In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct § Topological equivalence of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially.

To illustrate this directly: suppose that ${\displaystyle f}$ and ${\displaystyle g}$ are iterated functions, and there exists a homeomorphism ${\displaystyle h}$ such that

so that ${\displaystyle f}$ and ${\displaystyle g}$ are topologically conjugate. Then one must have

and so the iterated systems are topologically conjugate as well. Here, ${\displaystyle \circ }$ denotes function composition.

${\displaystyle f\colon X\to X,g\colon Y\to Y}$, and ${\displaystyle h\colon Y\to X}$ are continuous functions on topological spaces, ${\displaystyle X}$ and ${\displaystyle Y}$.

${\displaystyle f}$ being topologically semiconjugate to ${\displaystyle g}$ means, by definition, that ${\displaystyle h}$ is a surjection such that ${\displaystyle f\circ h=h\circ g}$.

${\displaystyle f}$ and ${\displaystyle g}$ being topologically conjugate means, by definition, that they are topologically semiconjugate and ${\displaystyle h}$ is furthermore injective, then bijective, and its inverse is continuous too; i.e. ${\displaystyle h}$ is a homeomorphism; further, ${\displaystyle h}$ is termed a topological conjugation between ${\displaystyle f}$ and ${\displaystyle g}$.

Similarly, ${\displaystyle \phi }$ on ${\displaystyle X}$, and ${\displaystyle \psi }$ on ${\displaystyle Y}$ are flows, with ${\displaystyle X,Y}$, and ${\displaystyle h\colon Y\to X}$ as above.