Fractals/Iterations in the complex plane/boettcher
The Böttcher function of which maps the complement of the Mandelbrot set ( Julia set) conformally to the complement of the closed unit disk.
Complex potential on the parameter plane
editComplex potential on dynamical plane
editLet be a polynomial with degree d and Then Green's function for polynomial p is:
Exterior or complement of filledin Julia set is :
 a basin of attraction of infinity ( superattracting fixed point) so dynamics in ttah basin is the same for all polynomials !!!
 one of components of the Fatou set
It can be analysed using
 escape time (simple but gives only radial values = escape time ) LSM/J,
 distance estimation ( more advanced, continuus, but gives only radial values = distance ) DEM/J
 Boettcher coordinate or complex potential ( the best )
" The dynamics of polynomials is much better understood than the dynamics of general rational maps" due to the Bottcher’s theorem^{[1]}
Superattracting fixed points
editFor complex quadratic polynomial there are many superattracting fixed point ( with multiplier = 0 ):
 infinity ( It is always is superattracting fixed point for polynomials ). In this case exterior of Julia set is superattracting basin
 interior of Julia set is a superattracting basin when c is a center ( nucleus) of any hyperbolic component of Mandelbrot set
 is finite superattracting fixed point for map
 and are two finite superattracting fixed points for map
 other centers

