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
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