On sets of pointwise recurrence and dynamically thick sets

TL;DR

This paper characterizes sets of pointwise recurrence and dynamically thick sets using combinatorial methods, introduces a local version called dynamical piecewise syndeticity, and explores their properties within dynamical systems.

Contribution

It provides new combinatorial characterizations of recurrence-related sets and introduces the concept of dynamical piecewise syndeticity, extending previous results in the field.

Findings

Characterization of sets of pointwise recurrence

Introduction of dynamical piecewise syndeticity

Proof that dynamically piecewise syndetic sets are piecewise syndetic

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.