# Fractals

This wikibook is about : how to make fractals (:-)) It covers only topics which are important for that (:-))

## Introduction

* Introduction
* Introductory Examples
* Mathematics for computer graphic: numbers, sequences, functions, numerical methods, fields, ...
* Programming computer graphic: files, plane, curves, ...
** plane transformations
* Fractal software
* Fractal links

## Fractals made by the iterations

Theory
* Definitions
* Iterations : forward and backward ( inverse ) and critical orbit
** Fractional iterations
* critical orbit
* Periodic points or cycle
** periodic points of complex quadratic map
** Period
* How to analyze map ? How to read location from the image?
* How to construct map with desired properities ?
* Algorithms ( graphical (coloring, transformations), numerical, symbolic, other)

### Iterations of real numbers : 1D

* (angle) doubling map
* logistic map
* real quadratic map
* tent map

### Iterations of complex numbers :2D

* complex-analytic formulas (like Mandelbrot set and Julia set)
* non-complex-analytic formulas (like Mandelbar and Burning Ship)

#### Rational maps

* Analysis
* Herman rings

##### Polynomials
###### Complex quadratic polynomials

Dynamic plane: Julia and Fatou sets
* coloring the dynamic plane and the Julia and the Fatou sets
* Julia set
** with an non-empty interior ( connected )
*** Hyperbolic Julia sets
**** attracting : filled Julia set have attracting cycle ( c is inside hyperbolic component )
**** superattracting : filled Julia set have superattracting cycle( c is in the center of hyperbolic component ). Examples : Airplane Julia set, Douady's Rabbit, Basillica.
*** Parabolic Julia set
*** Elliptic Julia set: Siegel disc - a linearizable irrationaly indifferent fixed point
** with empty interior
*** disconnected ( c is outside of Mandelbrot set )
*** connected ( c is inside Mandelbrot set )
**** Cremer Julia sets -a non-linearizable irrationaly indifferent fixed point
**** dendrits or Dendrite Julia sets ( Julia set is connected and locally connected ). Examples :
***** Misiurewicz Julia sets (c is a Misiurewicz point )
***** Feigenbaum Julia sets ( c is Generalized Feigenbaum point: the limit of the period-q cascade of bifurcations and landing points of parameter ray or rays with irrational angles )
***** others which have no description
* Fatou set
** exterior of all Julia sets = basin of attraction of superattracting fixed point (infinity)
*** Escape time
*** Boettcher coordinate
*** Orbit portraits and lamination of dynamical plane
*** Dynamic external rays
** Interior of Julia sets:
*** Basin of attraction of superattracting periodic/fixed point - Boettchers coordinate , c is a center of period n component of Mandelbrot set
**** Circle Julia set ( c = 0 is a center of period 1 component)
**** Basilica Julia set ( c = -1 is a center of period 2 component)
*** Basin of attraction of attracting periodic/fixed point - Koenigs coordinate
*** Local dynamics near indifferent fixed point/cycle
**** Local dynamics near rationally indifferent fixed point/cycle ( parabolic ). Leau-Fatou flower theorem
***** petal of the Leau-Fatou flower
***** Repelling and attracting directions
***** Rays landing on the parabolic fixed point
***** parabolic checkerboard
***** parabolic perturbation
***** Fatou_coordinate
****** Fatou_coordinate for f(z)=z/(1+z)
****** Fatou_coordinate for f(z)=z+z^2
****** Fatou_coordinate for f(z)=z^2 + c
**** Local dynamics near irrationally indifferent fixed point/cycle ( elliptic ) - Siegel disc
Parameter plane and Mandelbrot set
* Topological model of Mandelbrot set : Lavaurs algorithm and lamination of parameter plane
* structure of Mandelbrot set and ordering of hyperbolic components
** $F_{1/2}$ family: real slice of Mandelbrot set.
*** periodic part: period doubling cascade. Escape route 1/2
*** the Myrberg-Feigenbaum point of $F_{1/2}$ family
*** chaotic part main antenna is a shrub of $F_{1/2}$ family
* Transformations of parameter plane
* Sequences and orders on the parameter plane
* Parts of parameter plane
** exterior of the Mandelbrot set: escape time, Level Set Method ( LSM/M), Binary Decomposition Method (BDM/M)
*** External (Parameter) Rays of:
**** the wake ( root point)
**** the principle Misiurewicz points for the wake k/r of main cardioid
**** subwake (root points, tuning and internal address)
**** branch tips of the shrub ( Misiurewicz points)
**** islands ( root point, Douady tuning)
** Interior and the boundary : components
*** Number of the Mandelbrot set's components
*** Boundary of whole set and it's components
**** parabolic points: root points and cusps
**** unroll a closed curve and then stretch out into an infinite strip
**** Misiurewicz points
***** Devaney algorithm for principle Misiurewicz point
*** interior of hyperbolic components
**** centers of hyperbolic components = nuclesu of Mu-atoms
**** Internal rays
*** Islands
**** the biggest island of the wake
**** distortion of mini Mandelbrot sets
**** islands ( root point, Douady tuning)
** Points ( parameter of the iterated function)
* speed improvements
* coloring algorithm