Dynamics of Julia and Fatou sets

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Mandelbrot set carries no dynamics. It is a set of parameter values.
There are no orbits on parameter plane, one should not draw orbits on parameter plane. Orbit of critical point is on the dynamical plane
Julia set it is a dynamical system:

• complex ,
• nonlinear,
• deterministic,
• discrete .

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Classifications of periodic points Zp of period n with stability index = Abs(l)  = | l|

• super-attracting periodic point , when | l_n(Zp )| = 0
  super-attracting periodic point can be reached by forward iteration
  every super-attracting cycle is contained in Fatou set
• attracting ( but not super-attracting) periodic point, when 0 < | l_n(Zp )| < 1
  attracting periodic point can be reached by forward iteration
  every attracting cycle is contained in Fatou set
• indifferent = neutral periodic point , when | l_n(Zp )| = 1 = | e^2*Pi*i*angle|
• repelling periodic point , when | l_n(Zp )| > 1
  repelling periodic point can be reached by backward iteration ( IIM )
  Julia sets include cycles of repelling points

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• hyperbolic , when | l_n(Zp )| < > 1
• non-hyperbolic , when | l_n(Zp )| = 1= | e^2*Pi*i*angle|
  
  • rationally indifferent = parabolic, when angle is rational number
    Invariant set is Fatou-Leau flower
    every parabolic cycle is contained in Julia set
    parabolic parameter is a root point
  • irrationally indifferent = elliptic, when angle is irrational number
    
    • linearizable
    • semi-linearizable
    • non-linearizable

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Bifurcation sequences (scenarios ):

• sequence of period doubling bifurcations along real axis
• period tripling

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• Algebraic Geometry of Discrete Dynamics. The case of one variable Authors: V.Dolotin, A.Morozov

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effect of function w=f(z)=z*z+c

• conformal mappings by John Mathews
• ComplexFunPowerRoot by John Mathews

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• Discrete local holomorphic dynamics  by Marco Abate
• Symbolic Dynamics of Quadratic Polynomials by Henk Bruin , Dierk Schleicher
• The Mandelbrot and Julia sets Anatomy by Evgeny Demidov
• Conformal dynamics problem list edited by Ben Bielefeld  *.ps
• Frontiers in complex dynamics by C T McMullen
• The classification of conformal dynamical systems by Curtis T. McMullen
• Symbolic Dynamics in Mathematics, Physics, and Engineering submitted by Warren Weckesser
• Kneading Theory by Nicholas B. Tufillaro
• Dennis Pixton
• Dynamical Systems and Fractals Lecture Notes by David J. Wright
• Discrete local holomorphic dynamics. Informal notes by Marco Abte
• Lyubich, Mikhail; Yampolsky, Michael Dynamics of quadratic polynomials: complex bounds for real maps

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Autor: adammaj1-at-o2-dot--pl Adam Majewski Feel free to e-mail me. (:-)) About