# Fatou components

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

• attarcting ( 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

Types of dynamics
• 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

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