topological conjugation

Let X and Y be topological spacesMathworldPlanetmath, and let f:XX and g:YY be continuous functionsMathworldPlanetmathPlanetmath. We say that f is topologically semiconjugate to g, if there exists a continuous surjection h:YX such that fh=hg. If h is a homeomorphism, then we say that f and g are topologically conjugate, and we call h a topological conjugation between f and g.

Similarly, a flow φ on X is topologically semiconjugate to a flow ψ on Y if there is a continuous surjection h:YX such that φ(h(y),t)=hψ(y,t) for each yY, t. If h is a homeomorphism then ψ and φ are topologically conjugate.

0.1 Remarks

Topological conjugation defines an equivalence relationMathworldPlanetmath in the space of all continuous surjections of a topological space to itself, by declaring f and g to be related if they are topologically conjugate. This equivalence relation is very useful in the theory of dynamical systemsMathworldPlanetmathPlanetmath, since each class contains all functions which share the same dynamics from the topological viewpoint. In fact, orbits of g are mapped to homeomorphic orbits of f through the conjugationMathworldPlanetmath. Writing g=h-1fh makes this fact evident: gn=h-1fnh. 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 φ(,t) and ψ(,t) to be topologically conjugate for each t, which is requiring more than simply that orbits of φ be mapped to orbits of ψ homeomorphically. This motivates the definition of topological equivalence, which also partitionsPlanetmathPlanetmath the set of all flows in X into classes of flows sharing the same dynamics, again from the topological viewpoint.

We say that ψ and φ are topologically equivalent, if there is an homeomorphism h:YX, mapping orbits of ψ to orbits of φ homeomorphically, and preserving orientation of the orbits. This means that:

  1. 1.

    h(𝒪(y,ψ))={h(ψ(y,t)):t}={φ(h(y),t):t}=𝒪(h(y),φ) for each yY;

  2. 2.

    for each yY, there is δ>0 such that, if 0<|s|<t<δ, and if s is such that φ(h(y),s)=h(ψ(y,t)), then s>0.

Title topological conjugation
Canonical name TopologicalConjugation
Date of creation 2013-03-22 13:41:02
Last modified on 2013-03-22 13:41:02
Owner Koro (127)
Last modified by Koro (127)
Numerical id 14
Author Koro (127)
Entry type Definition
Classification msc 37C15
Classification msc 37B99
Defines topologically conjugate
Defines topological semiconjugation
Defines topologically semiconjugate
Defines topologically equivalent
Defines topological equivalence