Rotation number

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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.

Contents 1 Definition 2 Examples 3 See also 4 References

[edit] Definition

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

where fⁿ is the n 'th iterate of f. In almost all cases, the rotation number is independent of the starting point x.

[edit] Examples

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

[edit] See also

• Poincaré section
• Poincaré recurrence

[edit] References

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