Classification of rank-one actions via the cutting-and-stacking parameters
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910906260127744 |
|---|---|
| author | Danilenko, Alexandre I. Vieprik, Mykyta I. |
| author_facet | Danilenko, Alexandre I. Vieprik, Mykyta I. |
| contents | Let $G$ be a discrete countable infinite group. Let $T$ and $\widetilde T$ be two rank-one $σ$-finite measure preserving actions of $G$ and let $\mathcal T$ and $\widetilde {\mathcal T}$ be the cutting-and-stacking parameters that determine $T$ and $\widetilde T$ respectively. We find necessary and sufficient conditions on $\mathcal T$ and $\widetilde{\mathcal T}$ under which $T$ and $\widetilde T$ are isomorphic. We also show that the isomorphism equivalence relation is a $G_δ$-subset in the Cartesian square of the set of all admissible parameters $\mathcal T$ endowed with the natural Polish topology. If $G$ is amenable and $T$ and $\widetilde T$ are finite measure preserving then we also find necessary and sufficient conditioins on $\mathcal T$ and $\widetilde {\mathcal T}$ under which $\widetilde T$ is a factor of $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04987 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of rank-one actions via the cutting-and-stacking parameters Danilenko, Alexandre I. Vieprik, Mykyta I. Dynamical Systems Primary 37A05, 37A15, 37A40, secondary 37A20, 37B05 Let $G$ be a discrete countable infinite group. Let $T$ and $\widetilde T$ be two rank-one $σ$-finite measure preserving actions of $G$ and let $\mathcal T$ and $\widetilde {\mathcal T}$ be the cutting-and-stacking parameters that determine $T$ and $\widetilde T$ respectively. We find necessary and sufficient conditions on $\mathcal T$ and $\widetilde{\mathcal T}$ under which $T$ and $\widetilde T$ are isomorphic. We also show that the isomorphism equivalence relation is a $G_δ$-subset in the Cartesian square of the set of all admissible parameters $\mathcal T$ endowed with the natural Polish topology. If $G$ is amenable and $T$ and $\widetilde T$ are finite measure preserving then we also find necessary and sufficient conditioins on $\mathcal T$ and $\widetilde {\mathcal T}$ under which $\widetilde T$ is a factor of $T$. |
| title | Classification of rank-one actions via the cutting-and-stacking parameters |
| topic | Dynamical Systems Primary 37A05, 37A15, 37A40, secondary 37A20, 37B05 |
| url | https://arxiv.org/abs/2504.04987 |