An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set.[1]Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray.
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 also
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
^How to draw external rays by Wolf Jung
^Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies Tomoki Kawahira Archived 2016-03-03 at the Wayback Machine
^Douady Hubbard Parameter Rays by Linas Vepstas
^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