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 \(G\times X\longrightarrow X\), \((g,x)\longmapsto gx\). 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 \(A\in \mathfrak{F}\).