Rotation number.html

This article is about the rotation number, which is sometimes called the map winding number or simply winding number. There is another meaning for winding number, which appears in complex analysis.

In mathematics, the rotation number gives the asymptotic behaviour of an iterated function. The rotation number is sometimes termed the winding number. It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit.

1 Definition for a circle map
2 Examples
3 See also
4 References

Definition for a circle map

Given an iterated circle map f, the rotation number of the map is given by

$\omega(x)=\lim_{n\to\infty} \frac{f^n(x)-x}{n}.$

where $f n$ is the n 'th iterate of f. When f is invertible, the rotation number is independent of the starting point x.

For the special case of the circle group or the irrational rotation, the rotation number is simply the generator of the rotation.

Examples

The rotation number plays an important part in the analysis of the circle map.

References

Sebastian van Strien, Rotation Numbers and Poincaré's Theorem, (2001)

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Rotation number.html

Contents

Definition for a circle map

Examples

See also

References