# Definition

The Fatou set is called:
* the domain of normality
* the domain of equicontinuity

# Parts

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:
* the union of the sets $\;F_{i}\;$ is dense in the plane and
* $\;f(z)\;,$ 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 components)
* Number of components
* 2 domains ( basins) and 2 components, so domain = component
* 2 domains: 1 domain has one component, but the other domain has infinitely many components
* 1 domain and infinitely many components
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 )

## coordinate

Coordinate :
| Basin | coordinate | | speed of approach to fixed point |
| superattracting | Boettcher | | fast |
| attracting | Koenigs | | intermediate |
| parabolic | Fatou | | slow |
| Siegel disc | | irrational rotation so orbits are closed curves | point rotates around fixed point and never approaches it |
| Herman ring | | irrational rotation so orbits are closed curves | point rotates around fixed point and never approaches it |
Repelling basin is another name for
* superattracting basin for polynomials

# Local discrete complex dynamics

Julia set is connected ( 2 basins of attraction)
* 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
When Julia set is disconnected ther is no interior of Julia set ( critical fixed point is repelling ( or attracting to infinity) - onlu one basin of attraction
evolution of dynamics along escape route 0 ( parabolic implosion) :
| parameter c | location of c | Julia set | interior | type of critical orbit dynamics | critical point | fixed points | stability of alfa |
| c = 0 | center, interior | connected | exist | superattracting | atracted to alfa fixed point | fixed critical point equal to alfa fixed point, alfa is superattracting, beta is repelling | r = 0 |
| 0<c<1/4 | internal ray 0, interior | connected | exist | attracting | atracted to alfa fixed point | alfa is attracting, beta is repelling | 0 < r < 1.0 |
| c = 1/4 | cusp, boundary | connected | exist | parabolic | atracted to alfa fixed point | alfa fixed point equal to beta fixed point, both are parabolic | r = 1 |
| c>1/4 | external ray 0, exterior | disconnected | disappears | repelling | repelling to infinity | both finite fixed points are repelling | r > 1 |
Stability r is absolute value of multiplier at fixed point alfa:
* evolution of dynamics along internal/external ray 0
* center = superattracting
* attracting
* parabolic
* repelling

# Tests

Analysis of local dynamics :
* 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 | |
Tests
* for finite attractors ( radius = AR)
* for ininity ( bailout test with Escaping Radiud ( ER)

# Targets

Target set:
* trap for forward orbit
* it is a set which captures any orbit tending to fixed / periodic point\
* is always inside component containing fixed point
* parabolic case
* parabolic case

# References

* ↑ Beardon, Iteration of Rational Functions, Theorem 7.1.1.
* ↑ Beardon, Iteration of Rational Functions, Theorem 5.6.2.
* ↑ Campbell, J.T., Collins, J.T. Blowup Points and Baby Mandelbrot Sets for a Family of Singularly Perturbed Rational Maps. Qual. Theory Dyn. Syst. 16, 31–52 (2017). https://doi.org/10.1007/s12346-015-0169-5
* ↑ Maru Sarazola : On the cardinalities of the Fatou and Julia sets
* ↑ Complex Dynamics by Fionn´an Howard
* ↑ THE CLASSIFICATION OF POLYNOMIAL BASINS OF INFINITY by LAURA DEMARCO AND KEVIN PILGRIM
* ↑ A Topology Simplification Method For 2D Vector Fields by Xavier Tricoche, Gerik Scheuermann and Hans Hagen