Definition of Topological Conjugacy
Definition
Autonomous Systems
For topological spaces $X, Y$, let two functions $f : X \to X$ and $g : Y \to Y$ be continuous. If there exists a homeomorphism $h : Y \to X$ satisfying $h \circ g = f \circ h$, then the two autonomous systems $f$ and $g$ are said to be topologically conjugate to each other.
Flows
Suppose there are two flows $\phi : \mathbb{R} \times X \to X$ and $\psi : \mathbb{R} \times Y \to Y$. Likewise, if there exists a homeomorphism $h : Y \to X$ such that $\phi \left( t, h (y) \right) = h \left( \psi \left( t, y \right) \right)$ holds for all $y \in Y$ and $t \in \mathbb{R}$, then the two flows $\phi$ and $\psi$ are said to be topologically conjugate to each other.
Explanation
Since these definitions refer specifically to autonomous systems and flows, they are closely related to the notions of class equivalence and conjugacy between dynamical systems.