Factor Complexity of Infinite Words Associated with Non-Simple Parry Numbers

Abstract

The factor complexity of the infinite word u β canonically associated to a non-simple Parry number β is studied. Our approach is based on the notion of special factors introduced by Berstel and Cassaigne. At first, we give a handy method for determining infinite left special branches; this method is applicable to a broad class of infinite words which are fixed points of a primitive substitution. In the second part of the article, we focus on infinite words u β only. To complete the description of its special factors, we define and study (a, b)-maximal left special factors. This enables us to characterize non-simple Parry numbers β for which the word u β has affine complexity.

Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words

“…However, our result can indicate that there are connections between them, for the present waiting for their discovery. It has been recently shown in [12] that a fixed point of a canonical substitution associated with a non-simple cubic Parry number has affine factor complexity if and only if it belongs just to the class with which we have dealt in this paper. Therefore, briefly speaking, "if the factor complexity is affine, then the Abelian complexity has the optimal bound 7" holds in the cubic non-simple Parry case.…”

“…We deduce from Eqs. (12), (13), from |z (j) | ≥ p 2 + p − 1 and from the minimality of n that |ŵ (j) | ≥ p 2 + p, j = 1, 2.…”

“…These substitutions are canonical substitutions associated with cubic Pisot numbers β > 1, roots of polynomials x 3 − px 2 − x+ 1 (cf. [12]). Let us recall that the Tribonacci substitution is associated with a numeration system as well.…”

Critical exponent of infinite words coding beta-integers associated with non-simple Parry numbers

“…Corollary 4 claims that to get the critical exponent of u β , it suffices to go through all BS factors v and corresponding w (if it exists) such that (v, w) ∈ B(u β ). In what follows, we will take advantage of having described all BS factors of u β in [9]. In order to present the necessary results we need some more sophisticated notation for BS factors.…”

Abelian complexity and abelian co-decomposition

“…For example, it is known that the abelian complexity of the Tribonacci word t (recall that t is the fixed point of the substitution 0 → 01, 1 → 02, 2 → 0) satisfies AC t (n) ∈ {3, 4, 5, 6, 7} for all n, but only for the values 3 and 7 it is proved that they are attained infinitely many times, see [5]. Similarly, for u (p) being the fixed point of the substitution L → L p S, S → M , M → L p−1 S for an arbitrary p ≥ 2, it has been proved AC u (p) (n) ∈ {3, 4, 5, 6, 7}, but so far only the value 7 is known to be attained infinitely many times [6] (for additional information on those words see [8,9]).…”