Topologically equivalent

In mathematics, two functions are said to be topologically conjugate to one another if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy is important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterated function can be solved, then those for any topologically conjugate function follow trivially.

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

so that f and g are topologically conjugate. Then of course one must have

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

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