  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 $A\in \mathfrak{F}$.
