Rank-one nonsingular actions of countable groups and their odometer factors
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2024
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| _version_ | 1866929227293523968 |
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| author | Danilenko, Alexandre I. Vieprik, Mykyta I. |
| author_facet | Danilenko, Alexandre I. Vieprik, Mykyta I. |
| contents | For an arbitrary countable discrete infinite group $G$, nonsingular rank-one actions are introduced. It is shown that the class of nonsingular rank-one actions coincides with the class of nonsingular $(C,F)$-actions. Given a decreasing sequence $Γ_1\supsetneqΓ_2\supsetneq\cdots$ of cofinite subgroups in $G$ with $\bigcap_{n=1}^\infty\bigcap_{g\in G}gΓ_ng^{-1}=\{1_G\}$, the projective limit of the homogeneous $G$-spaces $G/Γ_n$ as $n\to\infty$ is a $G$-space. Endowing this $G$-space with an ergodic nonsingular nonatomic measure we obtain a dynamical system which is called a nonsingular odometer. Necessary and sufficient conditions are found for a rank-one nonsingular $G$-action to have a finite factor and a nonsingular odometer factor in terms of the underlying $(C,F)$-parameters. Similar conditions are also found for a rank-one nonsingular $G$-action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the nonsingular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the nonsingular noncommutative case from the classical finite measure preserving one: odometer actions which are not of rank one, factors of rank-one systems which are not of rank-one, however each probability preserving odometer is a factor of an infinite measure preserving rank-one system, etc. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16397 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rank-one nonsingular actions of countable groups and their odometer factors Danilenko, Alexandre I. Vieprik, Mykyta I. Dynamical Systems 37A40, 37A15 For an arbitrary countable discrete infinite group $G$, nonsingular rank-one actions are introduced. It is shown that the class of nonsingular rank-one actions coincides with the class of nonsingular $(C,F)$-actions. Given a decreasing sequence $Γ_1\supsetneqΓ_2\supsetneq\cdots$ of cofinite subgroups in $G$ with $\bigcap_{n=1}^\infty\bigcap_{g\in G}gΓ_ng^{-1}=\{1_G\}$, the projective limit of the homogeneous $G$-spaces $G/Γ_n$ as $n\to\infty$ is a $G$-space. Endowing this $G$-space with an ergodic nonsingular nonatomic measure we obtain a dynamical system which is called a nonsingular odometer. Necessary and sufficient conditions are found for a rank-one nonsingular $G$-action to have a finite factor and a nonsingular odometer factor in terms of the underlying $(C,F)$-parameters. Similar conditions are also found for a rank-one nonsingular $G$-action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the nonsingular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the nonsingular noncommutative case from the classical finite measure preserving one: odometer actions which are not of rank one, factors of rank-one systems which are not of rank-one, however each probability preserving odometer is a factor of an infinite measure preserving rank-one system, etc. |
| title | Rank-one nonsingular actions of countable groups and their odometer factors |
| topic | Dynamical Systems 37A40, 37A15 |
| url | https://arxiv.org/abs/2401.16397 |