In mathematics, the rotation number is an

invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

s of the

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

History

It was first defined by

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

in 1885, in relation to the

precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...

of the

perihelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any e ...

of a

planetary orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...

. Poincaré later proved a theorem characterizing the existence of

periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. Iterated functions Given ...

s in terms of rationality of the rotation number.

Definition

Suppose that

f:S^1 \to S^1

is an orientation-preserving

of the

S^1 = \R/\Z.

Then may be lifted to a

F: \R \to \R

of the real line, satisfying :

F(x + m) = F(x) +m

for every real number and every integer . The rotation number of is defined in terms of the iterates of : :

\omega(f)=\lim_ \frac.

proved that the limit exists and is independent of the choice of the starting point . The lift is unique modulo integers, therefore the rotation number is a well-defined element of Intuitively, it measures the average rotation angle along the

orbits In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...

of .

Example

If ''f'' is a rotation by ''2πθ'' (where ''0≤θ<1''), then :

F(x)=x+\theta,

then its rotation number is ''θ'' (cf

Irrational rotation In the mathematical theory of dynamical systems, an irrational rotation is a map : T_\theta : ,1\rightarrow ,1\quad T_\theta(x) \triangleq x + \theta \mod 1 where ''θ'' is an irrational number. Under the identification of a circle wit ...

Properties

The rotation number is invariant under

topological conjugacy In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated fun ...

, and even monotone topological semiconjugacy: if and are two homeomorphisms of the circle and :

h\circ f = g\circ h

for a monotone continuous map of the circle into itself (not necessarily homeomorphic) then and have the same rotation numbers. It was used by Poincaré and

Arnaud Denjoy Arnaud Denjoy (; 5 January 1884 – 21 January 1974) was a French mathematician. Biography Denjoy was born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. His integral was the first to be able to i ...

for topological classification of homeomorphisms of the circle. There are two distinct possibilities. * The rotation number of is a

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

(in the lowest terms). Then has a

, every periodic orbit has period , and the order of the points on each such orbit coincides with the order of the points for a rotation by . Moreover, every forward orbit of converges to a periodic orbit. The same is true for ''backward'' orbits, corresponding to iterations of , but the limiting periodic orbits in forward and backward directions may be different. * The rotation number of is an

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

. Then has no periodic orbits (this follows immediately by considering a periodic point of ). There are two subcases. :# There exists a dense orbit. In this case is topologically conjugate to the

irrational rotation In the mathematical theory of dynamical systems, an irrational rotation is a map : T_\theta : ,1\rightarrow ,1\quad T_\theta(x) \triangleq x + \theta \mod 1 where ''θ'' is an irrational number. Under the identification of a circle wit ...

by the angle and all orbits are

dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

. Denjoy proved that this possibility is always realized when is twice continuously differentiable. :# There exists a

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...

invariant under . Then is a unique minimal set and the orbits of all points both in forward and backward direction converge to . In this case, is semiconjugate to the irrational rotation by , and the semiconjugating map of degree 1 is constant on components of the complement of . The rotation number is ''continuous'' when viewed as a map from the group of homeomorphisms (with topology) of the circle into the circle.

References

* , also ''SciSpace'' for smaller file size i
pdf ver 1.3
* Sebastian van Strien,

' (2001)

External links