# On recurrence in G-spaces

## TL;DR

This paper introduces a generalized notion of recurrence in G-spaces, defining $rak{F}$-recurrence based on group actions and families of subsets, providing a new framework for analyzing recurrence phenomena.

## Contribution

It proposes a novel, general concept of recurrence in G-spaces using families of subsets and group actions, extending previous notions.

## Key findings

- Defined $rak{F}$-recurrence in G-spaces.
- Characterized properties of $rak{F}$-recurrent sets.
- Provided a framework for future recurrence analysis.

## Abstract

We introduce and analyze the following general concept of recurrence. Let $G$ be a group and let $X$ be a G-space with the action $G\times X\longrightarrow X$, $(g,x)\longmapsto gx$. For a family $\mathfrak{F}$ of subset of $X$ and $A\in \mathfrak{F}$, we denote $\Delta_{\mathfrak{F}}(A)=\{g\in G: gB\subseteq A$ for some $B\in \mathfrak{F}, \ B\subseteq A\}$, and say that a subset $R$ of $G$ is $\mathfrak{F}$-recurrent if $R\bigcap \Delta_{\mathfrak{F}} (A)\neq\emptyset$ for each $A\in \mathfrak{F}$.