Interpolation sets for dynamical systems
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909285528633344 |
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| author | Koutsogiannis, Andreas Le, Anh N. Moreira, Joel Pavlov, Ronnie Richter, Florian K. |
| author_facet | Koutsogiannis, Andreas Le, Anh N. Moreira, Joel Pavlov, Ronnie Richter, Florian K. |
| contents | Originating in harmonic analysis, interpolation sets were first studied in dynamics by Glasner and Weiss in the 1980s. A set $S \subset \mathbb{N}$ is an interpolation set for a class of topological dynamical systems $\mathcal{C}$ if any bounded sequence on $S$ can be extended to a sequence that arises from a system in $\mathcal{C}$. In this paper, we provide combinatorial characterizations of interpolation sets for:
$\bullet$ (totally) minimal systems;
$\bullet$ topologically (weak) mixing systems;
$\bullet$ strictly ergodic systems; and
$\bullet$ zero entropy systems.
Additionally, we prove some results on a slightly different notion, called weak interpolation sets, for several classes of systems. We also answer a question of Host, Kra, and Maass concerning the connection between sets of pointwise recurrence for distal systems and $IP$-sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15339 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interpolation sets for dynamical systems Koutsogiannis, Andreas Le, Anh N. Moreira, Joel Pavlov, Ronnie Richter, Florian K. Dynamical Systems Primary: 37B05, Secondary: 37B10 Originating in harmonic analysis, interpolation sets were first studied in dynamics by Glasner and Weiss in the 1980s. A set $S \subset \mathbb{N}$ is an interpolation set for a class of topological dynamical systems $\mathcal{C}$ if any bounded sequence on $S$ can be extended to a sequence that arises from a system in $\mathcal{C}$. In this paper, we provide combinatorial characterizations of interpolation sets for: $\bullet$ (totally) minimal systems; $\bullet$ topologically (weak) mixing systems; $\bullet$ strictly ergodic systems; and $\bullet$ zero entropy systems. Additionally, we prove some results on a slightly different notion, called weak interpolation sets, for several classes of systems. We also answer a question of Host, Kra, and Maass concerning the connection between sets of pointwise recurrence for distal systems and $IP$-sets. |
| title | Interpolation sets for dynamical systems |
| topic | Dynamical Systems Primary: 37B05, Secondary: 37B10 |
| url | https://arxiv.org/abs/2401.15339 |