# 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

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