Bidimensional Sturmian Sequences and Substitutions
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Invertible substitutions and Sturmian sequences
It is known on the one hand that a Sturmian sequence can be generated geometrically by the intersections of a straight line with the unit grid in the plane, and on the other hand that fixed points of invertible substitutions are Sturmian. We give a new characterization of invertible substitutions, which allows to determine the straight line which generates the fixed point of a given invertible ...
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For an infinite word α = α0α1α2 · · · over a finite alphabet, Kamae introduced a new notion of complexity called maximal pattern complexity defined by pα(k) := sup τ ]{αn+τ(0)αn+τ(1) · · ·αn+τ(k−1); n = 0, 1, 2, · · · } where the supremum is taken over all sequences of integers 0 = τ(0) < τ(1) < · · · < τ(k − 1) of length k. Kamae proved that α is aperiodic if and only if pα(k) ≥ 2k for every k...
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We provide a complete characterization of substitution invariant inhomogeneous bidirectional pointed Sturmian sequences. The result is analogous to that obtained by Berthé et al. [5] and Yasutomi [21] for one-directional Sturmian words. The proof is constructive, based on the geometric representation of Sturmian words by a cut-and-project scheme.
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Sturmian words are infinite words that have exactly n+ 1 factors of length n for every positive integer n. A Sturmian word sα,ρ is also defined as a coding over a two-letter alphabet of the orbit of point ρ under the action of the irrational rotation Rα : x 7→ x+ α (mod 1). A substitution fixes a Sturmian word if and only if it is invertible. The main object of the present paper is to investiga...
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