Infinitesimal containment and sparse factors of iid
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908703700025344 |
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| author | Frączyk, Mikołaj |
| author_facet | Frączyk, Mikołaj |
| contents | We introduce infinitesimal weak containment for measure-preserving actions of a countable group $Γ$: an action $(X,μ)$ is infinitesimally contained in $(Y,ν)$ if the statistics of the action of $Γ$ on small measure subsets of $X$ can be approximated inside $Y$. We show that the Bernoulli shift $[0,1]^Γ$ is infinitesimally contained in the left-regular action of $Γ$. For exact groups, this implies that sparse factor-of-iid subsets of $Γ$ are approximately hyperfinite. We use it to quantify a theorem of Chifan--Ioana on measured subrelations of the Bernoulli shift of an exact group. For the proof of infinitesimal containment we define \emph{entropy support maps}, which take a small subset $U$ of $\{0,1\}^I$ and assign weights to coordinates above every point of $U$, according to how ''important'' they are for the structure of the set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09301 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitesimal containment and sparse factors of iid Frączyk, Mikołaj Dynamical Systems Information Theory Probability 20P05 37A35 60G10 We introduce infinitesimal weak containment for measure-preserving actions of a countable group $Γ$: an action $(X,μ)$ is infinitesimally contained in $(Y,ν)$ if the statistics of the action of $Γ$ on small measure subsets of $X$ can be approximated inside $Y$. We show that the Bernoulli shift $[0,1]^Γ$ is infinitesimally contained in the left-regular action of $Γ$. For exact groups, this implies that sparse factor-of-iid subsets of $Γ$ are approximately hyperfinite. We use it to quantify a theorem of Chifan--Ioana on measured subrelations of the Bernoulli shift of an exact group. For the proof of infinitesimal containment we define \emph{entropy support maps}, which take a small subset $U$ of $\{0,1\}^I$ and assign weights to coordinates above every point of $U$, according to how ''important'' they are for the structure of the set. |
| title | Infinitesimal containment and sparse factors of iid |
| topic | Dynamical Systems Information Theory Probability 20P05 37A35 60G10 |
| url | https://arxiv.org/abs/2512.09301 |