# Fractals/Iterations in the complex plane/Fatou set

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# Fatou components[edit]

In case of discrete dynamical system based on complex quadratic polynomial Fatou set can consist of components :

- attracting ( basin of attraction of fixed point / cycle )
- superattracting ( Boettcher coordinate )
- attracting but not superattracting (

- parabolic (Leau-Fatou) basin ( Fatou coordinate ) Local dynamics near rationally indifferent fixed point/cycle ;
- elliptic basin = Siegel disc ( Local dynamics near irrationally indifferent fixed point/cycle )

# Local discrete dynamics[edit]

- attracting : hyperbolic dynamics
- superattracting : the very fast ( = exponential) convergence to periodic cycle ( fixed point )

- parabolic component = slow ( lazy ) dynamics = slow ( exponential slowdown) convergence to parabolic fixed point ( periodic cycle)
- Siegel disc component = rotation around fixed point and never reach the fixed point

# Tests[edit]

Analysis of local dynamics :

- drawing critical orbit(s)
- finding periodic points
- dividing complex move into simple paths
- topological graph,
^{[1]} - drawing grid ( polar or rectangular )

method | test | description | resulting sets | true sets | |
---|---|---|---|---|---|

binary escape time | bailout | abs(zn)>ER | escaping and not escaping | Escaping set contains fast escaping pixels and is a true exterior.
Not escaping set is treated as a filled Julia set ( interior and boundary) but it contains : - slow escaping points from exterior,
- Julia sets
- interior points
| |

discrete escape time = Level Set Method = LSM | bailout | Last iteration or final_n = n : abs(zn)>ER | escaping set is divided into subsets with the same n ( last iteration). This subsets are called Level Sets and create bands surrounding and approximating Julia set. Boundaries of level sets are called dwell-bands | ||

continous escape time | Example | Example | Example |