Flow views and infinite interval exchange transformations for recognizable 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 $S_{\ast }$, and a shift-invariant probability measure $\mu$. On the flow view, $\tau \in \Sigma$ is shown horizontally at a height of $\Phi \left(\tau \right)$ 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 $\Phi$ 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.