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arXiv · 2401.15339

Interpolation sets for dynamical systems

Andreas Koutsogiannis·Anh N. Le·Joel Moreira·Ronnie Pavlov·Florian K. Richter
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Abstract

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.

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Andreas Koutsogiannis, Anh N. Le, Joel Moreira, Ronnie Pavlov, Florian K. Richter. 2024-01-27. Interpolation sets for dynamical systems. https://arxiv.org/abs/2401.15339

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