# Difference between revisions of "Fractals/Iterations in the complex plane/parabolic"

 [checked revision] [checked revision]

"Most programs for computing Julia sets work well when the underlying dynamics is hyperbolic but experience an exponential slowdown in the parabolic case." ( Mark Braverman )[1]

# Key words

• parabolic chessboard
• parabolic implosion
• Ecalle cylinders or Ecalle-Voronin cylinders ( by Jean Ecalle [2])
• Julia-Lavaurs sets
• petal
• The Leau-Fatou flower theorem
• The horn map
• Blaschke product
• Inou and Shishikura's near parabolic renormalization

# eggbeater dynamics

Hand Egg beater
Here is real model of what happens in parabolic case
It is a dynamic plane for fc(z)=z^2 + 1/4. It is zoom around parabolic fixed point z=0.5. Orbits of some points inside Julia set are shown (white points)

Physical model : the behaviour of cake when one uses eggbeater.

The mathematical model : a 2D vector field with 2 centers ( second-order degenerate points ) [3]

The field is spinning about the centers, but does not appear to be diverging.

# Petal

There is no unified definition of petals.

Petal of a flower can be :

• attracting
• repelling

Attracting petal P is a :

• domain (topological disc ) containing parabolic periodic point p in its boundary :
${\displaystyle {\overline {P}}\ni \left\{p\right\}}$

• trap which captures any orbit tending to parabolic point [4]
• set contained inside component of filled-in Julia set

## 0/1

How the target set is changing along an internal ray 0

Cpp code by Wolf Jung see function parabolic from file mndlbrot.cpp ( program mandel ) [5]

C code :

 if (Zx>=0 && Zx <= 0.5 && (Zy > 0 ? Zy : -Zy)<= 0.5 - Zx)
{ if (Zy>0) data[i]=200; // show petal
else data[i]=150;}


Gnuplot code :

reset
f(x,y)=  x>=0 && x<=0.5 &&  (y > 0 ? y : -y) <= 0.5 - x
unset colorbox
set isosample 300, 300
set xlabel 'x'
set ylabel 'y'
set sample 300
set pm3d map
splot [-2:2] [-2:2] f(x,y)


# Flower

Flower with four petals
Critical orbit for internal angle 1/5 showing 5 directions

Sum of all petals creates a flower with center at parabolic periodic point.

Petals ${\displaystyle P_{j}}$ are symmetric with respect to the d-1 directions :

${\displaystyle arg(z)={\frac {2\Pi l}{d-1}}}$

where :

• d is a degree of complex quadratic map so d=2
• l is from 0 to d-2

Petals can have different size.

# Parabolic chessboard

Parabolic chessboard = parabolic checkerboard

See :

• Tiles: Tessellation of the Interior of Filled Julia Sets by T Kawahira[6]
• coloured califlower by A Cheritat [7]

Color points according to :[8]

• the integer part of Fatou coordinate
• the sign of imaginary part

# Cauliflower

Cauliflower or broccoli :[9]

• empty ( its interior is empty ) for c outside Mandelbrot set. Julia set is a totally disconnected (
• filled cauliflower for c=1/4 on boundary of the Mandelbrot set. Julia set is a Jordan curve ( quasi circle).

Pleae note that :

• size of image differs because of different z-planes.
• different algorithms are used so colours are hard to compare

# Bifurcation of the Cauliflower

How Julia set changes along real axis ( going from c=0 thru c=1/4 and futher ) :

Perturbation of a function ${\displaystyle f(z)}$ by complex ${\displaystyle \epsilon }$ :

${\displaystyle g(z)=f(z)+\epsilon }$

When one add epsilon > 0 ( move along real axis toward + infinity ) there is a bifurcation of parabolic fixed point :

• attracting fixed point ( epsilon<0 )
• one parabolic fixed point ( epsilon = 0 )
• one parabolic fixed point splits up into two conjugate repelling fixed points ( epsilon > 0 )

"If we slightly perturb with epsilon<0 then the parabolic fixed point splits up into two real fixed points on the real axis (one attracting, one repelling). "

See :

• demo 2 page 9 in program Mandel by Wolf Jung

# Fatou coordinate

Description at Hyperoperations Wiki

• what we call "Abel function", they call it "Fatou coordinates".[10]
• Fatou coordinates [11]
• Shishikura perturbed Fatou coordinates [12]
• Parabolic iteration [13][14]

# Gallery

critical orbits for internal angle from 1/1 to 1/10