Interpolation sets for dynamical systems

Koutsogiannis, Andreas; Le, Anh; Moreira, Joel; Pavlov, Ronnie; Richter, Florian

doi:10.1090/tran/9300

Interpolation sets for dynamical systems
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by Andreas Koutsogiannis, Anh N. Le, Joel Moreira, Ronnie Pavlov and Florian K. Richter;

Trans. Amer. Math. Soc. 378 (2025), 1373-1400

DOI: https://doi.org/10.1090/tran/9300

Published electronically: October 17, 2024

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Abstract:

Originating in harmonic analysis, interpolation sets were first studied in dynamics by Glasner and Weiss in the 1980s [Israel J. Math. 44 (1983), pp. 345–360]. 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:

(totally) minimal systems;

topologically (weak) mixing systems;

strictly ergodic systems; and

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 [Monatsh. Math. 179 (2016), pp. 57–89] concerning the connection between sets of pointwise recurrence for distal systems and $IP$-sets.

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Interpolation sets for dynamical systems
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Abstract: