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Words avoiding the morphic images of most of their factors

Pascal Ochem, Matthieu Rosenfeld

TL;DR

The paper studies imaged factors in infinite words, defining an imaged factor as a finite factor $f$ for which there exists a non-erasing, non-identity morphism $m$ with $m(f)$ appearing in the word. It proves that every infinite word contains an imaged factor of length at least $6$ (with $6$ optimal) and that every infinite binary word contains at least $36$ distinct imaged factors (with $36$ optimal). To obtain sharp results, the authors construct explicit morphic words and apply synchronization results for $q$-uniform morphisms (via Ochem's lemma), obtaining a length-7 avoidance result through a $37$-uniform morphism and a length-6 unavoidability via a case and computer-assisted backtracking argument. They further show that a binary word with at most $35$ imaged factors cannot be extended to an infinite word, while providing a binary word whose image structure yields exactly $36$ imaged factors, thus tightly characterizing the extremal behavior. The work blends combinatorial constructions with exhaustive computations to delineate the precise trade-off between avoidance and inevitability of imaged factors, offering both theoretical and algorithmic insights for morphic-word combinatorics.

Abstract

We say that a finite factor $f$ of a word $w$ is \emph{imaged} if there exists a non-erasing morphism $m$, distinct from the identity, such that $w$ contains $m(f)$. We show that every infinite word contains an imaged factor of length at least 6 and that 6 is best possible. We show that every infinite binary word contains at least 36 distinct imaged factors and that 36 is best possible.

Words avoiding the morphic images of most of their factors

TL;DR

The paper studies imaged factors in infinite words, defining an imaged factor as a finite factor for which there exists a non-erasing, non-identity morphism with appearing in the word. It proves that every infinite word contains an imaged factor of length at least (with optimal) and that every infinite binary word contains at least distinct imaged factors (with optimal). To obtain sharp results, the authors construct explicit morphic words and apply synchronization results for -uniform morphisms (via Ochem's lemma), obtaining a length-7 avoidance result through a -uniform morphism and a length-6 unavoidability via a case and computer-assisted backtracking argument. They further show that a binary word with at most imaged factors cannot be extended to an infinite word, while providing a binary word whose image structure yields exactly imaged factors, thus tightly characterizing the extremal behavior. The work blends combinatorial constructions with exhaustive computations to delineate the precise trade-off between avoidance and inevitability of imaged factors, offering both theoretical and algorithmic insights for morphic-word combinatorics.

Abstract

We say that a finite factor of a word is \emph{imaged} if there exists a non-erasing morphism , distinct from the identity, such that contains . We show that every infinite word contains an imaged factor of length at least 6 and that 6 is best possible. We show that every infinite binary word contains at least 36 distinct imaged factors and that 36 is best possible.

Paper Structure

This paper contains 5 sections, 7 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Avoiding imaged factors of length 7
  3. Imaged factors of length 6 are unavoidable
  4. Binary words with only 36 imaged factors
  5. Concluding remarks

Key Result

Theorem 1

Ng(2019) There exists an infinite binary word that avoids all factors of the form $fm(f)$ and $m(f)f$, for all non-erasing binary morphisms $m$, with $|f|\geqslant5$.

Theorems & Definitions (9)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • proof