There are no orbits on parameter plane, one should not draw orbits on parameter plane. Orbit of critical point is on the dynamical plane

- complex ,
- nonlinear,
- deterministic,
- discrete .

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

**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

- Bifurcation sequences (scenarios ):
- sequence of period doubling bifurcations along real axis
- period tripling

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

- 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