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Concept
Topological conjugacy
Summary
In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially.
To illustrate this directly: suppose that and are iterated functions, and there exists a homeomorphism such that
so that and are topologically conjugate. Then one must have
and so the iterated systems are topologically conjugate as well. Here, denotes function composition.
and are continuous functions on topological spaces, and .
being topologically semiconjugate to means, by definition, that is a surjection such that .
and being topologically conjugate means, by definition, that they are topologically semiconjugate and is furthermore injective, then bijective, and its inverse is continuous too; i.e. is a homeomorphism; further, is termed a topological conjugation between and .
Similarly, on , and on are flows, with , and as above.
being topologically semiconjugate to means, by definition, that is a surjection such that , for each , .
and being topologically conjugate means, by definition, that they are topologically semiconjugate and h is a homeomorphism.
The logistic map and the tent map are topologically conjugate.
The logistic map of unit height and the Bernoulli map are topologically conjugate.
For certain values in the parameter space, the Hénon map when restricted to its Julia set is topologically conjugate or semi-conjugate to the shift map on the space of two-sided sequences in two symbols.
Topological conjugation – unlike semiconjugation – defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring and to be related if they are topologically conjugate.
Official source
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