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 componentsEdit
Then there is a finite number of open sets , that are left invariant by and are such that:
- the union of the sets is dense in the plane and
- behaves in a regular and equal way on each of the sets
The last statement means that the termini of the sequences of iterations generated by the points of 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 are the Fatou domains of and their union is the Fatou set of
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 that is:
- a (finite) point z satisfying
- if the degree of the numerator is at least two larger than the degree of the denominator
- if for some c and a rational function satisfying this condition.
The complement of is the Julia set of
If all the critical points are preperiodic, that is they are not periodic but eventually land on a periodic cycle, then is all the sphere; otherwise, is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like is left invariant by and on this set the iteration is repelling, meaning that for all w in a neighbourhood of z [within ]. This means that 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.
- 0 ( Fatou set is empty, the whole Riemann sphere is a Julia set )
- 1 ( example , here is only one Fatou domains which consist of one component = full Fatou set)
- 2 ( example , here are 2 Fatou domains, both have one component )
- infinitely many ( example , here are 2 Fatou domains, one ( the exterior) has one component, the other ( interior) has infinitely many componnets)
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 (
- superattracting ( Boettcher coordinate )
- 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 dynamicsEdit
- 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
- 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 :
|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|
- 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