# Fractals/Iterations in the complex plane/parabolic

"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)
In other words it means that one can need days for making a good picture of parabolic Julia set with standard / naive algorithms.
There are 2 problems here:
* slow (= lazy) local dynamics (in the neighbourhood of a parabolic fixed point)
* some parts are very thin (hard to find using standard plane scanning)

# Planes
## Dynamic plane

Dynamic plane = complex z-plane $\mathbb {C} =J_{f}\cup F_{f}$:
* Julia set $J_{f}\subset \mathbb {C}$ is a common boundary: $J_{f}\,=\partial A_{f}(p)=\partial A_{f}(\infty )$
* Fatou set $F_{f}\subset \mathbb {C}$
** exterior of Julia set = basin of attraction to infinity: $A_{f}(\infty )\ {\overset {\underset {\mathrm {def} }}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to \infty \ as\ k\to \infty \}.$
** interior of Julia set = basin of attraction of finite, parabolic fixed point p: $A_{f}(p)\ {\overset {\underset {\mathrm {def} }}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to p\ as\ k\to \infty \}.$
*** immediate basin = sum of components which have parabolic fixed point p on it's boundary ; the immediate parabolic basin of p is the union of periodic connected components of the parabolic basin.
**** attracting Lea-Fatou flower = sum of n attracting petals = sum of 2*n attracting sepals
***** petal = part of the flower. Each petal contains part of 2 sepals,
***** sepals (Let 1 be an invariant curve in the first quadrant and L 1 the region enclosed by 1 ∪ {0}, called a sepal.)
$\mathbb {C} \supset F(f)\supset A_{f}(p)$
See also:
* Filled Julia set $K_{f}$

# Key words

* parabolic chessboard or checkerboard
* parabolic implosion
* Fatou coordinate
* Hawaiian earring
** in wikipedia = "The plane figure formed by a sequence of circles that are all tangent to each other at the same point and such that the sequence of radii converges to zero." (Barile, Margherita in MathWorld)
** commons:Category:Hawaiian earrings
* Gevrey symptotic expansions
** the Écalle-Voronin invariants of (7.1) at the origin which have Gevrey- 1/2 asymptotic expansions
* germ
** germ of the function: Taylor expansion of the function
* multiplicity
* Julia-Lavaurs sets
* The Leau-Fatou flower theorem: repelling or attracting flower. Flower consist of petals
** Leau-Fatou flower
* Parabolic Linearization theorem
* The horn map
* Blaschke product
* Inou and Shishikura's near parabolic renormalization
* complex polynomial vector field
* numbers
** "a positive integer ν, the parabolic degeneracy with the following property: there are νq attracting petals and νq repelling petals, which alternate cyclically around the fixed point."
** combinatorial rotation number
* a Poincaré linearizer of function f at parabolic fixed point
* "the parabolic pencil. This is the family of circles which all have one common point, and thus are all tangent to each other, either internally or externally."

## Leau-Fatou flower theorem

The Leau-Fatou flower theorem states that, if function $f^{n}$ has the Taylor expansion
$f^{n}(z)=z+az^{n+1}+O(z^{n+2})$
then the complex number $v$ on the unit circle $v\in S^{1}$ describes unit vector ( direction):
* an repulsion vector (from origin to v) for f at the origin if $nav^{n}=1$
* an attraction vector ( from v to origin ) if $nav^{n}=-1$
There are n equally spaced attracting directions, separated by n equally spaced repelling directions. The integer n+1 is called the multiplicity of fixed point

### Examples

Function expansion: z+z^5
So :
* a = 1
* n = 4
Compute attracting directions:
Function m*z+z^2
So :
* a = 1
* n = 1

## Ecalle cylinder

Ecalle cylinders or Ecalle-Voronin cylinders (by Jean Ecalle)
"... the quotient of a petal P under the equivalence relation identifying z and f (z) if both z and f (z) belong to P. This quotient manifold is called the Ecalle cilinder, and it is conformally isomorphic to the infinite cylinder C/Z"

## eggbeater dynamics

* Hand Egg beater
* Here is real model of what happens in parabolic case
Physical model: the behaviour of cake when one uses eggbeater.
The mathematical model: a 2D vector field with 2 centers (second-order degenerate points)
The field is spinning about the centers, but does not appear to be diverging.
Maybe better description of parabolic dynamics will be Hawaiian earrings

## parabolic germ

Germ:
* z+z^2
* z+z^3
* z+z^{k+1}
* z+a_{k+1}z^{k+1}
* z+a_{k+1}z^{k+1}
* "a germ $f(x)=a_{1}x+a_{2}x^{2}+\ldots$ with $\vert a_{1}\vert \neq 0,1$ is holomorphically conjugated to its linear part $g(x)=a_{1}z$" (Sylvain Bonnot)
germ of vector field

## The horn map

"the horn map h = Φ ◦ Ψ, where Φ is a shorthand for Φattr and Ψ for Ψrep (extended Fatou coordinate and parameterizations)."

## Petal

Petal
* "The petals are Jordan domains invariant by $f^{q}$" R PEREZ-MARCO
* Petal is trap which captures any orbit tending to parabolic point
* petal is a part of the flower

## Sepal

Definitions:
* A sepal is the intersection of an attracting and repelling petal.A parabolic Pommerenke-Levin-Yoccoz inequality. by Xavier Buf and Adam L. Epstein There does not seem to be an official definition of sepal.
* "Let l be an invariant curve in the first quadrant and L1 the region enclosed by l ∪ {0}, called a sepal."
* interior of component with parabolic fixed point on it's boundary

## Flower

* Lea-Fatu flower
* Flower with four petals and parabolic point in the center
* Critical orbit for internal angle 1/5 showing 5 attracting directions
Sum of all petals creates a flower with center at parabolic periodic point.

## Cauliflower

Cauliflower or broccoli:
* 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).
* c = 0 < 1/4
* c = 0.25 = 1/4 t he root of the main cardioid. Julia set is a cauliflower.
* c = 0.255 > 1/4 "imploded cauliflower"
* c = 0.258 > 1/4
* c = [0.285, 0.01]
Please 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 further):
Perturbation of a function $f(z)$ by complex $\epsilon$:
$g(z)=f(z)+\epsilon$
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

## parabolic implosion

Parabolic imposion
* on the parameter plane
** point c moves from interior of the component, through the boundary to the exterior of Mandelbrot set
** nucleus ( c=0), along internal ray 0, parabolic point ( c= 0.25), along external ray 0
* on the dynamic plane
** connected Julia set ( with interior) imploedes and comes disconnected ( without interior)
** fixed point moves from interior to Julia set ( parabolic)
** one basin ( interior) disappears
evolution of dynamics along escape route 0 ( parabolic implosion) :
| parameter c | location | Julia set | interior | type of dynamics | critical point | fixed points |
| c = 0 | center, interior | connected | exist | superattracting | attracted to alfa fixed point | fixed critical point equal to alfa fixed point, alfa is superattracting, beta is repelling |
| 0<c<1/4 | internal ray 0, interior | connected | exist | attracting | attracted to alfa fixed point | alfa is attracting, beta is repelling |
| c = 1/4 | cusp, boundary | connected | exist | parabolic | attracted to alfa fixed point | alfa fixed point equal to beta fixed point, both are parabolic |
| c>1/4 | external ray 0, exterior | disconnected | disappears | - | repelling to infinity | both finite fixed points are repelling |
Video on YouTube
* Parabolic implosion : from circle thru cauliflower to imploded cauliflower
* c=0
* c=1/4
* c= 1/4 + 0.05
* c = 1/4 + 0.029
* c = 1/4 + 0.035

## Vector field

* 2D vector field and its

### singularity

singularity types:
* center type: "In this case, one can find a neighborhood of the singular point where all integral curves are closed, inside one another, and contain the singular point in their interior"
* non-center type: neighborhood of singularity is made of several curvilinear sectors:
"A curvilinear sector is defined as the region bounded by a circle C with arbitrary small radius and two streamlines S and S! both converging towards singularity. One then considers the streamlines passing through the open sector g in order to distinguish between three possible types of curvilinear sectors."

# dynamics

Dynamics:
* global
* local
Local dynamics:
* in the exterior of Julia set
* on the Julia set
* near parabolic fixed point (inside Julia set)
See also
* youtube playlist: ODTWÓRZ WSZYSTKIE Corinna Ulcigrai - Parabolic dynamics and renormalization from Institut des Hautes Études Scientifiques (IHÉS)

## Near parabolic fixed point

Why analyze f^p not f ?
Forward orbit of f near parabolic fixed point is composite. It consist of 2 motions:
* around fixed point
* toward / away from fixed point

# How to compute parabolic c values

* Computing parameters using boundary equations
Type of parabolic parameters:
* root points
* cusps
Parabolic points of period 1 component of Mandelbrot set (parameter plane) :
| n | Internal angle (rotation number) t = 1/n | The root point c = parabolic parameter | Two external angles of parameter rays landing on the root point c (1/(2^n+1); 2/(2^n+1) | fixed point $z_{\alpha }$ | external angles of dynamic rays landing on fixed point $z_{\alpha }$ |
| 1 | 1/1 | 0.25 | (0/1 ; 1/1) | 0.5 | (0/1 = 1/1) |
| 2 | 1/2 | -0.75 | (1/3; 2/3) | -0.5 | (1/3; 2/3) |
| 3 | 1/3 | 0.64951905283833*%i-0.125 | (1/7; 2/7) | 0.43301270189222*%i-0.25 | (1/7; 2/7; 3/7) |
| 4 | 1/4 | 0.5*%i+0.25 | (1/15; 2/15) | 0.5*%i | (1/15; 2/15; 4/15; 8/15) |
| 5 | 1/5 | 0.32858194507446*%i+0.35676274578121 | (1/31; 2/31) | 0.47552825814758*%i+0.15450849718747 | (1/31; 2/31; 4/31; 8/31; 16/31) |
| 6 | 1/6 | 0.21650635094611*%i+0.375 | (1/63; 2/63) | 0.43301270189222*%i+0.25 | (1/63; 2/63; 4/63; 8/63; 16/63; 32/63) |
| 7 | 1/7 | 0.14718376318856*%i+0.36737513441845 | (1/127; 2/127) | 0.39091574123401*%i+0.31174490092937 | (1/127; 2/127, 4/127; 8/127; 16/127; 32/127, 64/127) |
| 8 | 1/8 | 0.10355339059327*%i+0.35355339059327 | | 0.35355339059327*%i+0.35355339059327 | |
| 9 | 1/9 | 0.075191866590218*%i+0.33961017714276 | | 0.32139380484327*%i+0.38302222155949 | |
| 10 | 1/10 | 0.056128497072448*%i+0.32725424859374 | | 0.29389262614624*%i+0.40450849718747 | |
For internal angle n/p parabolic period p cycle consist of one z-point with multiplicity p and multiplier = 1.0 . This point z is equal to fixed point $z_{alfa}$

## Period 1

One can easily compute boundary point c
$c=c_{x}+c_{y}*i$
of period 1 hyperbolic component (main cardioid) for given internal angle (rotation number) t using this cpp code by Wolf Jung
or this Maxima CAS code:

## period d
# How to draw parabolic Julia set

* 5 methods of drawing
* Checking iteration
* modified DEM
* checking angle
* checking if zn is in target set
* checking if zn is in triangle target set
All points of interior of filled Julia set tend to one periodic orbit (or fixed point). This point is in Julia set and is weakly attracting. One can analyse only behavior near parabolic fixed point. It can be done using critical orbits.
There are two cases here: easy and hard.
If the Julia set near parabolic fixed point is like n-th arm star (not twisted) then one can simply check argument of zn, relative to the fixed point. See for example z+z^5. This is an easy case.
In the hard case Julia set is twisted around fixed.

