Definition 40.1.1 (Topological Dynamical System).label Let $G$ be a group and $X$ be a (non-empty) compact Hausdorff space, then a $G$-flow/topological dynamical system is a homomorphism $\alpha: G \to \text{Aut}(X)$. In which case, the system is denoted by $G \curvearrowright X$.