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On sets of pointwise recurrence and dynamically thick sets

Daniel Glasscock, Anh N. Le

TL;DR

This work provides a comprehensive set of combinatorial and dynamical characterizations for sets of pointwise recurrence and dynamically thick sets, introducing dynamical piecewise syndeticity and a structural description via robustly syndetic collections. The authors establish equivalences linking pointwise recurrence to thick-intersection conditions and show dynamically thick sets can be represented as unions of robustly syndetic blocks intersected with thick components, with a parallel development for dynamically piecewise syndetic sets. They prove the non-sigma-compactness of duals to the dynamically thick and pointwise recurrence families, indicating intrinsic complexity and blocking simple splitting results. The paper also connects these dynamical notions to central/IP/PS families, derives translation/dilation stability, and demonstrates Brauer-type polynomial configurations within dynamically central piecewise syndetic sets, highlighting both the depth and the breadth of the dynamical/combinatorial landscape. The results rely on ultrafilter dynamics in $eta $, the algebra of family translations, and companion insights on dynamically syndetic sets, broadening understanding of recurrence phenomena in minimal systems and their combinatorial fingerprints.

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.

On sets of pointwise recurrence and dynamically thick sets

TL;DR

This work provides a comprehensive set of combinatorial and dynamical characterizations for sets of pointwise recurrence and dynamically thick sets, introducing dynamical piecewise syndeticity and a structural description via robustly syndetic collections. The authors establish equivalences linking pointwise recurrence to thick-intersection conditions and show dynamically thick sets can be represented as unions of robustly syndetic blocks intersected with thick components, with a parallel development for dynamically piecewise syndetic sets. They prove the non-sigma-compactness of duals to the dynamically thick and pointwise recurrence families, indicating intrinsic complexity and blocking simple splitting results. The paper also connects these dynamical notions to central/IP/PS families, derives translation/dilation stability, and demonstrates Brauer-type polynomial configurations within dynamically central piecewise syndetic sets, highlighting both the depth and the breadth of the dynamical/combinatorial landscape. The results rely on ultrafilter dynamics in , the algebra of family translations, and companion insights on dynamically syndetic sets, broadening understanding of recurrence phenomena in minimal systems and their combinatorial fingerprints.

Abstract

A set is a set of pointwise recurrence if for all minimal dynamical systems , all , and all open neighborhoods of , there exists a time such that . The set is dynamically thick if the same holds for all non-empty, open sets . 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.
Paper Structure (54 sections, 52 theorems, 99 equations, 1 figure)

This paper contains 54 sections, 52 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Characterizing sets of pointwise recurrence and dynamically thick sets
  3. Motivation: set and point recurrence in dynamics
  4. Sets of pointwise recurrence in the literature
  5. Main results: combinatorial characterizations
  6. More on dynamically thick sets
  7. Novel examples
  8. The structure of dynamical thick sets
  9. The splitting problem and sigma-compactness
  10. Dynamically piecewise syndetic sets
  11. Relations with other established families
  12. Applications of \ref{['thm:relations_C_dPS_PS-intro']}
  13. A companion paper on dynamically syndetic sets
  14. Organization of the paper
  15. Acknowledgments
  16. ...and 39 more sections

Key Result

Theorem 1

Let $A \subseteq \mathbb{N}$. The following are equivalent.

Figures (1)

  • Figure 1: Containment amongst the families appearing in this paper. An arrow $\mathscr{F} \to \mathscr{G}$ indicates that $\mathscr{F} \subsetneq \mathscr{G}$. There are eight families and their duals, yielding sixteen families in all, two pairs of which coincide. The diagram's symmetry is explained by the family duality described in \ref{['sec_dual_families']}.

Theorems & Definitions (104)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Lemma 2.1
  • proof
  • ...and 94 more