Fractals/Iterations in the complex plane/Fatou set
Definition
editThe Fatou set is called:
 the domain of normality
 the domain of equicontinuity
Parts
editFatou set, domains and components 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.^{[1]}
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.
Complement
editThe 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.
components
editNumber of Fatou set's components in case of rational map:^{[2]}
 0 ( Fatou set is empty, the whole Riemann sphere is a Julia set )^{[3]}
 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 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 consists of the whole complex sphere = Fatou set is empty^{[4]}^{[5]}
domains
editIn 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 ^{[6]}
 attracting but not superattracting (
 superattracting ( Boettcher coordinate )
 parabolic (LeauFatou) basin ( Fatou coordinate ) Local dynamics near rationally indifferent fixed point/cycle ;
 elliptic basin = Siegel disc ( Local dynamics near irrationally indifferent fixed point/cycle )
coordinate
editBasin  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
editJulia 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
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:
c = 0.0000000000000000+0.0000000000000000*I m(c) = 0.0000000000000000+0.0000000000000000*I r(m) = 0.0000000000000000 t(m) = 0.0000000000000000 period = 1 c = 0.0250000000000000+0.0000000000000000*I m(c) = 0.0513167019494862+0.0000000000000000*I r(m) = 0.0513167019494862 t(m) = 0.0000000000000000 period = 1 c = 0.0500000000000000+0.0000000000000000*I m(c) = 0.1055728090000841+0.0000000000000000*I r(m) = 0.1055728090000841 t(m) = 0.0000000000000000 period = 1 c = 0.0750000000000000+0.0000000000000000*I m(c) = 0.1633399734659244+0.0000000000000000*I r(m) = 0.1633399734659244 t(m) = 0.0000000000000000 period = 1 c = 0.1000000000000000+0.0000000000000000*I m(c) = 0.2254033307585166+0.0000000000000000*I r(m) = 0.2254033307585166 t(m) = 0.0000000000000000 period = 1 c = 0.1250000000000000+0.0000000000000000*I m(c) = 0.2928932188134524+0.0000000000000000*I r(m) = 0.2928932188134524 t(m) = 0.0000000000000000 period = 1 c = 0.1500000000000000+0.0000000000000000*I m(c) = 0.3675444679663241+0.0000000000000000*I r(m) = 0.3675444679663241 t(m) = 0.0000000000000000 period = 1 c = 0.1750000000000000+0.0000000000000000*I m(c) = 0.4522774424948338+0.0000000000000000*I r(m) = 0.4522774424948338 t(m) = 0.0000000000000000 period = 1 c = 0.2000000000000000+0.0000000000000000*I m(c) = 0.5527864045000419+0.0000000000000000*I r(m) = 0.5527864045000419 t(m) = 0.0000000000000000 period = 1 c = 0.2250000000000000+0.0000000000000000*I m(c) = 0.6837722339831620+0.0000000000000000*I r(m) = 0.6837722339831620 t(m) = 0.0000000000000000 period = 1 c = 0.2500000000000000+0.0000000000000000*I m(c) = 0.9999999894632878+0.0000000000000000*I r(m) = 0.9999999894632878 t(m) = 0.0000000000000000 period = 1 c = 0.2750000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.3162277660168377*I r(m) = 1.0488088481701514 t(m) = 0.0487455572605341 period = 1 c = 0.3000000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.4472135954999579*I r(m) = 1.0954451150103321 t(m) = 0.0669301182003075 period = 1 c = 0.3250000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.5477225575051662*I r(m) = 1.1401754250991381 t(m) = 0.0797514300099943 period = 1 c = 0.3500000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.6324555320336760*I r(m) = 1.1832159566199232 t(m) = 0.0897542589928440 period = 1

center = superattracting

attracting

parabolic

repelling
Tests
edit drawing critical orbit(s)
 finding periodic points
 dividing complex move into simple paths
 topological graph,^{[7]}
 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 dwellbands  
continous escape time  Example  Example  Example 
Tests
 for finite attractors ( radius = AR)
 for ininity ( bailout test with Escaping Radiud ( ER)
Targets
edit 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
edit ↑ 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/s1234601501695
 ↑ 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