Rotation number
The rotation number is an invariant of self-portrayals of the circle , which was first examined by Henri Poincaré in 1885 in his work on celestial mechanics . Homeomorphisms of circles occur there as Poincaré maps (return maps) of 2-dimensional rivers and the number of rotations of the Poincaré map provides information about the long-term behavior of the 2-dimensional flow .
definition
Let it be an orientation- preserving homeomorphism of the circle (see circle group ). Then there is an elevation from to a homeomorphism of the number line with
for every real number and every integer .
The rotation number of is defined using the iteration of as:
- .
Henri Poincaré proved that the limit value exists and does not depend on the choice of the starting point .
The elevation is only clearly defined modulo integer displacements, therefore the rotation number is a well-defined element . It clearly measures the average angle of rotation along the orbit of .
Examples
- If the rotation is about the angle , then is .
- If has at least one fixed point, then is . If has no fixed points, then is .
- is rational if and only if has a periodic point . If is a rational number , then all periodic points have the period .
properties
- The rotation number is invariant under conjugation : if is a homeomorphism, then is .
- The number of rotations continuously depends on , i. H. if a sequence converges uniformly to , then converges to .
Applications
- Poincaré's classification theorem : If is irrational, then there is a monotonic, continuous mapping with
- ,
- where denotes the rotation through the angle . is a homeomorphism if and only if the action of under iteration is transitive. Examples of non-transitory homeomorphisms with irrational rotation numbers were constructed by Denjoy.
- If is rational and orientation preserving, then there are two possible types of periodic orbits:
- If has exactly one periodic orbit, then every other point is below heteroclinic to two points on the periodic orbit.
- If has multiple periodic orbits, then every other point below is heteroclinic to two points on different periodic orbites.
Generalizations
There are several generalizations of the rotation number, all of which (including Poincaré's classic definition) fit into the following approach.
Let it be a locally compact topological group and a bounded -value Borel cohomology class. Be it . Because the restriction of the corresponding real-valued cohomology class to the subgroup disappears, there is because of the exact sequence
a clear continuous homomorphism in the archetype of . The rotation number of is then defined as
- .
- The classic rotation number is obtained for and for the Euler class .
- For and the image from below we get the symplectic rotation number.
- For the automorphism groups of symmetrical regions of the tube type one obtains the Clerc-Koufany rotation number.
literature
- Katok, Anatole; Hasselblatt, Boris: Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995. ISBN 0-521-34187-6 (Chapter 11)
- Aranson, S. Kh .; Belitsky, GR; Zhuzhoma, EV: Introduction to the qualitative theory of dynamical systems on surfaces. Translated from the Russian manuscript by HH McFaden. Translations of Mathematical Monographs, 153. American Mathematical Society, Providence, RI, 1996. ISBN 0-8218-0369-7
- Herman, Michael-Robert: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Inst. Hautes Études Sci. Publ. Math. No. 49: 5-233 (1979). doi : 10.1007 / BF02684798
Web links
- Misiurewicz: Rotation theory
- Menon: Circle maps
Individual evidence
- ↑ is the cohomology of the -invariants of the complex of restricted Borel measurable maps
- ↑ Burger, Marc; Iozzi, Alessandra; Wienhard, Anna: Surface group representations with maximally Toledo invariant. Ann. of Math. (2) 172 (2010), no. 1, 517-566. pdf (chapter 7)
- ^ Ghys, Étienne: Groupes d'homéomorphismes du cercle et cohomologie bornée. The Lefschetz centennial conference, Part III (Mexico City, 1984), 81-106, Contemp. Math., 58, III, Amer. Math. Soc., Providence, RI, 1987.
- ↑ Barge, J .; Ghys, É .: Cocycles d'Euler et de Maslov. Math. Ann. 1992, 294, no. 2, 235-265. pdf
- ↑ Clerc, Jean-Louis; Koufany, Khalid: Primitive du cocycle de Maslov généralisé. Math. Ann. 337 (2007), no. 1, 91-138. pdf