhibit an explicit example for such a sequence.

MSC:

37B10 Symbolic dynamics
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11J70 Continued fractions and generalizations
37A25 Ergodicity, mixing, rates of mixing
68R15 Combinatorics on words
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