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Relative position in binary substitutions

Michael Coons, Christopher Ramsey, Nicolae Strungaru

Abstract

Given an infinite word ${\bf w}$ on a finite alphabet, an immediate question arises:~can we understand the frequency of letters in ${\bf w}$\,? For words that are the fixed points of substitutions, the answer to this question is often `yes' -- the details and methods of these answers have been well-documented. In this paper, toward a better-understanding of the fixed points of binary substitutions, we delve deeper by investigating, in fine detail, the position of letters by defining various position functions and proving results about their behavior. Our analysis reveals new information about the Fibonacci substitution and the extended Pisa family of substitutions, as well as a new characterization of the Thue--Morse sequence.

Relative position in binary substitutions

Abstract

Given an infinite word on a finite alphabet, an immediate question arises:~can we understand the frequency of letters in \,? For words that are the fixed points of substitutions, the answer to this question is often `yes' -- the details and methods of these answers have been well-documented. In this paper, toward a better-understanding of the fixed points of binary substitutions, we delve deeper by investigating, in fine detail, the position of letters by defining various position functions and proving results about their behavior. Our analysis reveals new information about the Fibonacci substitution and the extended Pisa family of substitutions, as well as a new characterization of the Thue--Morse sequence.

Paper Structure

This paper contains 6 sections, 56 theorems, 152 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mean values of (relative) position functions
  4. The Fibonacci and extended Pisa family of substitutions
  5. Substitutions with exactly, and asymptotically, linear relative position
  6. The Thue--Morse substitution and related words

Key Result

Lemma 2.3

Let $p :\mathbb{N} \to \mathbb{Z}_{\geqslant 0}$. The following hold.

Theorems & Definitions (109)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5: Reconstruction algorithm
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Corollary 2.8
  • ...and 99 more