Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

WELLDOC property for words generated by morphisms

Svetlana Puzynina, Vladimir Schavelev

Abstract

In this paper, we study an abelian-type property of infinite words called well distributed occurrences, or WELLDOC for short. An infinite word $w$ on a $d$-ary alphabet has the WELLDOC property if, for each factor $u$ of $w$, positive integer $m$, and vector $v\in \mathbb{N}^d$, there is an occurrence of $u$ such that the Parikh vector of the prefix of $w$ preceding such occurrence is congruent to $v$ modulo $m$. The Parikh vector of a finite word $v$ on an alphabet has its $i$-th component equal to the number of occurrences of the $i$-th letter in $v$. We provide a criterion of the WELLDOC property for words generated by morphisms.

WELLDOC property for words generated by morphisms

Abstract

In this paper, we study an abelian-type property of infinite words called well distributed occurrences, or WELLDOC for short. An infinite word on a -ary alphabet has the WELLDOC property if, for each factor of , positive integer , and vector , there is an occurrence of such that the Parikh vector of the prefix of preceding such occurrence is congruent to modulo . The Parikh vector of a finite word on an alphabet has its -th component equal to the number of occurrences of the -th letter in . We provide a criterion of the WELLDOC property for words generated by morphisms.
Paper Structure (10 sections, 23 theorems, 13 equations)

This paper contains 10 sections, 23 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Definitions and main results
  3. Algebraic statements
  4. Sufficiency of the condition for the WELLDOC property
  5. Proof of sufficiency of the condition for the WELLDOC property
  6. Example: episturmian words
  7. Decidability of the sufficient condition
  8. Necessity of the condition for the WELLDOC property
  9. Recognizable morphisms
  10. Proof of necessity of the condition for the WELLDOC property

Key Result

Theorem 1

Let $w$ be an infinite recurrent binary word generated by a morphism $\phi$. Then $w$ satisfies the WELLDOC property if and only if $\det A_\phi = \pm 1$.

Theorems & Definitions (62)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Remark 1
  • Remark 2
  • Definition 5
  • Definition 6
  • Proposition 1
  • ...and 52 more