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where denotes the extended complex plane.
Let denote the Boettcher map.[14] is a uniformizing map of the basin of attraction of infinity, because it conjugates on the complement of the filled Julia set to on the complement of the unit disk:
External rays for angles of the form : n / ( 21 - 1) (0/1; 1/1) landing on the point c= 1/4, which is cusp of main cardioid ( period 1 component)
External rays for angles of the form : n / ( 22 - 1) (1/3, 2/3) landing on the point c= - 3/4, which is root point of period 2 component
External rays for angles of the form : n / ( 23 - 1) (1/7,2/7) (3/7,4/7) landing on the point c= -1.75 = -7/4 (5/7,6/7) landing on the root points of period 3 components.
External rays for angles of form : n / ( 24 - 1) (1/15,2/15) (3/15, 4/15) (6/15, 9/15) landing on the root point c= -5/4 (7/15, 8/15) (11/15,12/15) (13/15, 14/15) landing on the root points of period 4 components.
External rays for angles of form : n / ( 25 - 1) landing on the root points of period 5 components
internal ray of main cardioid of angle 1/3: starts from center of main cardioid c=0, ends in the root point of period 3 component, which is the landing point of parameter (external) rays of angles 1/7 and 2/7
Internal ray for angle 1/3 of main cardioid made by conformal map from unit circle
Mini Mandelbrot set with period 134 and 2 external rays
mjwinq program by Matjaz Erat written in delphi/windows without source code ( For the external rays it uses the methods from quad.c in julia.tar by Curtis T McMullen)
^Atela, Pau (1992). "Bifurcations of dynamic rays in complex polynomials of degree two". Ergodic Theory and Dynamical Systems. 12 (3): 401–423. doi:10.1017/S0143385700006854. S2CID123478692.
^A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).
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.)
John Milnor, Dynamics in One Complex Variable, Third Edition, Princeton University Press, 2006, ISBN0-691-12488-4