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)
RatioField by Gert Buschmann, for windows with Pascal source code for Dev-Pascal 1.9.2 (with Free Pascal compiler )
Mandelbrot program by Milan Va, written in Delphi with source code
Power MANDELZOOM by Robert Munafo
ruff by Claude Heiland-Allen
See alsoedit
Wikimedia Commons has media related to Category:External rays.
^J. Kiwi : Rational rays and critical portraits of complex polynomials. Ph. D. Thesis SUNY at Stony Brook (1997); IMS Preprint #1997/15. Archived 2004-11-05 at the Wayback Machine
^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. S2CID 123478692.
^Holomorphic Dynamics: On Accumulation of Stretching Rays by Pia B.N. Willumsen, see page 12
^The iteration of cubic polynomials Part I : The global topology of parameter by BODIL BRANNER and JOHN H. HUBBARD
^Stretching rays for cubic polynomials by Pascale Roesch
^Komori, Yohei; Nakane, Shizuo (2004). "Landing property of stretching rays for real cubic polynomials" (PDF). Conformal Geometry and Dynamics. 8 (4): 87–114. Bibcode:2004CGDAM...8...87K. doi:10.1090/s1088-4173-04-00102-x.
^A. Douady, J. Hubbard: Etude dynamique des polynˆomes complexes. Publications math´ematiques d’Orsay 84-02 (1984) (premi`ere partie) and 85-04 (1985) (deuxi`eme partie).
^Schleicher, Dierk (1997). "Rational parameter rays of the Mandelbrot set". arXiv:math/9711213.
^Video : The beauty and complexity of the Mandelbrot set by John Hubbard ( see part 3 )
^Yunping Jing : Local connectivity of the Mandelbrot set at certain infinitely renormalizable points Complex Dynamics and Related Topics, New Studies in Advanced Mathematics, 2004, The International Press, 236-264
^POLYNOMIAL BASINS OF INFINITY LAURA DEMARCO AND KEVIN M. PILGRIM
^Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira Archived 2016-03-03 at the Wayback Machine
^John H. Ewing, Glenn Schober, The area of the Mandelbrot Set
^Irwin Jungreis: The uniformization of the complement of the Mandelbrot set. Duke Math. J. Volume 52, Number 4 (1985), 935-938.
^Adrien Douady, John Hubbard, Etudes dynamique des polynomes complexes I & II, Publ. Math. Orsay. (1984-85) (The Orsay notes)
^Bielefeld, B.; Fisher, Y.; Vonhaeseler, F. (1993). "Computing the Laurent Series of the Map Ψ: C − D → C − M". Advances in Applied Mathematics. 14: 25–38. doi:10.1006/aama.1993.1002.
^Weisstein, Eric W. "Mandelbrot Set." From MathWorld--A Wolfram Web Resource
^An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
^http://www.mrob.com/pub/muency/externalangle.html External angle at Mu-ENCY (the Encyclopedia of the Mandelbrot Set) by Robert Munafo
^Computation of the external argument by Wolf Jung
^A. DOUADY, Algorithms for computing angles in the Mandelbrot set (Chaotic Dynamics and Fractals, ed. Barnsley and Demko, Acad. Press, 1986, pp. 155-168).
^Adrien Douady, John H. Hubbard: Exploring the Mandelbrot set. The Orsay Notes. page 58
^Exploding the Dark Heart of Chaos by Chris King from Mathematics Department of University of Auckland
^Topological Dynamics of Entire Functions by Helena Mihaljevic-Brandt
^Dynamic rays of entire functions and their landing behaviour by Helena Mihaljevic-Brandt
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, ISBN 0-691-12488-4
Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002
External linksedit
Wikibooks has a book on the topic of: Fractals
Hubbard Douady Potential, Field Lines by Inigo Quilez [permanent dead link]
Intertwined Internal Rays in Julia Sets of Rational Maps by Robert L. Devaney
Extending External Rays Throughout the Julia Sets of Rational Maps by Robert L. Devaney With Figen Cilingir and Elizabeth D. Russell
John Hubbard's presentation, The Beauty and Complexity of the Mandelbrot Set, part 3.1 Archived 2008-02-26 at the Wayback Machine
videos by ImpoliteFruit
Milan Va. "Mandelbrot set drawing". Retrieved 2009-06-15.[permanent dead link]