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Flow views and infinite interval exchange transformations for recognizable substitutions

Natalie Priebe Frank

Abstract

A flow view is the graph of a measurable conjugacy $Φ$ between a substitution or S-adic subshift $(Σ,σ, μ)$ and an exchange of infinitely many intervals in $([0,1], F, m)$, where $m$ is Lebesgue measure. A natural refining sequence of partitions of $Σ$ is transferred to $([0,1],m)$ using a canonical addressing scheme, a fixed dual substitution, and a shift-invariant probability measure $μ$. On the flow view, $T \in Σ$ is shown horizontally at a height of $Φ(T)$ 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 $Φ\in L^2(Σ, μ),$ is of particular interest. As an example of utility, some spectral results for constant-length substitutions are included.

Flow views and infinite interval exchange transformations for recognizable substitutions

Abstract

A flow view is the graph of a measurable conjugacy between a substitution or S-adic subshift and an exchange of infinitely many intervals in , where is Lebesgue measure. A natural refining sequence of partitions of is transferred to using a canonical addressing scheme, a fixed dual substitution, and a shift-invariant probability measure . On the flow view, is shown horizontally at a height of using colored unit intervals to represent the letters. The infinite interval exchange transformation 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 is self-similar. We discuss why the spectral type of is of particular interest. As an example of utility, some spectral results for constant-length substitutions are included.

Paper Structure

This paper contains 17 sections, 16 theorems, 50 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Setting: symbolic and substitution dynamical systems
  3. Symbolic dynamics
  4. Substitutions and their dynamical systems (see FoggQueffelec)
  5. The address scheme
  6. Canonical isomorphisms
  7. The infinite interval exchange transformation
  8. Self-similarity of some $\mathfrak{F}$
  9. Spectral analysis of $({\Sigma},\mathop{\mathrm{\sigma}}\nolimits,\mu)$
  10. Review: spectral analysis in a general system (see e.g. Petersen.book)
  11. The spectral measure of $\Phi$
  12. Special case: constant-length substitutions
  13. The non-constant length case.
  14. Self-similar tilings, S-adic sequences and fusion tilings of $\mathbb{R}$
  15. S-adic systems
  16. ...and 2 more sections

Key Result

Proposition 1

Let $\mathcal{S}$ be a recognizable substitution whose subshift $({\Sigma}, \mu)$ is minimal and let $\Phi: ({\Sigma}, \mu) \to ([0, 1], m)$ be a canonical isomorphism. Then $\Phi$ is measure preserving, well-defined and uniformly continuous everywhere, bijective almost everywhere, and at most $2|{\

Figures (19)

  • Figure 1: (Approximations to the) canonical IIETs for Fibonacci (left) and Thue--Morse (right) substitutions.
  • Figure 2: Flow views for Fibonacci (top) and Thue--Morse (bottom) subshifts. The red line highlights the $\pmb{\tau}\in {\Sigma}$ for which $\Phi(\pmb{\tau}) ={1}/{e}$.
  • Figure 3: Building a supertile from an address string.
  • Figure 4: Our choice of initial partition for $\mathcal{S}_{PD}$.
  • Figure 5: The the level-1 address diagram and flow view for $\mathcal{S}_{PD}$.
  • ...and 14 more figures

Theorems & Definitions (46)

  • Proposition : see Proposition \ref{['Pf:mtcong']}
  • Theorem : \ref{['thm:iiet']}
  • Corollary : \ref{['cor:iietest']}
  • Proposition : \ref{['prop:self-similarilty']}
  • Proposition : \ref{['prop:constantlengthfinite']}
  • Theorem : \ref{['thm:coincthm']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • ...and 36 more