Return time complexity of Sturmian sequences

A 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...

Language Complexity of Rotations and Sturmian Sequences

Given a rotation of the circle, we study the complexity of formal languages that are generated by the itineraries of interval covers. These languages are regular iff the rotation is rational. In the case of irrational rotations, our study reduces to that of the language complexity of the corresponding Sturmian sequences. We show that for a large class of irrationals, including e, all quadratic ...

Powers in Sturmian sequences

Initial Powers of Sturmian Sequences

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 irrationa...

The Index of Sturmian Sequences