Flow views and infinite interval exchange transformations for recognizable substitutions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913433313607680 |
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| author | Frank, Natalie Priebe |
| author_facet | Frank, Natalie Priebe |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_07997 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Flow views and infinite interval exchange transformations for recognizable substitutions Frank, Natalie Priebe Dynamical Systems 37A05, 37B10 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. |
| title | Flow views and infinite interval exchange transformations for recognizable substitutions |
| topic | Dynamical Systems 37A05, 37B10 |
| url | https://arxiv.org/abs/2103.07997 |