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 .

Overlay of the circle by the number line . Every continuous mapping can be lifted up to a continuous mapping with for all . ${\displaystyle X=S^{1}=\mathbb {R} /\mathbb {Z} }$ ${\displaystyle Y=\mathbb {R} }$ ${\displaystyle f\colon S^{1}\to S^{1}}$ ${\displaystyle F\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle F(x+m)=F(x)+m}$ ${\displaystyle m\in \mathbb {Z} }$

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 ${\displaystyle f\colon S^{1}\to S^{1}}$ ${\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }$ ${\displaystyle f}$ ${\displaystyle F\colon \mathbb {R} \to \mathbb {R} }$

${\displaystyle F(x+m)=F(x)+m}$

for every real number and every integer . ${\displaystyle x}$ ${\displaystyle m}$

The rotation number of is defined using the iteration of as: ${\displaystyle f}$ ${\displaystyle F}$

${\displaystyle \omega (f)=\lim _{n\to \infty }{\frac {F^{n}(x)-x}{n}}}$.

Henri Poincaré proved that the limit value exists and does not depend on the choice of the starting point . ${\displaystyle x}$

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 . ${\displaystyle F}$ ${\displaystyle \mathbb {R} /\mathbb {Z} }$ ${\displaystyle f}$

Examples

Rotations of the circle

• If the rotation is about the angle , then is . ${\displaystyle f=R_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \omega (f)={\frac {1}{2\pi }}\alpha }$
• If has at least one fixed point, then is . If has no fixed points, then is . ${\displaystyle f}$ ${\displaystyle \omega (f)=0}$ ${\displaystyle f}$ ${\displaystyle \omega (f)\not =0}$
• ${\displaystyle \omega (f)}$is rational if and only if has a periodic point . If is a rational number , then all periodic points have the period . ${\displaystyle f}$ ${\displaystyle \omega (f)}$ ${\displaystyle {\frac {p}{q}}}$ ${\displaystyle q}$

properties

• The rotation number is invariant under conjugation : if is a homeomorphism, then is . ${\displaystyle h\colon S^{1}\to S^{1}}$ ${\displaystyle \omega (hfh^{-1})=\omega (f)}$
• The number of rotations continuously depends on , i. H. if a sequence converges uniformly to , then converges to . ${\displaystyle f}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle \omega (f_{n})}$ ${\displaystyle \omega (f)}$

Applications

• Poincaré's classification theorem : If is irrational, then there is a monotonic, continuous mapping with ${\displaystyle \omega (f)}$ ${\displaystyle h}$

${\displaystyle h\circ f=R_{\omega (f)}\circ h}$,

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. ${\displaystyle R_{\omega (f)}}$ ${\displaystyle \omega (f)}$ ${\displaystyle h}$ ${\displaystyle f}$

• If is rational and orientation preserving, then there are two possible types of periodic orbits: ${\displaystyle \omega (f)}$ ${\displaystyle f}$
  • If has exactly one periodic orbit, then every other point is below heteroclinic to two points on the periodic orbit. ${\displaystyle f}$ ${\displaystyle f^{q}}$
  • If has multiple periodic orbits, then every other point below is heteroclinic to two points on different periodic orbites. ${\displaystyle f}$ ${\displaystyle f^{q}}$

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 ${\displaystyle G}$ ${\displaystyle \kappa \in {\widehat {H}}_{cb}^{2}(G,\mathbb {Z} )}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle g\in G}$ ${\displaystyle \kappa _{\mathbb {R} }\in {\widehat {H}}_{cb}^{2}(G,\mathbb {R} )}$ ${\displaystyle B:={\overline {\langle g\rangle }}\subset G}$

${\displaystyle 0\to \operatorname {Hom} _{c}(B,\mathbb {R} /Z)\to {\widehat {H}}_{cb}^{2}(B,\mathbb {Z} )\to {\widehat {H}}_{cb}^{2}(B,\mathbb {R} )\to 0}$

a clear continuous homomorphism in the archetype of . The rotation number of is then defined as ${\displaystyle f\colon B\to \mathbb {R} /\mathbb {Z} }$ ${\displaystyle \kappa }$ ${\displaystyle g}$

${\displaystyle \operatorname {Rot} _{\kappa }(g):=f(g)}$.

• The classic rotation number is obtained for and for the Euler class . ${\displaystyle G=\operatorname {Homeo} ^{+}(S^{1})}$ ${\displaystyle \kappa =e}$
• For and the image from below we get the symplectic rotation number. ${\displaystyle G=\operatorname {Sp} (n,\mathbb {R} )}$ ${\displaystyle \kappa }$ ${\displaystyle u(k)=\det(k)^{2}}$ ${\displaystyle \operatorname {Hom} _{c}(U(n),\mathbb {R} /Z)\to {\widehat {H}}_{cb}^{2}(U(n),\mathbb {Z} )={\widehat {H}}_{cb}^{2}(\operatorname {Sp} (2n,\mathbb {R} ),\mathbb {Z} )}$
• 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

1. ↑ is the cohomology of the -invariants of the complex of restricted Borel measurable maps ${\displaystyle {\widehat {H}}_{cb}^{*}(G,\mathbb {Z} )}$ ${\displaystyle G}$ ${\displaystyle G^{n+1}\to \mathbb {Z} }$
2. ↑ 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)
3. ^ 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.
4. ↑ Barge, J .; Ghys, É .: Cocycles d'Euler et de Maslov. Math. Ann. 1992, 294, no. 2, 235-265. pdf
5. ↑ Clerc, Jean-Louis; Koufany, Khalid: Primitive du cocycle de Maslov généralisé. Math. Ann. 337 (2007), no. 1, 91-138. pdf