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Quasi-Sturmian colorings on regular trees

Dong Han Kim, Seul Bee Lee, Seonhee Lim, Deokwon Sim

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.

Quasi-Sturmian colorings on regular trees

Abstract

Quasi-Sturmian words, which are infinite words with factor complexity eventually 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.

Paper Structure

This paper contains 9 sections, 24 theorems, 39 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Quotient graphs of quasi-Sturmian colorings
  3. Quotient graphs of quasi-Sturmian colorings of bounded type
  4. Quasi-Sturmian colorings of unbounded type
  5. Evolution of factor graphs
  6. Prelimiary
  7. Acyclic quasi-Sturmian colorings
  8. Quasi-Sturmian colorings of bounded type
  9. Recurrence functions of colorings of trees

Key Result

Theorem 1.3

If $\phi$ is a quasi-Sturmian coloring, then the quotient graph is one of the following graphs. More precisely, the quotient graph of a coloring of bounded type is the first graph, where as the quotient graph of a coloring of unbounded type is a geodesic ray or a biinfinite geodesic.

Figures (5)

  • Figure 1: The evolution of Rauzy graphs of a quasi-Sturmian word (above) and The evolution of $\mathcal{G}_n$ of a quasi-Sturmian coloring on a tree (below)
  • Figure 2:
  • Figure 3: The evolution of $\mathcal{G}_{n_k}$ along the path (I) $\to$ (II) $\to \cdots \to$ (II) $\to$ (I). The vertex $\circ$ represents either $S_{n_k}$ or the extensions of $S_{n_k}$.
  • Figure :
  • Figure :

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Quotient graphs of quasi-Sturmian colorings
  • Theorem 1.4: Induction algorithm
  • Theorem 1.5: Quotient graphs of colorings of bounded type
  • Theorem 1.6: Bounds of $R"(n)$
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • ...and 42 more