Flow views and infinite interval exchange transformations for recognizable substitutions

TL;DR

This paper introduces a flow view framework linking substitution subshifts to infinite interval exchange transformations, enabling numeric analysis and exploring self-similarity and spectral properties.

Contribution

It develops a method to represent substitution subshifts as flow views and approximates infinite interval exchanges, highlighting conditions for self-similarity and spectral analysis.

Findings

Infinite interval exchange transformations can be approximated by finite exchanges.

A dual substitution guarantees self-similarity of the transformation.

Spectral properties are analyzed for constant-length substitutions.

Abstract

A flow view is the graph of a measurable conjugacy $\Phi$ between a substitution or S-adic subshift $(\Sigma,\sigma, \mu)$ and an exchange of infinitely many intervals in $([0,1], F, m)$, where $m$ is Lebesgue measure. A natural refining sequence of partitions of $\Sigma$ is transferred to $([0,1],m)$ using a canonical addressing scheme, a fixed dual substitution, and a shift-invariant probability measure $\mu$. On the flow view, $T \in \Sigma$ is shown horizontally at a height of $\Phi(T)$ using colored unit intervals to represent the letters. The infinite interval exchange transformation $F$ is well approximated by exchanges of finitely many intervals, making numeric and graphic methods possible. We prove that in certain cases a choice of dual substitution guarantees that $F$ is self-similar. We discuss why the spectral type of $\Phi \in L^2(\Sigma, \mu),$ is of particular interest. As an example of utility, some spectral results for constant-length substitutions are included.