On sets of pointwise recurrence and dynamically thick sets

Abstract

A set $A\subseteq \mathbb{N}$ is a set of pointwise recurrence if for all minimal dynamical systems $(X,T)$, all $x\in X$, and all open neighborhoods $U\subseteq X$ of $x$, there exists a time $n\in A$ such that $T^{n}x\in U$. The set $A$ is dynamically thick if the same holds for all non-empty, open sets $U\subseteq X$. Our main results give combinatorial characterizations of sets of pointwise recurrence and dynamically thick sets that allow us to answer questions of Host, Kra, Maass and Glasner, Tsankov, Weiss, and Zucker. We also introduce and study a local version of dynamical thickness called dynamical piecewise syndeticity. We show that dynamically piecewise syndetic sets are piecewise syndetic, generalizing results of Dong, Glasner, Huang, Shao, Weiss, and Ye. The proofs involve the algebra of families of large sets, dynamics on the space of ultrafilters, and our recent characterization of dynamically syndetic sets.

48 pages. This paper is the second part of our study of dynamically syndetic sets and sets of pointwise recurrence. The first part can be found in arXiv:2408.12785