# Fractals/Iterations in the complex plane/Fatou set

The Fatou set is called:

• the domain of normality
• the domain of equicontinuity

# Fatou set, domains and components

Then there is a finite number of open sets $\;F_{1},...,F_{r}\;$ , that are left invariant by $\;f(z)\;,$  and are such that:

1. the union of the sets $\;F_{i}\;$  is dense in the plane and
2. $\;f(z)\;,$  behaves in a regular and equal way on each of the sets $\;F_{i}\ ;.$

The last statement means that the termini of the sequences of iterations generated by the points of $\;F_{i}\;$  are

• either precisely the same set, which is then a finite cycle
• or they are finite cycles of circular or annular shaped sets that are lying concentrically.

In the first case the cycle is attracting, in the second it is neutral.

These sets $\;F_{i}\;$  are the Fatou domains of $\;f(z)\;,$  and their union is the Fatou set $\;\operatorname {F} (f)\;$  of $\;f(z)\;.$

Each domain of the Fatou set of a rational map can be classified into one of four different classes.

Each of the Fatou domains contains at least one critical point of $\;f(z)\;,$  that is:

• a (finite) point z satisfying $\;f'(z)=0\;,$
• $\;f(z)=\infty \;,$
• if the degree of the numerator $\;p(z)\;$  is at least two larger than the degree of the denominator $\;q(z)\;,$
• if $\;f(z)=1/g(z)+c\;$  for some c and a rational function $\;g(z)\;$  satisfying this condition.

## Complement

The complement of $\;\operatorname {F} (f)\;$  is the Julia set $\;\operatorname {J} (f)\;$  of $\;f(z)\;.$

If all the critical points are preperiodic, that is they are not periodic but eventually land on a periodic cycle, then $\;\operatorname {J} (f)\;$  is all the sphere; otherwise, $\;\operatorname {J} (f)\;$  is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like $\;\operatorname {F} (f)\;,$  $\;\operatorname {J} (f)\;$  is left invariant by $\;f(z)\;,$  and on this set the iteration is repelling, meaning that $\;|f(z)-f(w)|>|z-w|\;$  for all w in a neighbourhood of z [within $\;\operatorname {J} (f)\;$ ]. This means that $\;f(z)\;$  behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitesimal part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.

## components

Number of Fatou set's components in case of rational map:

• 0 ( Fatou set is empty, the whole Riemann sphere is a Julia set )
• 1 ( example $f(z)=z^{2}-2$ , here is only one Fatou domains which consist of one component = full Fatou set)
• 2 ( example $f(z)=z^{2}$ , here are 2 Fatou domains, both have one component )
• infinitely many ( example $f(z)=z^{2}-1$ , here are 2 Fatou domains, one ( the exterior) has one component, the other ( interior) has infinitely many componnets)

the Julia set for the The Samuel Lattes function $l(z)={\frac {{(z^{2}+1)}^{2}}{4z(z^{2}-1)}}$  consists of the whole complex sphere = Fatou set is empty

## domains

In case of discrete dynamical system based on complex quadratic polynomial Fatou set can consist of domains ( basins) :

• attracting ( basin of attraction of fixed point / cycle )
• superattracting ( Boettcher coordinate )
• basin of infinity 
• 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

• 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

• drawing critical orbit(s)
• finding periodic points
• dividing complex move into simple paths
• topological graph,
• 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