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To Symbolic Dynamics Through The Thue-Morse Sequence

Diyath Pannipitiya

Abstract

The celebrated Thue-Morse sequence, or the Prouhet-Thue-Morse sequence (A010060 in the OEIS), has a number of interesting properties and is a rich source to many (counter)examples. We introduce two different square-free sequences on three letters with one of them is equivalent (up-to permutations of letters) to Thue's original square-free sequence on three letters. Then we use them to introduce an explicit method to construct infinitely many number of recurrent points in ${\{0,1\}^{\mathbb{N}}}$ whose orbit closures under the shift map is minimal, uncountable, and for any two distinct such points their orbit closures are disjoint.

To Symbolic Dynamics Through The Thue-Morse Sequence

Abstract

The celebrated Thue-Morse sequence, or the Prouhet-Thue-Morse sequence (A010060 in the OEIS), has a number of interesting properties and is a rich source to many (counter)examples. We introduce two different square-free sequences on three letters with one of them is equivalent (up-to permutations of letters) to Thue's original square-free sequence on three letters. Then we use them to introduce an explicit method to construct infinitely many number of recurrent points in whose orbit closures under the shift map is minimal, uncountable, and for any two distinct such points their orbit closures are disjoint.
Paper Structure (5 sections, 11 theorems, 15 equations)

This paper contains 5 sections, 11 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Background and revisiting Thue
  3. Two Square-free Sequences
  4. Morse-type minimal points
  5. Acknowledgements

Key Result

Theorem 2.1

M is overlap-free.

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 2.1: Thue
  • Proof 1
  • Theorem 2.2: Thue
  • Proof 2
  • ...and 24 more