Subword complexity and Sturmian colorings of regular trees

TL;DR

This paper investigates the subword complexity of colorings on regular trees, characterizing Sturmian colorings with minimal unbounded complexity and classifying their structures via quotient graphs.

Contribution

It introduces a classification of Sturmian colorings on regular trees using type sets and quotient graphs, extending the understanding of minimal unbounded subword complexity.

Findings

Sturmian colorings are lifts of colorings on specific quotient graphs.

A complete classification of quotient graphs for eventually periodic Sturmian colorings.

Characterization of bounded subword complexity colorings.

Abstract

In this article, we study subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and study Sturmian colorings, which are colorings of minimal unbounded subword complexity. We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray with loops possibly attached, thus a lifting of an "infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones.