Denjoy's theorem

The title of this article is ambiguous. For more meanings, see Denjoy's Theorem (disambiguation) .

In mathematics , Denjoy's theorem is a fundamental result from the theory of dynamic systems . It says that twice continuously differentiable self-mappings of the circle cannot leave a Cantor set invariant, and further that twice continuously differentiable self-mappings of the circle with an irrational rotation number are always topologically conjugated to a rotation. Both results are wrong for self-maps that are only continuously differentiable once.

Invariant sets

For an orientation-preserving diffeomorphism , every minimal non-empty closed invariant subset of either ${\displaystyle f\colon S^{1}\to S^{1}}$ ${\displaystyle S^{1}}$

• a finite set, or
• whole , or ${\displaystyle S^{1}}$
• a Cantor crowd.

The following example shows that the third case for -diffeomorphisms is indeed possible. However, Denjoy's theorem says that the third case cannot occur for -diffeomorphisms. In particular, in the case of irrational rotation numbers, all orbites are then dense in . ${\displaystyle C^{1}}$ ${\displaystyle C^{2}}$ ${\displaystyle S^{1}}$

Denjoy's example

There is an orientation- preserving -diffeomorphism that has no periodic points but leaves a Cantor set invariant. ${\displaystyle C^{1}}$ ${\displaystyle f\colon S^{1}\to S^{1}}$

For every irrational number one can construct such a Denjoy diffeomorphism which is semi-conjugated to the rotation . ${\displaystyle \alpha }$ ${\displaystyle R_{\alpha }}$

The idea is roughly as follows. Take the orbit of an irrational rotation, this is a countable dense set . Cut open at the points of . For each one fill in an interval , the sum of the interval lengths should be finite so that the space constructed in this way is homeomorphic to the circle again. A continuous monotonic mapping is constructed, which is the rotation outside of the inserted intervals and which maps continuously and strictly monotonically to . One obtains a homeomorphism (a space that is too homeomorphic) which has no periodic points, leaves a Cantor set invariant and is conjugated to an irrational rotation. You can modify it to become. ${\displaystyle R_{\alpha }^{n}(z_{0})}$ ${\displaystyle A\subset S^{1}}$ ${\displaystyle S^{1}}$ ${\displaystyle A}$ ${\displaystyle R_{\alpha }^{n}(z_{0})\in A}$ ${\displaystyle I_{n}}$ ${\displaystyle R_{\alpha }}$ ${\displaystyle I_{n}}$ ${\displaystyle I_{n+1}}$ ${\displaystyle S^{1}}$ ${\displaystyle C^{1}}$

The mapping torus of the restriction of this mapping to the invariant Cantor set is called the Denjoy solenoid .

Denjoy's theorem

If is an orientation- preserving -diffeomorphism, then every minimal non-empty closed invariant subset of is either a finite set, or whole . ${\displaystyle f\colon S^{1}\to S^{1}}$ ${\displaystyle C^{2}}$ ${\displaystyle S^{1}}$ ${\displaystyle S^{1}}$

As a consequence of Denjoy's theorem, one obtains that every orientation- preserving -diffeomorphism with an irrational rotation number is topologically conjugated to a rotation. (This sentence is also known as Denjoy's sentence . With this sentence Arnaud Denjoy refuted a conjecture by Henri Poincaré .) ${\displaystyle C^{2}}$

In general, the conjugate map is only continuous. Under certain arithmetic requirements for the number of rotations, however, a differentiable conjugation map is obtained.

Foliage

Suspension of a figure gives a scroll on the 2-dimensional torus . Denjoy's example shows that a -following can have an exceptional minimum set and it follows from Denjoy's theorem that a -following of a surface has no exceptional minimum sets . ${\displaystyle S^{1}\to S^{1}}$ ${\displaystyle C^{1}}$ ${\displaystyle C^{2}}$

On the other hand, there are examples of -following of codimension 1 on higher-dimensional manifolds that have an exceptional minimum amount. The higher-dimensional generalization of Denjoy's theorem is instead the Sacksteder theorem : an exceptional minimal set of foliage of codimension 1 on a compact manifold always contains a leaf with nontrivial linearized holonomy . ${\displaystyle C^{\infty }}$ ${\displaystyle C^{2}}$

literature

• Arnaud Denjoy: Sur les courbes définies par les equations différentielles à la surface du tore , J. Math. Pures Appl. (9) 11: 333-375 (1932). online (pdf)
• VV Nemyckiĭ, V. V, Stepanov: Kačestvennaya Teoriya Differencialʹnyh Uravneniĭ. (Russian), OGIZ, Moscow-Leningrad (1947).
• Harold Rosenberg: Un contre-example à la conjecture de Seifert (d'après P. Schweitzer). Séminaire Bourbaki, 25ème année (1972/1973), Exp. 434, pp. 294-306. Lecture Notes in Math., Vol. 383, Springer, Berlin, 1974. online (pdf)
• Gilbert Hector, Ulrich Hirsch: Introduction to the geometry of foliations. Part A. Foliations on compact surfaces, fundamentals for arbitrary codimension, and holonomy. Second edition. Aspects of Mathematics, 1st Friedr. Vieweg & Sohn, Braunschweig, 1986. ISBN 3-528-18501-5
• Alberto Candel, Lawrence Conlon: Foliations. I. Graduate Studies in Mathematics, 23rd American Mathematical Society, Providence, RI, 2000. ISBN 0-8218-0809-5

Web links

• John Milnor: Introductory Dynamics Lecture , Chapter 15: Denjoy's Theorem (pdf)

Individual evidence

1. ↑ Hector-Hirsch, Chapter 4
2. ↑ Candel-Conlon, Example 4.1.10
3. ↑ Hector-Hirsch, Chapter 5.2
4. ↑ Rosenberg
5. ↑ Nemyckiĭ-Stepanov
6. ^ Hector-Hirsch, Corollary 5.3.3.
7. ↑ Candel-Conlon, Exercise 9.2.19
8. ↑ Michel Herman: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Inst. Hautes Études Sci. Publ. Math. No. 49: 5-233 (1979). online (pdf)
9. ↑ Milnor, §15C
10. ↑ Hector-Hirsch, Chapter VI