Definition 43.1.1 (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 $G \curvearrowright X$.
Let $G \curvearrowright X$ and $G \curvearrowright$ be $G$-flows, then a morphism of $G$-flows is a continuous map $\pi: X \to Y$ such that the following diagram commutes
\[\xymatrix{ X \ar@{->}[r]^{\pi} \ar@{->}[d]_{g} & Y \ar@{->}[d]^{g} \\ X \ar@{->}[r]_{\pi} & Y }\]
for all $g \in G$.
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