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Infinite Words with very Low Factor Complexity: an introduction to Combinatorics on Words

Mélodie Andrieu

Abstract

These lecture notes provide an introduction to combinatorics on words and its interactions with dynamics, algebra, and arithmetic. The central theme is the notion of low factor complexity for infinite words. We investigate the following guiding questions: What is the minimal complexity of a non-trivial infinite word over a binary, ternary, or more generally finite alphabet? How should ''non-triviality'' be formalized? Which words achieve this minimal complexity? Are there many? Are they interesting? In exploring these questions, we introduce classical objects and tools from combinatorics on words -- such as Sturmian words and Rauzy graphs -- as well as little-known and new results. In particular, the third chapter is devoted to a theorem by R. Tijdeman from 1999, which generalizes a seminal result of M. Morse and G. Hedlund from 1938. We provide a new, algebraic proof of this theorem (due to J. Cassaigne and the author, 2022) and develop its consequences.

Infinite Words with very Low Factor Complexity: an introduction to Combinatorics on Words

Abstract

These lecture notes provide an introduction to combinatorics on words and its interactions with dynamics, algebra, and arithmetic. The central theme is the notion of low factor complexity for infinite words. We investigate the following guiding questions: What is the minimal complexity of a non-trivial infinite word over a binary, ternary, or more generally finite alphabet? How should ''non-triviality'' be formalized? Which words achieve this minimal complexity? Are there many? Are they interesting? In exploring these questions, we introduce classical objects and tools from combinatorics on words -- such as Sturmian words and Rauzy graphs -- as well as little-known and new results. In particular, the third chapter is devoted to a theorem by R. Tijdeman from 1999, which generalizes a seminal result of M. Morse and G. Hedlund from 1938. We provide a new, algebraic proof of this theorem (due to J. Cassaigne and the author, 2022) and develop its consequences.
Paper Structure (40 sections, 28 theorems, 91 equations, 4 figures)

This paper contains 40 sections, 28 theorems, 91 equations, 4 figures.

Table of Contents

  1. Acknowledgements
  2. A seminal theorem by Morse and Hedlund (1938)
  3. Preliminaries: infinite words, complexity
  4. The seminal theorem by Morse and Hedlund
  5. The class of Sturmian words
  6. Basic properties
  7. Interlude: Crash course on the continued fraction
  8. Connection between Sturmian words and continued fractions
  9. An efficient construction of Sturmian words
  10. Equivalent dynamical and geometrical descriptions of Sturmian words
  11. Billiard words.
  12. Digitization of straight lines.
  13. Coding a linear flow on the torus.
  14. Coding of rotations.
  15. Commented bibliography
  16. ...and 25 more sections

Key Result

Theorem 1.3

The following assertions are equivalent:

Figures (4)

  • Figure 1: Top-left: billiard trajectory. Top-right: digitization of a straight line. Bottom-left: linear flow on the torus. Bottom-right: rotation of the circle.
  • Figure 2: From the linear flow on the 2D torus to a rotation on a circle.
  • Figure 3: The extension graph of every factor in $w_1=11111...$ (left), and of the empty word in $w_2=112211221122...$ (right).
  • Figure 4: Example of a partition of the extension graph.

Theorems & Definitions (69)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.3: M. Morse and G. Hedlund, 1938
  • proof
  • Remark 1.4
  • Definition 1.6
  • Example 1.7
  • Proposition 1.9
  • proof
  • Proposition 1.11
  • ...and 59 more