Initial Powers of Sturmian Sequences
نویسندگان
چکیده
We investigate powers of prefixes in Sturmian sequences. We obtain 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 u, in terms of its S-adic representation. This formula is based on Ostrowski’s numeration system. We characterize those irrational slopes α of which there exists a Sturmian sequence ω beginning in only finitely many words of the form u for every fixed ε > 0, 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.
منابع مشابه
Characterization of Repetitions in Sturmian Words: A New Proof
We present a new, dynamical way to study powers (or repetitions) in Sturmian words based on results from Diophantine approximation theory. As a result, we provide an alternative and shorter proof of a result by Damanik and Lenz characterizing powers in Sturmian words [Powers in Sturmian Sequences, European J. Combin. 24 (2003), 377–390]. Further, as a consequence, we obtain a previously known f...
متن کاملOn powers of words occurring in binary codings of rotations
We discuss combinatorial properties of a class of binary sequences generalizing Sturmian sequences and obtained as a coding of an irrational rotation on the circle with respect to a partition in two intervals. We give a characterization of those having a finite index in terms of a two-dimensional continued fraction like algorithm, the so-called D-expansion. Then, we discuss powers occurring at ...
متن کاملA Note on the Return Time of Sturmian Sequences
Let Rn be the the first return time to its initial nword. Then the Ornstein-Weiss first return time theorem implies that log Rn divided by n converges to entropy. We consider the convergence of log Rn for Sturmian sequences which has the lowest complexity. In this case, we normalize the logarithm of the first return time by log n. We show that for any numbers 1 ≤ α, β ≤ ∞, there is a Sturmian s...
متن کامل