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

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)

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


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

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

=References=