In mathematics, one speaks of topological conjugation when there is a homeomorphism that conjugates one continuous map to another. The concept is more important in the analysis of dynamic systems , especially when considering discrete systems .
Let X and Y be two metric spaces and well as two continuous maps. Then be called and topologically conjugated if there is a homeomorphism such that
If it is only a continuous surjective mapping , then we say that and are topologically semiconjugate .
Analogously, we say that two flows on and on are topologically conjugated (topologically semiconjugated) if a homeomorphism (a continuous surjective mapping) exists such that
The concept of the topological conjugation of two mappings is particularly important when analyzing the dynamic systems given by them . Because there are a number of topological invariants , i.e. topological properties of a mapping , which are invariant under the topological conjugation. In this sense, the topological conjugation can be viewed as a kind of coordinate transformation .
From the above definition we inductively see that
With this we can conclude that orbits of a dynamic system under the topological conjugation are mapped to the orbits of the topologically conjugated dynamic system, namely periodic to periodic orbits and non-periodic to non-periodic orbits.
Much more important for the analysis of the dynamics, however, is the observation that chaos is also a topological invariant. Because for the two topologically conjugated maps and the following applies: is chaotic if and only if is chaotic.
Other invariants under the topological conjugation are, for example, topological transitivity , sensitive dependence on the initial values and topological entropy .
the logistic mapping . With the help of the topological conjugation it can now be shown that for parameter values of the Cantor set inductively defined as follows operates chaotically:
- Werner Krabs: Dynamic systems: controllability and chaotic behavior . BGTeubner, Leipzig 1998, ISBN 3-519-02638-4 .