- Topological conjugacy
In
mathematics , two functions are said to be topologically conjugate to one another if there exists ahomeomorphism that will conjugate the one into the other. Topological conjugacy is important in the study ofiterated function s and more generallydynamical 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 thetent map are topologically conjugate. Furthermore, the logisitic map of unit height and theBernoulli map are topologically conjugate.Definition
Let and be
topological space s, and let and becontinuous function s. We say that is topologically semiconjugate to , if there exists a continuoussurjection such that . If is ahomeomorphism , then we say that and are topologically conjugate, and we call a topological conjugation between and .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.
Discussion
Topological conjugation defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring and to be related if they are topologically conjugate. This equivalence relation is very useful in the theory of
dynamical system s, since each class contains all functions which share the same dynamics from the topological viewpoint. For example, orbits of 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.
Topological Equivalence
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
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