## topology

Julia set of root point is topologically the same as the Julia set of the child period center, but
* the center ( nucleus) Julia set is very easy to draw ( superattracting basin = very fast dynamics because critical point is also periodic point)
* while the root Julia set ( parabolic) is hard to draw ( parabolic basin and lasy dynamics)
Examples:
* t = 1/2
** Julia set of the root point = fat Basilica Julia set: c = -3/4 = - 0.75
** Julia set of period 2 center = (slim) Basicica Julia set: c = -1
* t = 1/3
** Julia set of the root point = fat Douady Rabbit: c = -0.125000000000000 +0.649519052838329i
** Julia set of period 3 center = (slim) Doudy Rabbit Julia set: c = -0.122561166876654 +0.744861766619744i period = 3

## Estimation from exterior
### Escape time

Description

### Long iteration or big step method

Description
* Long iteration method
* The big steps algorithm
Steps
* choose fuction $f$which has fixed parabolic point at origin ( z=0)
* choose internal angle $\theta ={\frac {m}{k}}$
* compute $F_{k}$ which is an aproximation of higher iterates of function $f$ for z close to zero using power series centered at zero ( Taylor series = Maclaurin series )
* find how many terms of power series to use and on which annuli to use specific $F_{k}$ experimentally
each $F_{k}$ will be used on an annulus
${\frac {1}{2^{k+K+1}}}<|z|<{\frac {1}{2^{k+K}}}$
where K is fixed

#### example

Lambda form of complex quadratic polynomial which has an indifferent fixed point with multiplier $\lambda$ at the origin
$f_{\lambda }(z)=\lambda z+z^{2}$
where:
* multiplier of fixed point $\lambda =e^{2\pi \theta i}$
* internal angle is an rational number and proper fraction $\theta ={\frac {m}{k}}$
Choose
* $\theta ={\frac {1}{7}}$ so $\lambda =e^{(2\pi i)/7}$
* 30 terms of power series
* approximated function for annuli k=4,5,...,10 and default function f^n for larger values of z ( outside annuli)
* delta for function equal to 10^-5
Mathematical Functions of Wolfram language ( :
* Series[f,{x,x0,n}] generates a power series expansion for f about the point $x=x_{0}$ to order $(x-x_{0})^{n}$, where n is an explicit integer
* N[expr] gives the numerical value of expr
* Chop[expr,delta] replaces numbers smaller in absolute magnitude than delta by 0
* Normal[expr] converts expr to a normal expression from a variety of special forms.
* Do[expr,n] evaluates expr n times
* Do[expr,{i,imin,imax}] evaluates expr and starts with i=imin
* With[{x=x0,y=y0,…},expr] specifies that all occurrences of the symbols x, y, … in expr should be replaced by x0, y0, ….
* Compile[{{x1,t1},…},expr] assumes that xi is of a type that matches ti.
* Which[test1,value1,test2,value2,…] evaluates each of the testi in turn, returning the value of the valuei corresponding to the first one that yields True
* Nest[f,expr,n] gives an expression with f applied n times to expr.

### Dynamic rays

One can use periodic dynamic rays landing on parabolic fixed point to find narrow parts of exterior.
Let's check how many backward iterations needs point on periodic ray with external radius = 4 to reach distance 0.003 from parabolic fixed point:
| period | Inverse iterations | time |
| 1 | 340 | 0m0.021s |
| 2 | 55 573 | 0m5.517s |
| 3 | 8 084 815 | 13m13.800s |
| 4 | 1 059 839 105 | 1724m28.990s |
One can use only argument of point z of external rays and its distance to alfa fixed point (see code from image). It works for periods up to 15 (maybe more ...).

## Estimation from interior

Julia set is a boundary of filled-in Julia set Kc.
* find points of interior of Kc
* find boundary of interior of Kc using edge detection
If components of interior are lying very close to each other then find components using:
For parabolic components between parent and child component:
where denominator is a denominator of internal angle of parent component of Mandelbrot set.

