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单词 TopologicalConjugation
释义

topological conjugation


Let X and Y be topological spacesMathworldPlanetmath, and let f:XX and g:YYbe continuous functionsMathworldPlanetmathPlanetmath. We say that f istopologically semiconjugate to g, if there exists a continuoussurjection h:YX such that fh=hg. If h is a homeomorphism,then we say that f and g are topologically conjugate, and we callh 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 thatφ(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 thespace of all continuous surjections of a topological space to itself,by declaring f and g to be related if they are topologicallyconjugate. This equivalence relation is very useful in the theory ofdynamical systemsMathworldPlanetmathPlanetmath, since each class contains all functions whichshare the same dynamics from the topological viewpoint. In fact, orbitsof 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.

Titletopological conjugation
Canonical nameTopologicalConjugation
Date of creation2013-03-22 13:41:02
Last modified on2013-03-22 13:41:02
OwnerKoro (127)
Last modified byKoro (127)
Numerical id14
AuthorKoro (127)
Entry typeDefinition
Classificationmsc 37C15
Classificationmsc 37B99
Definestopologically conjugate
Definestopological semiconjugation
Definestopologically semiconjugate
Definestopologically equivalent
Definestopological equivalence
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