Return time sets and product recurrence
Jian Li, Xianjuan Liang, Yini Yang
TL;DR
This work characterizes return time sets of piecewise syndetic recurrent points for actions of a countable group $G$ by linking them to quasi-central sets of $\mathbb{N}_0$, establishing a precise combinatorial-dynamical bridge. It develops a robust framework using Furstenberg families with properties (P1) and (P2), and leverages hulls in the Stone-Čech compactification $\beta G$ to translate recurrence into essential $\mathcal{F}$-sets. For compact metric $G$-systems, the authors prove that $\mathcal{F}$-recurrence yields return-time sets that are essential, and conversely that essential $\mathcal{F}$-sets arise as return sets for some $\mathcal{F}$-recurrent point; in particular, distal points are exactly those that are $\mathcal{F}_{\text{ps}}$-product recurrent. Extending to $\beta G$-actions, they show that when a closed subsemigroup $S$ contains the smallest ideal $K(\beta G)$, distality is equivalent to $S$-product recurrence (and to weak $S$-product recurrence), partially answering Auslander–Furstenberg-type questions. The results unify combinatorial density notions (central, IP, D-sets) with dynamical recurrence and provide a framework applicable to both amenable and general countable groups.
Abstract
Let $G$ be a countable infinite discrete group. We show that a subset $F$ of $G$ contains a return time set of some piecewise syndetic recurrent point $x$ in a compact Hausdorff space $X$ with a $G$-action if and only if $F$ is a quasi-central set. As an application, we show that if a nonempty closed subsemigroup $S$ of the Stone-Čech compactification $βG$ contains the smallest ideal $K(βG)$ of $βG$ then $S$-product recurrent is equivalent to distality, which partially answers a question of Auslander and Furstenberg (Trans. Amer. Math. Soc. 343, 1994, 221--232).