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).
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.
- 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, Zbl 0448.58019
- Kornfeld, Sinai, Fomin, Ergodic theory.