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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.27899 |
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| _version_ | 1866915898183385088 |
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| author | Kousek, Ioannis Saavedra-Araya, Vicente |
| author_facet | Kousek, Ioannis Saavedra-Araya, Vicente |
| contents | Given a semigroup $G$ and a bounded function $f: G \to \mathbb{C}$, a topological Furstenberg system of $f$ is a topological dynamical system $\mathbb{X}=(X, (T_g)_{g \in G})$ that encodes the dynamical behaviour of $f$. We show that $\mathbb{X}$ is unique up to topological isomorphism, thus providing a topological analogue of the measurable case established by Bergelson and Ferré Moragues for amenable semigroups. We also provide necessary and sufficient conditions for subsets of a group to have isomorphic Furstenberg systems. In addition, we study sets with minimal Furstenberg systems and identify them as a special subclass of dynamically syndetic sets. Moreover, we use this notion to obtain a new characterization of sets of topological recurrence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27899 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniqueness of a topological Furstenberg system Kousek, Ioannis Saavedra-Araya, Vicente Dynamical Systems 37B05 (Primary) 37B10, 05D10 (Secondary) Given a semigroup $G$ and a bounded function $f: G \to \mathbb{C}$, a topological Furstenberg system of $f$ is a topological dynamical system $\mathbb{X}=(X, (T_g)_{g \in G})$ that encodes the dynamical behaviour of $f$. We show that $\mathbb{X}$ is unique up to topological isomorphism, thus providing a topological analogue of the measurable case established by Bergelson and Ferré Moragues for amenable semigroups. We also provide necessary and sufficient conditions for subsets of a group to have isomorphic Furstenberg systems. In addition, we study sets with minimal Furstenberg systems and identify them as a special subclass of dynamically syndetic sets. Moreover, we use this notion to obtain a new characterization of sets of topological recurrence. |
| title | Uniqueness of a topological Furstenberg system |
| topic | Dynamical Systems 37B05 (Primary) 37B10, 05D10 (Secondary) |
| url | https://arxiv.org/abs/2603.27899 |