External ray

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In complex analysis, particularly in complex dynamics and geometric function theory, external rays are associated to a compact, full, connected subset <math>K\,</math> of the complex plane as the images of radial rays under the Riemann map of the complement of <math>K\,</math>. Equivalently, they are the gradient lines of the Green's function of <math>K\,</math> or field lines of Douady-Hubbard potential .</br> External rays together with equipotential lines of Douady-Hubbard potential form a new polar coordinate system for exterior ( complement ) of <math>K\,</math>.</br> External rays are particularly useful in the dynamical study of complex polynomials, where they were introduced in Douady and Hubbard's study of the Mandelbrot set. External rays of (connected) Julia sets of polynomials are often called dynamic rays, while external rays of the Mandelbrot set (and similar one-dimensional connectedness loci) are called parameter rays.

1 Formal definition
2 References
3 Programs that can draw external rays:
4 External links

[edit] Formal definition

Let

<math>\phi:\mathbb{C}\setminus \overline{\mathbb{D}}\to\mathbb{C}\setminus K</math>

(where <math>\mathbb{D}\,</math> denotes the unit disk) be the unique conformal isomorphism whose leading Laurent coefficient at infinity is real and positive.

Then the external ray of angle <math>\theta\,</math> is the curve

<math>\mathcal{R}^K_{\theta}:(1,\infty)\to\mathbb{C}; t\mapsto \phi\left(te^{2\pi i \theta}\right).</math></br>

Angle <math>\theta</math> is named external angle or external argument.</br>

[edit] References

Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993
Adrien Douady and John H. Hubbard, Etude dynamique des polynômes complexes, Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)
John W. Milnor, Periodic Orbits, External Rays and the Mandelbrot Set: An Expository Account; Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque No. 261 (2000), 277–333. (First appeared as a Stony Brook IMS Preprint in 1999, available as arXiV:math.DS/9905169.)