### Angle

"if the iterate zn of tends to a fixed parabolic point, then the initial seed z0 is classified according to the argument of zn−z0, the classification being provided by the flower theorem" (Mark McClure)

### Attraction time

Interior of filled Julia set consist of components. All comonents are preperiodic, some of them are periodic (immediate basin of attraction).
In other words:
* one iteration moves z to another component (and whole component to another component)
* all point of components have the same attraction time (number of iteration needed to reach target set around attractor)
It is possible to use it to color components. Because in the parabolic case the attractor is weak (weakly attracting) it needs a lot of iterations for some points to reach it.
Here are some example values:
where:
Note that attraction time i is proportional to denominator.
Now you see what means weakly attracting.
One can:
* use brutal force method (Attracting radius < pixelSize; iteration Max big enough to let all points from interior reach target set; long time or fast computer)
* find better method (:-)) if time is to long for you

### Trap
## Estimation from interior and exterior

Julia set is a common boundary of filled-in Julia set and basin of attraction of infinity.
* find points of interior/components of Kc
* find escaping points
* find boundary points using Sobel filter
It works for denominator up to 4.

## Inverse iteration of repelling points

Inverse iteration of alfa fixed point. It works good only for cutting point (where external rays land). Other points still are not hitten.

## Bof61

* Using bof61 for coloring interior
* example

# Gallery

* Critical orbits in case of one critical point
* critical orbits for internal angle from 1/1 to 1/10. True attracting directions
* N-th arm stars for n from 1 to 10. Schematic attracting directions
* critical orbit, attracting and repelling vectors for internal angle 1/3
* from period 1 thru ...
* 1/1 cauliflower
* 1/2 San Marco fractal
* 1/3 Douady fat rabbit
* 1/4
* 1/5
* 1/7
* 1/10
* 1/15
* 1/20
* 1/30
* from period 2 thru ...
* from 2 thru internal ray 1/1 ; c=-0.75 Julia set is a San Marco fractal
* from period 2 thru 1/2
* from period 2 thru 1/3
* from period 2 thru 1/4
* from period 3 thru ...
* from period 3 thru 1/3
External examples:
* A parabolic point with 377 petalsparabolic renormalization of z^3+c by YANG Fei
* A quad. Julia sets with 987 and 89 petals by A Cheritat
For other polynomial maps see here

## See also

* Image: Nonstandard Parabolic by Cheritat
* Julia set of parabolic case in Maxima CAS
* The parabolic Mandelbrot set Pascale ROESCH (joint work with C. L. PETERSEN
* PARABOLIC IMPLOSION A MINI-COURSE by ArnaudCheritat
* Workshop on parabolic implosion 2010
* The renormalization for parabolic fixed points... Mitsushiro Shishikura and Hiroyuki Inou
* Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets by Artem Dudko, arxiv.org: Dudko_A
* Minicourse "Analytic classification of germs of generic families unfolding a parabolic point by Christiane Rousseau
* Perspectives on Parabolic Points in Holomorphic Dynamics - The Banff International Research Station for Mathematical Innovation and Discovery (BIRS)
* Richard Oudkerk: The Parabolic Implosion for f_0(z) = z + z^{nu+1} + O(z^{nu+2})
* Perspectives on Parabolic Points in Holomorphic Dynamics Videos from BIRS Workshop 15w5082
* Generic Perturbation of Parabolic Points Having More Than One Attracting Petal by Arnaud Chéritat
* Parabolic fixed points - Mathematica Notebook by Mark McClure

