Multidimensional Generalized Automatic Sequences and Shape-symmetric Morphic Words

An infinite word is S-automatic if, for all n>=0, its (n + 1)st letter is the output of a deterministic automaton fed with the representation of n in the considered numeration system S. In this extended abstract, we consider an analogous definition in a multidimensional setting and present the connection to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for d>=2, we state that a multidimensional infinite word x : N^d \to \Sigma over a finite alphabet \Sigma is S-automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word

Asymptotic properties of free monoid morphisms

Motivated by applications in the theory of numeration systems and recognizable sets of integers, this paper deals with morphic words when erasing morphisms are taken into account. Cobham showed that if an infinite word $w =g(f^\omega(a))$ is the image of a fixed point of a morphism $f$ under another morphism $g$, then there exist a non-erasing morphism $\sigma$ and a coding $\tau$ such that $w =\tau(\sigma^\omega(b))$. Based on the Perron theorem about asymptotic properties of powers of non-negative matrices, our main contribution is an in-depth study of the growth type of iterated morphisms when one replaces erasing morphisms with non-erasing ones. We also explicitly provide an algorithm computing $\sigma$ and $\tau$ from $f$ and $g$.Comment: 25 page

Abstract numeration systems on bounded languages and multiplication by a constant

A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with at least three letters, we show that multiplication by an integer $\lambda\ge2$ does not preserve $S$-recognizability, meaning that there always exists a $S$-recognizable set $X$ such that $\lambda X$ is not $S$-recognizable. The main tool is a bijection between the representation of an integer over a bounded language and its decomposition as a sum of binomial coefficients with certain properties, the so-called combinatorial numeration system

Abelian bordered factors and periodicity

A finite word u is said to be bordered if u has a proper prefix which is also a suffix of u, and unbordered otherwise. Ehrenfeucht and Silberger proved that an infinite word is purely periodic if and only if it contains only finitely many unbordered factors. We are interested in abelian and weak abelian analogues of this result; namely, we investigate the following question(s): Let w be an infinite word such that all sufficiently long factors are (weakly) abelian bordered; is w (weakly) abelian periodic? In the process we answer a question of Avgustinovich et al. concerning the abelian critical factorization theorem.Comment: 14 page

Structural properties of bounded languages with respect to multiplication by a constant

peer reviewedWe consider the preservation of recognizability of a set of integers after multiplication by a constant for numeration systems built over a bounded language. As a corollary we show that any nonnegative integer can be written as a sum of binomial coefficients with some prescribed properties

Multidimensional generalized automatic sequences and shape-symmetric morphic words

peer reviewedAn infinite word is S-automatic if, for all n ≥ 0, its (n + 1)st letter is the output of a deterministic automaton fed with the representation of n in the considered numeration system S. In this paper, we consider an analogous definition in a multidimensional setting and study the relationship with the shape-symmetric infinite words as introduced by Arnaud Maes. Precisely, for d ≥ 2, we show that a multidimensional infinite word x : N^d → Σ over a finite alphabet Σ is S-automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word