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

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File:Julia z2+0,25.png | from period 1 thru internal ray '''1/1'''; c = 0.25 = 1/4 the root of the main cardioid. Julia set is a '''cauliflower''' |
File:Julia z2+0,25.png | from period 1 thru internal ray '''1/1'''; c = 0.25 = 1/4 the root of the main cardioid. Julia set is a '''cauliflower''' |
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− | File:Parabolic julia set c=-0.75.png | from period 1 to 2 thru internal ray '''1/2''' ; c=-0.75 |
+ | File:Parabolic julia set c=-0.75.png | from period 1 to 2 thru internal ray '''1/2''' ; c=-0.75 Julia set is a San Marco fractal<ref>[http://planetmath.org/encyclopedia/SanMarcoDragon.html planetmath : San Marco fractal]</ref> |

File:Douady fat rabbit.jpg| from period 1 thru internal ray '''1/3''' ; Juia set is a '''fat Douady rabbit''' |
File:Douady fat rabbit.jpg| from period 1 thru internal ray '''1/3''' ; Juia set is a '''fat Douady rabbit''' |
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File:Parabolic julia set c = -1.125 + 0.21650635094611*i.png | from period 2 thru 1/3 |
File:Parabolic julia set c = -1.125 + 0.21650635094611*i.png | from period 2 thru 1/3 |

## Revision as of 17:09, 6 March 2012

"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

**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 :

```
```

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

## 0/1

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

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

Petals are symmetric with respect to the d-1 directions :

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 by complex :

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

# parabolic implosion

# 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

# See also

- Image : Nonstandard Parabolic by Cheritat
^{[16]} - Julia set of parabolic case in Maxima CAS
^{[17]} - The parabolic Mandelbrot set Pascale ROESCH (joint work with C. L. PETERSEN
- PARABOLIC IMPLOSION A MINI-COURSE by ArnaudCheritat
- Workshop on parabolic implosion 2010

# References

- ↑ Mark Braverman : On efficient computation of parabolic Julia sets
- ↑ Jean Ecalle home page
- ↑ MODULUS OF ANALYTIC CLASSIFICATION FOR UNFOLDINGS OF GENERIC PARABOLIC DIFFEOMORPHISMSby P. Mardesic, R. Roussarie¤ and C. Rousseau
- ↑ IMPLOSION A MINI-COURSE by ARNAUD CHERITAT
- ↑ Program Mandel by Wolf Jung
- ↑ tiles by T Kawahira
- ↑ Coloured califlower by A Cheritat
- ↑ Applications of near-parabolic renormalization by Mitsuhiro Shishikura
- ↑ cauliflower at MuEncy by Robert Munafo
- ↑ new results from complex dynamics at Tetration Forum
- ↑ Fatou coordinate at Hyperoperations Wiki
- ↑ Shishikura perturbed Fatou coordinates
- ↑ Tetration Forum : Parabolic Iteration
- ↑ Tetration Forum : Parabolic Iteration, again
- ↑ planetmath : San Marco fractal
- ↑ Image : Nonstandard Parabolic by Cheritat
- ↑ Julia set of parabolic case in Maxima CAS