In complex analysis, particularly in complex dynamics and geometric function theory, external rays are associated to a compact, full, connected subset of the complex plane as the images of radial rays under the Riemann map of the complement of . Equivalently, they are the gradient lines of the Green's function of . Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Complex dynamics is the study of dynamical systems for which the phase space is a complex manifold. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the... In mathematics, a Greens function is a type of function used to solve inhomogeneous differential equations subject to boundary conditions. ...

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. Adrien Douady (born 1935) is a French mathematician. ... Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. ... In one-dimensional complex dynamics, the connectedness locus in a parameter space of polynomials or rational functions consists of those parameters for which the corresponding Julia set is connected. ...

Formal definition

Let

(where denotes the unit disk) be the unique conformal isomorphism whose leading Laurent coefficient at infinity is real and positive. A disc of unit radius on a plane is called a unit disc. ... The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the...

Then the external ray of angle is the curve

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.)