## References

* ↑ Mark Braverman: On efficient computation of parabolic Julia sets
* ↑ Note on dynamically stable perturbations of parabolics by Tomoki Kawahira
* ↑ Filled Julia set in wikipedia
* ↑ Barile, Margherita. "Hawaiian Earring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/HawaiianEarring.html
* ↑ Augustin Fruchard, Reinhard Sch¨afke. Composite Asymptotic Expansions and Difference Equations. Revue Africaine de la Recherche en Informatique et Math´ematiques Appliqu´ees, INRIA, 2015, 20, pp.63-93. <hal-01320625>
* ↑ wikipedia: Germ(mathematics)
* ↑ Fixed points of diffeomorphisms, singularities of vector fields and epsilon-neighborhoods of their orbits by Maja Resman
* ↑ The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point by Colin Christopher, Christiane Rousseau
* ↑ wikipedia: Multiplicity (mathematics)
* ↑ Dynamics of surface homeomorphisms Topological versions of the Leau-Fatou flower theorem and the stable manifold theorem by Le Roux, F
* ↑ The Dynamics of Complex Polynomial Vector Fields in C by Kealey Dias
* ↑ LIMITS OF DEGENERATE PARABOLIC QUADRATIC RATIONAL MAPS by XAVIER BUFF, JEAN ECALLE, AND ADAM EPSTEIN
* ↑ Poincaré linearizers in higher dimensionsby Alastair Fletcher
* ↑ pencil of circles by James King
* ↑ math.stackexchange question: what-is-the-shape-of-parabolic-critical-orbit
* ↑ Théorie des invariants holomorphes. Thèse d'Etat, Orsay, March 1974
* ↑ Jean Ecalle in french wikipedia
* ↑ Jean Ecalle home page
* ↑ Lukas Geyer - Normal forms via uniformization (10/28/2016). This is the draft of the proof of local normal forms at attracting, repelling, and parabolic fixed points using the uniformization theorem, handed out in class. It will eventually be incorporated into the lecture notes.
* ↑ mappings by Luna Lomonaco
* ↑ MODULUS OF ANALYTIC CLASSIFICATION FOR UNFOLDINGS OF GENERIC PARABOLIC DIFFEOMORPHISMSby P. Mardesic, R. Roussarie¤ and C. Rousseau
* ↑ mathoverflow questions: the functional equation ffxxfx2
* ↑ Germ in wikipedia
* ↑ MODULUS OF ANALYTIC CLASSIFICATION FOR UNFOLDINGS OF GENERIC PARABOLIC DIFFEOMORPHISMS by P. Mardesic , R. Roussarie and C. Rousseau
* ↑ The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point Colin Christopher, Christiane Rousseau
* ↑ Mathoverflow: infinitesimal classification of functions near a fixed point upto conjugation
* ↑ Near parabolic renormalization for unisingular holomorphic maps by Arnaud Cheritat
* ↑ Ricardo Pérez-Marco. "Fixed points and circle maps." Acta Math. 179 (2) 243 - 294, 1997. https://doi.org/10.1007/BF02392745
* ↑ Note on dynamically stable perturbations of parabolics by Tomoki Kawahira
* ↑ Parabolic fixed points: The Leau-Fatou Flower by Davide Legacci March 18, 2021
* ↑ wikipedia: Rose (topology)
* ↑ cauliflower at MuEncy by Robert Munafo
* ↑ Circle Implodes Into Flames - video by sinflrobot
* ↑ A Topology Simplification Method For 2D Vector Fields by Xavier Tricoche Gerik Scheuermann Hans Hagen
* ↑ encyclopedia of math: Sector_in_the_theory_of_ordinary_differential_equations
* ↑ wikipedia: Multiplicity in mathematics
* ↑ Mandel: software for real and complex dynamics by Wolf Jung
* ↑ Local dynamics at a fixed point by Evgeny Demidov
* ↑ Parabolic Julia Sets are Polynomial Time Computable Mark Braverman
* ↑ Mark McClure class Spring2019: ComplexDynamics , see code "Iteration near parabolic points" ( Mathematica notebook )
* ↑ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
* ↑ The fixed points and periodic orbits by Evgeny Demidov
* ↑ Src code of c program for drawing parabolic Julia set
* ↑ stackexchange questions: what-is-the-shape-of-parabolic-critical-orbit
* ↑ planetmath: San Marco fractal
* ↑ wikipedia: Douady rabbit
* ↑ planetmath: San Marco fractal
* ↑ Image: Nonstandard Parabolic by Cheritat
* ↑ Julia set of parabolic case in Maxima CAS