On recurrence in G-spaces

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 $G\times X⟶X$, $(g,x)⟼gx$. 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 $R\bigcap {\Delta }_{\mathfrak{F}}(A)\ne \varnothing$ for each $A\in \mathfrak{F}$.

$G$-space, recurrent subset, ultrafilters, Stone-$\check{C}$ech compactification