Denjoy's theorem on rotation number

In mathematics, the Denjoy theorem gives a sufficient condition for a diffeomorphism of the circle to be topologically conjugate to a diffeomorphism of a special kind, namely an irrational rotation. Denjoy (1932) proved the theorem in the course of his topological classification of homeomorphisms of the circle. He also gave an example of a C1 diffeomorphism with an irrational rotation number that is not conjugate to a rotation.

Statement of the theorem

Let ƒ: S1 → S1 be an orientation-preserving diffeomorphism of the circle whose rotation number θ = ρ(ƒ) is irrational. Assume that it has positive derivative ƒ ′(x) > 0 that is a continuous function with bounded variation on the interval [0,1). Then ƒ is topologically conjugate to the irrational rotation by θ. Moreover, every orbit is dense and every nontrivial interval I of the circle intersects its forward image ƒ°q(I), for some q > 0 (this means that the non-wandering set of ƒ is the whole circle).

Complements

If ƒ is a C2 map, then the hypothesis on the derivative holds; however, for any irrational rotation number Denjoy constructed an example showing that this condition cannot be relaxed to C1, continuous differentiability of ƒ.

Vladimir Arnold showed that the conjugating map need not be smooth, even for an analytic diffeomorphism of the circle. Later Michel Herman proved that nonetheless, the conjugating map of an analytic diffeomorphism is itself analytic for "most" rotation numbers, forming a set of full Lebesgue measure, namely, for those that are badly approximable by rational numbers. His results are even more general and specify differentiability class of the conjugating map for Cr diffeomorphisms with any r ≥ 3.

gollark: There would be ethical problems with simulating civilizations accurately enough.
gollark: Possibly not a shame since some of them would end horribly... still though.
gollark: It's a shame we can't just set up "test civilizations" somewhere and see how well each thing works.
gollark: I mean. Maybe it could work in small groups. But small tribe-type setups scale poorly.
gollark: 1. Is that seriously how you read what I was saying? I was saying: fix our minds' weird ingroup/outgroup division.2. That is very vague and does not sound like it could actually work.

See also
  • Circle map

References
  • Denjoy, Arnaud (1932), "Sur les courbes definies par les équations différentielles à la surface du tore.", Journal de Mathématiques Pures et Appliquées (in French), 11: 333–375, Zbl 0006.30501
  • Herman, M.R. (1979), "Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations", Publ. Math. IHES (in French), 49: 5–234, doi:10.1007/BF02684798, Zbl 0448.58019
  • Kornfeld, Sinai, Fomin, Ergodic theory.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.