Words and morphisms with Sturmian erasures

TL;DR

This paper studies words over a three-letter alphabet where removing any one letter yields a Sturmian word, explores their connection to billiard trajectories, and proves the associated morphism monoid is not finitely generated.

Contribution

It introduces the concept of words with Sturmian erasures, links them to billiard trajectories, and proves the non-finite generation of their morphism monoid.

Findings

Words with Sturmian erasures are linked to billiard trajectories in the cube.

The monoid of morphisms preserving these words is not finitely generated.

The paper characterizes a large family of such words.

Abstract

We say $x \in \{0,1,2 \}^{\NN}$ is a word with Sturmian erasures if for any $a\in \{0,1,2 \}$ the word obtained erasing all $a$ in $x$ is a Sturmian word. A large family of such words is given coding trajectories of balls in the game of billiards in the cube. We prove that the monoid of morphisms mapping all words with Sturmian erasures to words with Sturmian erasures is not finitely generated.