In other words, it is a level set of a rotation number with nonempty interior.

The rotation number of f is an irrational number θ.

One may also define a rotation number.

The Bartels rotation number of the Sun based on a regular 27 day cycle is named after him.

The rotation number of f is a rational number p/q (in the lowest terms).

The term winding number may also refer to the rotation number of an iterated map.

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

He also gave an example of a C diffeomorphism with an irrational rotation number that is not conjugate to a rotation.

It consists of parameters that have an attracting cycle of period and combinatorial rotation number .

This article is about the rotation number, which is sometimes called the map winding number or simply winding number.

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

The values of (K,Ω) in one of these regions will all result in a motion such that the rotation number .

Poincaré later proved a theorem characterizing the existence of periodic orbits in terms of rationality of the rotation number.

V. Afraimovich and T. Young, Relative density of irrational rotation numbers in families of circle di eomorphisms.

These can be detected by considering their Thurston-Bennequin invariants and rotation number, which are together known as the "classical invariants" of Legendrian knots.

According to the Denjoy theorem, every orientation-preserving C-diffeomorphism of the circle with an irrational rotation number θ is topologically conjugate to T.

For example, all values of (K,Ω) in the large V-shaped region in the bottom-center of the figure correspond to a rotation number of .

The relative contact homology has been shown to be a strictly more powerful invariant than the "classical invariants", namely Thurston-Bennequin number and rotation number (within a class of smooth knots).

One reason the term "locking" is used is that the individual values can be perturbed by rather large random disturbances (up to the width of the tongue, for a given value of K), without disturbing the limiting rotation number.

If ƒ is a C map, then the hypothesis on the derivative holds; however, for any irrational rotation number Denjoy constructed an example showing that this condition cannot be relaxed to C, continuous differentiability of ƒ.

In mathematics, particularly in dynamical systems theory, an Arnold tongue of a finite-parameter family of circle maps, named after Vladimir Arnold, is a region in the space of parameters where the map has locally-constant rational rotation number.

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.

Later Michel Herman proved that nonetheless, the conjugating map of an analytic diffeomorphism is itself analytic for "most" rotation numbers, forming a set of full Lebesgue measure, namely, for those that are badly approximable by rational numbers.