, 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 f and g are iterated functions, and there exists an 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, denotes function composition.
As examples, the logistic map and the tent map are topologically conjugate. Furthermore, the logisitic map of unit height and the Bernoulli map are topologically conjugate.
be topological spaces
, and let
be continuous functions
. We say that
is topologically semiconjugate
, if there exists a continuous surjection
is a homeomorphism
, then we say that
are topologically conjugate
, and we call
a topological conjugation
Similarly, a flow on is topologically semiconjugate to a flow on if there is a continuous surjection such that for each , . If is a homeomorphism then and are topologically conjugate.
Topological conjugation defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring
to be related if they are topologically conjugate. This equivalence relation is very useful in the theory of dynamical systems
, since each class contains all functions which share the same dynamics from the topological viewpoint. For example, orbits
are mapped to homeomorphic orbits of
through the conjugation. Writing
makes this fact evident:
. Speaking informally, topological conjugation is a “change of coordinates” in the topological sense.
However, the analogous definition for flows is somewhat restrictive. In fact, we are requiring the maps and to be topologically conjugate for each , which is requiring more than simply that orbits of be mapped to orbits of homeomorphically. This motivates the definition of topological equivalence, which also partitions the set of all flows in into classes of flows sharing the same dynamics, again from the topological viewpoint.
We say that and are topologically equivalent, if there is an homeomorphism , mapping orbits of to orbits of homeomorphically, and preserving orientation of the orbits. In other words, letting denote an orbit, one has
for each . In addition, one must line up the flow of time: for each , there exists a such that, if