Topological conjugacy

From Topological conjugation is called in mathematics, if there is a homeomorphism, the conjugate is a continuous map to another. The concept is in the analysis of dynamical systems of greater importance, namely here, especially when considering discrete systems.

Definition

Let X and Y be two metric spaces and continuous maps as well as two. Then hot and topologically conjugate if there is a homeomorphism such that

If only a surjective map, then we say that and are semikonjugiert topologically.

We say analog, on two rivers and are ( topologically semikonjugiert ) topologically conjugate if ( a surjective map ) there is a homeomorphism, so that

Discussion

The concept of the topological conjugation two images is particularly useful for the analysis of the dynamic system represented by it is of great importance. Because there are a number of topological invariants, ie topological properties of an image that are invariant under topological conjugation. In this sense, one can consider the topological conjugation as a kind of coordinate transformation.

We see from the above definition inductively immediately that

This allows us to conclude that orbits of a dynamic system shown under the topological conjugation on the orbits of the topologically conjugate dynamical system, namely periodic and non-periodic to periodic orbits on non-periodic orbits.

More significant for the analysis of the dynamics, however, is the finding that even chaos is a topological invariant. Because for the two topologically conjugated pictures and the following applies: if and only messy when is chaotic.

Other invariants under the topological conjugation, for example topological transitivity, sensitive dependence on initial values ​​and the topological entropy.

Example

It should be

The logistic map. It can now show with the help of topological conjugation that for parameter values ​​of chaotic operates on the defined inductively as follows Cantor set

And