Morphic words and equidistributed sequences

TL;DR

This paper constructs a unique equidistributed sequence associated with infinite words over ordered alphabets, especially those generated by specific morphisms, and introduces software for computing these sequences.

Contribution

It provides a method to explicitly construct the sequence for morphic words and extends this to all binary morphisms, including a software tool for practical computation.

Findings

Sequence construction for morphic words is possible and explicit.

Extension of the construction to all binary morphisms.

Development of software for sequence computation.

Abstract

The problem we consider is the following: Given an infinite word $w$ on an ordered alphabet, construct the sequence $\nu_w=(\nu[n])_n$, equidistributed on $[0,1]$ and such that $\nu[m]<\nu[n]$ if and only if $\sigma^m(w)<\sigma^n(w)$, where $\sigma$ is the shift operation, erasing the first symbol of $w$. The sequence $\nu_w$ exists and is unique for every word with well-defined positive uniform frequencies of every factor, or, in dynamical terms, for every element of a uniquely ergodic subshift. In this paper we describe the construction of $\nu_w$ for the case when the subshift of $w$ is generated by a morphism of a special kind; then we overcome some technical difficulties to extend the result to all binary morphisms. The sequence $\nu_w$ in this case is also constructed with a morphism. At last, we introduce a software tool which, given a binary morphism $\varphi$, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of $\varphi$.