Fractals/Iterations in the complex plane/Fatou set

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:

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 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

Types of local dynamics ( speed)

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

Types of 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:

$r=|m(z_{\alpha })|$

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

• 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