In this paper we undertake the general study of the Abelian complexity of an infinite word on a finite alphabet. We investigate both similarities and differences between the Abelian complexity and the usual subword complexity. While the Thue-Morse minimal subshift is neither characterized by its Abelian complexity nor by its subword complexity alone, we show that the subshift is completely characterized by the two complexity functions together. We give an affirmative answer to an old question of G. Rauzy by exhibiting a class of words whose Abelian complexity is everywhere equal to 3. We also investigate links between Abelian complexity and the existence of Abelian powers. Using van der Waerden's Theorem, we show that any minimal subshift having bounded Abelian complexity contains Abelian k-powers for every positive integer k. In the case of Sturmian words we prove something stronger: For every Sturmian word ω and positive integer k, each sufficiently long factor of ω begins in an Abelian k-power.

G. Rauzy showed that the Tribonacci minimal subshift generated by the morphism $\tau: 0\mapsto 01, 1\mapsto 02 and 2\mapsto 0$ is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in $R^2$, each domain being translated by the same vector modulo a lattice. In this paper we study the Abelian complexity AC(n) of the Tribonacci word $t$ which is the unique fixed point of $\tau$. We show that $AC(n)\in {3,4,5,6,7}$ for each $n\geq 1$, and that each of these five values is assumed. Our proof relies on the fact that the Tribonacci word is 2-balanced, i.e., for all factors $U$ and $V$ of $t$ of equal length, and for every letter $a \in {0,1,2}$, the number of occurrences of $a$ in $U$ and the number of occurrences of $a$ in $V$ differ by at most 2. While this result is announced in several papers, to the best of our knowledge no proof of this fact has ever been published. We offer two very different proofs of the 2-balance property of $t$. The first uses the word combinatorial properties of the generating morphism, while the second exploits the spectral properties of the incidence matrix of $\tau$.Comment: 20 pages, 1 figure. This is an extended version of 0904.2872v

The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian k-powers for some integer k, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian k-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian k-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo-Varricchio and Halbeisen-Hungerbühler.

Abstract. We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a Sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a byproduct of our results, we give several new proofs and tightenings of well-known properties of Sturmian words.Mathematics Subject Classification. 68R15.

ABSTRACT. It is a fundamental property of non-letter Lyndon words that they can be expressed as a concatenation of two shorter Lyndon words. This leads to a naive lower bound ⌈log 2 (n)⌉ + 1 for the number of distinct Lyndon factors that a Lyndon word of length n must have, but this bound is not optimal. In this paper we show that a much more accurate lower bound is ⌈log φ (n)⌉ + 1, where φ denotes the golden ratio (1 + √ 5)/2. We show that this bound is optimal in that it is attained by the Fibonacci Lyndon words. We then introduce a mapping Lx that counts the number of Lyndon factors of length at most n in an infinite word x. We show that a recurrent infinite word x is aperiodic if and only if Lx ≥ L f , where f is the Fibonacci infinite word, with equality if and only if x is in the shift orbit closure of f .

No abstract

No abstract

No abstract