Quasi-Sturmian colorings on regular trees

TL;DR

This paper explores quasi-Sturmian colorings on regular trees, classifying them into bounded and unbounded types, and develops an induction algorithm to analyze their properties based on recurrence functions.

Contribution

It introduces a classification of quasi-Sturmian colorings on regular trees and presents an induction algorithm for their analysis, extending properties known from Sturmian words.

Findings

Two types of quasi-Sturmian colorings identified: bounded and unbounded.

An induction algorithm similar to Sturmian colorings is developed.

Recurrence functions distinguish between the two types.

Abstract

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this paper, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.