Relative position in binary substitutions

TL;DR

This paper investigates the detailed positional structure of letters in fixed points of binary substitutions, revealing new insights into well-known sequences like Fibonacci, Pisa, and Thue–Morse.

Contribution

It introduces new position functions and characterizations that deepen understanding of binary substitution fixed points beyond frequency analysis.

Findings

New position functions for fixed points of binary substitutions

Characterization of the Thue–Morse sequence based on position analysis

Enhanced understanding of Fibonacci and Pisa substitution sequences

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.