Dynamically syndetic sets and the combinatorics of syndetic, idempotent filters

TL;DR

This paper characterizes dynamically central syndetic sets as members of syndetic, idempotent filters, linking dynamical systems and combinatorics, and addresses open questions on topological recurrence.

Contribution

It provides a global characterization of dynamically central syndetic sets using syndetic, idempotent filters, extending the understanding of the dynamics-combinatorics connection.

Findings

Characterization of dynamically central syndetic sets as members of syndetic, idempotent filters

Answering open questions on sets of pointwise topological recurrence

Establishing a global analogue to Furstenberg's local characterization of central sets

Abstract

A subset of the positive integers is dynamically central syndetic if it contains the times that a point returns to a neighborhood of itself in a minimal topological dynamical system. These sets are part of the highly-influential link between dynamics and combinatorics forged by Furstenberg and Weiss in the 1970's. Our main result is a characterization of dynamically central syndetic sets as precisely those sets that belong to syndetic, idempotent filters. This gives a "global" analogue to the well-known "local" characterization of Furstenberg's central sets as members of minimal, idempotent ultrafilters. Applying the main result, we answer two open questions posed by Host, Kra, and Maass concerning sets of pointwise topological recurrence.