Topological conjugation

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 .

definition

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 ${\displaystyle f\colon X\rightarrow X}$ ${\displaystyle g\colon Y\rightarrow Y}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle h\colon X\rightarrow Y}$

${\displaystyle h\circ f=g\circ h.}$

If it is only a continuous surjective mapping , then we say that and are topologically semiconjugate . ${\displaystyle h}$ ${\displaystyle g}$ ${\displaystyle f}$

Analogously, we say that two flows on and on are topologically conjugated (topologically semiconjugated) if a homeomorphism (a continuous surjective mapping) exists such that ${\displaystyle \varphi }$ ${\displaystyle X}$ ${\displaystyle \psi }$ ${\displaystyle Y}$ ${\displaystyle h\colon Y\to X}$

${\displaystyle \varphi (h(y),t)=h\psi (y,t),\ t\in \mathbb {R} .}$

discussion

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 . ${\displaystyle f}$

From the above definition we inductively see that

${\displaystyle f=h^{-1}\circ g\circ h{\text{ und }}f^{n}=h^{-1}\circ g^{n}\circ h.}$

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. ${\displaystyle f\colon X\to X}$ ${\displaystyle g\colon Y\to Y}$ ${\displaystyle f}$ ${\displaystyle g}$

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

example

Be it

${\displaystyle {\begin{aligned}G_{r}\colon \mathbb {R} &\to \mathbb {R} \\x&\mapsto rx(1-x)\end{aligned}}}$

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: ${\displaystyle G_{r}}$ ${\displaystyle r>2+{\sqrt {5}}}$

${\displaystyle {\begin{aligned}A_{0}&=\left\{x\in [0,1]\mid G_{r}(x)>1\right\},\\A_{n}&=\left\{x\in [0,1]\mid G_{r}^{i}(x)\in A_{i-1}\ \forall \ i=1,\ldots ,n\right\}\end{aligned}}}$

and

${\displaystyle A=[0,1]\setminus (\bigcup _{n=0}^{\infty }A_{n}).}$

literature

• Werner Krabs: Dynamic systems: controllability and chaotic behavior . BGTeubner, Leipzig 1998, ISBN 3-519-02638-4 .