Some characterizations of finite Sturmian words (original) (raw)

The number of runs in Sturmian words

Denote by S the class of standard Sturmian words. It is a class of highly compressible words extensively studied in combinatorics of words, including the well known Fibonacci words. The suffix automata for these words have a very particular structure. This implies a simple characterization (described in the paper by the Structural Lemma) of the periods of runs (maximal repetitions) in Sturmian words. Using this characterization we derive an explicit formula for the number ρ(w) of runs in words w ∈ S, with respect to their recurrences (directive sequences). We show that ρ(w) |w| ≤ 4 5 for each w ∈ S, and there is an infinite sequence of strictly growing words w k ∈ S such that lim k→∞ ρ(w k ) |w k | = 4 5 . The complete understanding of the function ρ for a large class S of complicated words is a step towards better understanding of the structure of runs in words. We also show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation (recurrences describing the word). This is an example of a very fast computation on texts given implicitly in terms of a special grammar-based compressed representation (usually of logarithmic size with respect to the explicit text).

Lyndon factorization of sturmian words

Discrete Mathematics, 2000

We express any general characteristic sturmian word as a unique infinite non-increasing product of Lyndon words. Using this identity, we give a new ω-division for characteristic sturmian words. We also give a short proof of a result by Berstel and de Luca (Sturmian words, Lyndon words and trees, Theoret. Comput. Sci., 178 (1997) 171–203.); more precisely, we show that the set of factors of sturmian words that qualify as Lyndon words is the set of primitive Christoffel words.

ON THE WORDS BY kkk TO kkk INSERTION OF A LETTER IN STURMIAN WORDS

International Journal of Apllied Mathematics

We study the classical complexity of k to k insertion words of a letter in Sturmian words. Then, we determine the Abelian complexity and palindromic complexity of these words. Finally, we show that the k to k insertion of a letter x in Sturmian words preserves the palindromic richness of Sturmian words if and only if k = 1.

Initial Powers of Sturmian Words

2001

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 p , 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 2+ε 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.

Usefulness of directed acyclic subword graphs in problems related to standard Sturmian words

The class of finite Sturmian words consists of words having particularly simple compressed representation, which is a generalization of the Fibonacci recurrence for Fibonacci words. The subword graphs of these words (especially their compacted versions) have a very special regular structure. In this paper we investigate this structure in more detail than in the previous papers and show how several syntactical properties of Sturmian words follow from their graph properties. Consequently simple alternative graph-based proofs of several known facts are presented. The very special structure of subword graphs leads also to special easy algorithms computing some parameters of Sturmain words: the number of subwords, the critical factorization point, lexicographically maximal suffixes, occurrences of subwords of a fixed length, and right special factors. These algorithms work in linear time with respect to n, the size of the compressed representation of the standard word, though the words themselves can be of exponential size with respect to n. Some of the computed parameters can be also of exponential size, however we provide their linear size compressed representations. This gives more examples of fast computations for highly compressed words. We introduce also a new concept related to standard words: Ostrowski automata.

Sturmian words with balanced construction

In this paper we define Sturmian words with balanced construction. We formulate a fixed-point theorem for Sturmian words and analyze the set of all fixed points. The inspiration for this work came from the Kolakoski word and the general idea of self-reading sequences by Pȃun and Salomaa. The basis for this article is the author's earlier research on the influence of the continued fraction elements in the expansion of a ∈ ]0, 1[\Q on the construction of runs for the upper mechanical word with slope a and intercept 0.

Generalized balances in Sturmian words

Discrete Applied Mathematics, 2002

An inÿnite word x on the {0; 1} alphabet is balanced if, given two factors of x; w and w , having the same length, the di erence between the number of 0's in w (denoted by |w|0) and the number of 0's in w is at most 1, i.e. ||w|0 − |w |0| 6 1. It is well known that an aperiodic word is Sturmian if and only if it is balanced.