In mathematics, the **rotation number** is an invariant of homeomorphisms of the circle.

It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincaré later proved a theorem characterizing the existence of periodic orbits in terms of rationality of the rotation number.

Suppose that *f*: *S*^{1} → *S*^{1} is an orientation preserving homeomorphism of the circle *S*^{1} = **R**/**Z**. Then *f* may be lifted to a homeomorphism *F*: **R** → **R** of the real line, satisfying

for every real number *x* and every integer *m*.

The **rotation number** of *f* is defined in terms of the iterates of *F*:

Henri Poincaré proved that the limit exists and is independent of the choice of the starting point *x*. The lift *F* is unique modulo integers, therefore the rotation number is a well-defined element of **R**/**Z**. Intuitively, it measures the average rotation angle along the orbits of *f*.

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

then its rotation number is *θ* (cf Irrational rotation).

The rotation number is invariant under topological conjugacy, and even monotone topological **semiconjugacy**: if *f* and *g* are two homeomorphisms of the circle and

for a monotone continuous map *h* of the circle into itself (not necessarily homeomorphic) then *f* and *g* have the same rotation numbers. It was used by Poincaré and Arnaud Denjoy for topological classification of homeomorphisms of the circle. There are two distinct possibilities.

- The rotation number of
*f*is a rational number*p*/*q*(in the lowest terms). Then*f*has a periodic orbit, every periodic orbit has period*q*, and the order of the points on each such orbit coincides with the order of the points for a rotation by*p*/*q*. Moreover, every forward orbit of*f*converges to a periodic orbit. The same is true for*backward*orbits, corresponding to iterations of*f*^{−1}, but the limiting periodic orbits in forward and backward directions may be different. - The rotation number of
*f*is an irrational number*θ*. Then*f*has no periodic orbits (this follows immediately by considering a periodic point*x*of*f*). There are two subcases.

- There exists a dense orbit. In this case
*f*is topologically conjugate to the irrational rotation by the angle*θ*and all orbits are dense. Denjoy proved that this possibility is always realized when*f*is twice continuously differentiable. - There exists a Cantor set
*C*invariant under*f*. Then*C*is a unique minimal set and the orbits of all points both in forward and backward direction converge to*C*. In this case,*f*is semiconjugate to the irrational rotation by*θ*, and the semiconjugating map*h*of degree 1 is constant on components of the complement of*C*.

- There exists a dense orbit. In this case

The rotation number is *continuous* when viewed as a map from the group of homeomorphisms (with topology) of the circle into the circle.

In the mathematical field of topology, a **homeomorphism**, **topological isomorphism**, or **bicontinuous function** is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called **homeomorphic**, and from a topological viewpoint they are the same. The word *homeomorphism* comes from the Greek words *ὅμοιος* (*homoios*) = similar or same and *μορφή* (*morphē*) = shape, form, introduced to mathematics by Henri Poincaré in 1895.

In mathematics, a **Lie group** is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.

In topology, a branch of mathematics, two continuous functions from one topological space to another are called **homotopic** if one can be "continuously deformed" into the other, such a deformation being called a **homotopy** between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

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The **dyadic transformation** is the mapping

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In mathematics, particularly in dynamical systems, **Arnold tongues** are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamical system, or other related invariant property thereof, changes according to two or more of its parameters. The regions of constant rotation number have been observed, for some dynamical systems, to form geometric shapes that resemble tongues, in which case they are called Arnold tongues.

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 *C*^{1} diffeomorphism with an irrational rotation number that is not conjugate to a rotation.

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In mathematics, the **topological entropy** of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Their definition was modelled after the definition of the Kolmogorov–Sinai, or metric entropy. Later, Dinaburg and Rufus Bowen gave a different, weaker definition reminiscent of the Hausdorff dimension. The second definition clarified the meaning of the topological entropy: for a system given by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates. An important **variational principle** relates the notions of topological and measure-theoretic entropy.

In mathematics, **Smale's axiom A** defines a class of dynamical systems which have been extensively studied and whose dynamics is relatively well understood. A prominent example is the Smale horseshoe map. The term "axiom A" originates with Stephen Smale. The importance of such systems is demonstrated by the chaotic hypothesis, which states that, 'for all practical purposes', a many-body thermostatted system is approximated by an Anosov system.

In mathematics, two functions are said to be **topologically conjugate** to one another if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy is important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterated function can be solved, then those for any topologically conjugate function follow trivially.

In the mathematical theory of dynamical systems, an **irrational rotation** is a map

In mathematics, especially in the area of mathematical analysis known as dynamical systems theory, a **linear flow on the torus** is a flow on the *n*-dimensional torus

In mathematics, the **Douady–Earle extension**, named after Adrien Douady and Clifford Earle, is a way of extending homeomorphisms of the unit circle in the complex plane to homeomorphisms of the closed unit disk, such that the extension is a diffeomorphism of the open disk. The extension is analytic on the open disk. The extension has an important equivariance property: if the homeomorphism is composed on either side with a Möbius transformation preserving the unit circle the extension is also obtained by composition with the same Möbius transformation. If the homeomorphism is quasisymmetric, the diffeomorphism is quasiconformal. An extension for quasisymmetric homeomorphisms had previously been given by Lars Ahlfors and Arne Beurling; a different equivariant construction had been given in 1985 by Pekka Tukia. Equivariant extensions have important applications in Teichmüller theory, for example they lead to a quick proof of the contractibility of the Teichmüller space of a Fuchsian group.

In mathematics, and more specifically in the theory of C*-algebras, the **noncommutative tori***A*_{θ}, also known as **irrational rotation algebras** for irrational values of θ, form a family of noncommutative C*-algebras which generalize the algebra of continuous functions on the 2-torus. Many topological and geometric properties of the classical 2-torus have algebraic analogues for the noncommutative tori, and as such they are fundamental examples of a noncommutative space in the sense of Alain Connes.

- M.R. Herman,
*Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations*, Publ. Math. IHES, 49 (1979) pp. 5–234 - Sebastian van Strien,
*Rotation Numbers and Poincaré's Theorem*(2001)

- Michał Misiurewicz (ed.). "Rotation theory".
*Scholarpedia*. - Weisstein, Eric W. "Map Winding Number." From MathWorld--A Wolfram Web Resource

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