Invertible substitutions and Sturmian sequences

On substitution invariant Sturmian words: an application of Rauzy fractals

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...

Bidimensional Sturmian Sequences and Substitutions

A Characterization of Sturmian Morphisms

A one-sided infinite word is balanced if the difference of the number of occurrences of a letter in two factors of the same length never exceeds one. It is Sturmian if it is balanced and not ultimately periodic. Sturmian words have a long history. A clear exposition of early work by J. Bernoulli, Christoffel, and A. A. Markov is given in the book by Venkov [22]. The term "Sturmian" has been use...

Decomposition theorem for invertible substitutions on three - letter alphabet ∗

We study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as Iw ◦ σ1 ◦ σ2 ◦ · · · ◦ σk, where Iw is the inner automorphism associated with w, and σj ∈ S for 1 ≤ j ≤ k. As a consequence, M is the matrix of an invertible substitution if and only if it is a f...

Maximal pattern complexity for discrete systems

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...