> [!definition] > > Let $X$ be a [[Compactness|compact]] [[Metric Space|metric space]], and $T: X \to X$ be a [[Homeomorphism|homeomorphism]], then the pair $(X, T)$ is a **topological system**. The topological system $(X, T)$ is **ergodic** if the following equivalent conditions are satisfied: > 1. $(X, T)$ has no proper subsystems. > 2. Every orbit is [[Dense|dense]] in $X$. > [!theorem] > > Let $(X, S), (Y, T)$ be topological systems, then they are isomorphic if there exists a homeomorphism $f: X \to Y$ such that $S = f^{-1} \circ T \circ f$.