Words avoiding the morphic images of most of their factors

TL;DR

This paper investigates the occurrence of factors in infinite words that are images under non-trivial morphisms, establishing minimal lengths and counts of such factors in binary and general infinite words.

Contribution

It proves the minimal length of imaged factors in infinite words and the minimal number of such factors in binary words, providing optimal bounds.

Findings

Every infinite word contains an imaged factor of length at least 6.

Every infinite binary word contains at least 36 distinct imaged factors.

The bounds of 6 and 36 are proven to be optimal.

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.