On higher order regionally proximal relations and topological characteristic factors for group actions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914527638978560 |
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| author | Álvarez, Axel |
| author_facet | Álvarez, Axel |
| contents | We study several aspects of higher-order regionally proximal relations for group actions. First, we develop an algebraic approach to study higher-order regionally proximal relations. To this end, we introduce a new topology on a subgroup of the universal minimal system, which can be seen as a higher-order analogue of the classical $τ$-topology. Using this topology, we obtain an algebraic characterization of the relation $\mathbf{RP}^{[d]}$ for abelian actions. Then, we study higher-order regionally proximal relations via recurrence sets, extending results of Huang, Shao, and Ye for $\mathbb{Z}$-actions to more general group actions under suitable assumptions. We then study topological characteristic factors and prove, modulo almost one-to-one factors, that the maximal factor of order $d-1$ is the topological characteristic factor of order d for cubic configurations for arbitrary group actions, and for arithmetic progressions for finitely generated abelian group actions. As a consequence, we show that $\mathbf{RP}^{[d]}$ and $\mathbf{AP}^{[d]}$ coincide on minimal points for finitely generated abelian group actions, and we apply this to obtain results on independence along arithmetic progressions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02304 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On higher order regionally proximal relations and topological characteristic factors for group actions Álvarez, Axel Dynamical Systems Primary: 37B05, Secondary: 37B02, 37B20 We study several aspects of higher-order regionally proximal relations for group actions. First, we develop an algebraic approach to study higher-order regionally proximal relations. To this end, we introduce a new topology on a subgroup of the universal minimal system, which can be seen as a higher-order analogue of the classical $τ$-topology. Using this topology, we obtain an algebraic characterization of the relation $\mathbf{RP}^{[d]}$ for abelian actions. Then, we study higher-order regionally proximal relations via recurrence sets, extending results of Huang, Shao, and Ye for $\mathbb{Z}$-actions to more general group actions under suitable assumptions. We then study topological characteristic factors and prove, modulo almost one-to-one factors, that the maximal factor of order $d-1$ is the topological characteristic factor of order d for cubic configurations for arbitrary group actions, and for arithmetic progressions for finitely generated abelian group actions. As a consequence, we show that $\mathbf{RP}^{[d]}$ and $\mathbf{AP}^{[d]}$ coincide on minimal points for finitely generated abelian group actions, and we apply this to obtain results on independence along arithmetic progressions. |
| title | On higher order regionally proximal relations and topological characteristic factors for group actions |
| topic | Dynamical Systems Primary: 37B05, Secondary: 37B02, 37B20 |
| url | https://arxiv.org/abs/2605.02304 |