On recurrence in $G$-spaces

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 $\Delta_{\mathfrak{F}}(A)=\{g\in G\colon gB\subseteq A$ 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 $A\in \mathfrak{F}$.