Limits of Rauzy graphs of languages with subexponential complexity
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917706687578112 |
|---|---|
| author | Leemann, Paul-Henry Nagnibeda, Tatiana Skripchenko, Alexandra Veprev, Georgii |
| author_facet | Leemann, Paul-Henry Nagnibeda, Tatiana Skripchenko, Alexandra Veprev, Georgii |
| contents | To a subshift over a finite alphabet, one can naturally associate an infinite family of finite graphs, called its Rauzy graphs. We show that for a subshift of subexponential complexity the Rauzy graphs converge to the line $\mathbf{Z}$ in the sense of Benjamini-Schramm convergence if and only if its complexity function $p(n)$ is unbounded and satisfies $\lim_n\frac{p(n+1)}{p(n)} = 1$. We then apply this criterion to many examples of well-studied dynamical systems. If the subshift is moreover uniquely ergodic then we show that the limit of labelled Rauzy graphs if it exists can be identified with the unique invariant measure. In addition we consider an example of a non uniquely ergodic system recently studied by Cassaigne and Kaboré and identify a continuum of invariant measures with subsequential limits of labelled Rauzy graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15877 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limits of Rauzy graphs of languages with subexponential complexity Leemann, Paul-Henry Nagnibeda, Tatiana Skripchenko, Alexandra Veprev, Georgii Dynamical Systems Combinatorics 37B10 To a subshift over a finite alphabet, one can naturally associate an infinite family of finite graphs, called its Rauzy graphs. We show that for a subshift of subexponential complexity the Rauzy graphs converge to the line $\mathbf{Z}$ in the sense of Benjamini-Schramm convergence if and only if its complexity function $p(n)$ is unbounded and satisfies $\lim_n\frac{p(n+1)}{p(n)} = 1$. We then apply this criterion to many examples of well-studied dynamical systems. If the subshift is moreover uniquely ergodic then we show that the limit of labelled Rauzy graphs if it exists can be identified with the unique invariant measure. In addition we consider an example of a non uniquely ergodic system recently studied by Cassaigne and Kaboré and identify a continuum of invariant measures with subsequential limits of labelled Rauzy graphs. |
| title | Limits of Rauzy graphs of languages with subexponential complexity |
| topic | Dynamical Systems Combinatorics 37B10 |
| url | https://arxiv.org/abs/2402.15877 |