p = 1

p=2

p=3 rabbit

p=3 airplane

p=4 rabbit

p=17
Description
editNear^{[2]} super attracting fixed point (for example infinity) the behaviour of discrete dynamical system :
based on complex quadratic polynomial is similar to
based on
It can be treated as one dynamical system viewed in two coordinate systems :
 easy ( w )
 hard to analyse( z )
^{[3]}
In other words map is conjugate ^{[4]} to map near infinity.^{[5]}
Names
edit is Boettcher coordinate
 is Boettcher function^{[6]}
 Boettcher Functional Equation:^{[7]}^{[8]}
where :
Complex potential or Boettcher coordinate has :
 radial values ( real potential ) LogPhi = CPM/J = value of the Green's function
 angular values ( external angle ) ArgPhi
Both values can be used to color with 2D gradient.
Computation
editnumerical solution
editMethods
editOriginal approach
editThe Boettcher functional equation: ^{[9]}
where:
 f is a given function which is analytic in a neighborhood of its fixed point x
 F is the Boettcher's function
When the fixed point x is superattracting:
a “basic algorithm”:
Let be a function such that
Then the general solution to the functional equation is given by:
where Q is an arbitrary constant.
Mathemathica
editFormula from Mathemathica^{[10]}
To compute Boettcher coordinate use this formula ^{[11]}
It looks "simple", but :
 square root of complex number gives two values so one have to choose one value
 more precision is needed near Julia set
PDE's approach
editTo construct a Riemann map explicitly on a given domain use the following PDE's approach^{[12]}
 first, translate the domain so that it contains the origin
 next, use a numerical method to construct a harmonic function satisfying for all , and let
Then , , and is harmonic, so is the radial component (i.e. modulus) of a Riemann map on .
The angular component can now be determined by the fact that its level curves are perpendicular to the level curves of , and have equal angular spacing near the origin.
Examples
editThe Basilica Julia set
edit is a center of period 2 hyperbolic component of Mandelbrot set
 fixed point
 superattracting 2point cycle (limit cycle) : ( period is 2 )
 2 external rays : which land on fixed point , which is a root point of the component of filled Julia set with center z=1
 The pinch points for the Basilica are points have external rays at angles that are rational numbers of the form and for some k,n∈Nk, n ∈ Nk,n∈N.
Central component
editDescription
 the central bounded Fatou component
 large central component, which contains the origin of the complex plane
 it is a quotient of the central gap in the Basilica lamination, i.e. the gap containing the center point of the disk
 "It is obtained from the central gap by collapsing each of the infinitely many central arcs that surround this gap. We can label these central arcs using the dyadic rationals. Each central arc corresponds to a single point in the boundary ∂C of C the central component. These points are dense in ∂C, the labeling of the central arcs by dyadics induces a welldefined homeomorphism from ∂C to the unit circle"^{[13]}
Points:
 alpha fixed point and it's preimages^{[14]}
Boettchers equation
edit
Maxima CAS src code :
(%i1) kill(all); (%o0) done (%i1) remvalue(all); (%o1) [] (%i2) display2d:false; (%o2) false (%i3) define(f(z), z^21); (%o3) f(z):=z^21 (%i4) a:factor(f(f(z))); (%o4) z^2*(z^22) (%i5) taylor(a,z,0,14); (%o5) (2*z^2)+z^4
It gives Böttcher's functional equation^{[15]}:
Solve above functional equation for the Boettcher map
The Riemann map for the central component for the Basilica
editmethods by James M. Belk for:
 drawing external rays
 Riemann map on the central component of Basilica^{[16]}
Description
 choose starting points with known Bottcher coordinates: the dyadic points on the boundary of central component
 compute a very large number of inverse images of these points
 draw curves through the right sequences of points
Here is a summary of the procedure for drawing equipotentials for a quadratic Julia set:
 Start with a large number (say 3 * 2^13) of equally spaced sample points on a circle of large radius (say R = 2^16) centered at the origin. Note that this circle is basically an equipotential for the Julia set, since the radius is so large.
 Compute an equal number of points on the inverse image of this circle (which is again basically an equipotential whose radius is the square root of R). Note that each of the original points has two preimages, so it must be worked out in each case which preimage to use.
 Iterate step 2 to produce sample points on a large number (say 50) of equipotentials that converge to the Julia set.
If you want to draw the external rays, it works to draw a piecewiselinear path between corresponding points on different equipotentials. Of course, the resulting external rays will be straight between the sample points, but you can fix this by using more than one orbit of equipotentials.
The Riemann map for the central component for the Basilica was drawn in essentially the same way, except that instead of starting with points on a big circle, I started with sample points on a circle of small radius (e.g. 0.00001) around the origin.
The Bubble bath Julia set
editthe Bottcher map is well defined (up to a choice of which ray to map to the angle zero) by the theorem on the previous page pp. 42
For the Bubble bath Julia set^{[17]} :
A commutative diagram ( the picture describing function composition):
Here:
 is the middle bubble ( component), whose center is z=0
 is the open unit disc, whose center is w=0
 is the function which maps unit disk to itself and it's equation is
 is the function which maps middle bubble to itself and it's equation is
 is the Boettcher map which maps middle buble to unit disk
Maxima CAS code
a:factor(f(f(f(z)))); (%o10)(z^4*(z^44*z^2+2))/(2*z^21)^2 taylor(a,z,0,14); (%o11) 2*z^4+4*z^6+9*z^8+20*z^10+44*z^12+96*z^14
In the above square diagram, commutativity of the diagram means:
.
It gives Böttcher's functional equation^{[18]}:
Solve above functional equation for the Boettcher map
explicit closed form solution
edit power maps
 Chebyshev polynomials
c = 2
editFor c = 2 Julia set is a horizontal segment between 2 and 2 on the real axis:^{[19]}
Now:
 equipotential lines are elipses
 field lines are hyperbolas
 Boettcher map and it's inverse have explicit equations ( closedform expression^{[20]} ):
where branch cut is taken to coincide with
See also:
 charged line segment
 Joukowsky transformation^{[21]}
 math.stackexchange question: explicitlycalculatinggreensfunctionincomplexdynamics
History
editIn 1904 LE Boettcher:^{[22]}
 solved Schröder functional equation^{[23]}^{[24]} in case of supperattracting fixed point^{[25]}^{[26]}
 "proved the existence of an analytic function near which conjugates the polynomial with , that is " ( Alexandre Eremenko ) ^{[27]}
LogPhi  DouadyHubbard potential  real potential  radial component of complex potential
editAproximates distance to the fractal^{[28]}
 it is smooth
 it is 0 at the fractal and logz at infinity
The potential function and the real iteration number
editThe Julia set for is the unit circle, and on the outer Fatou domain, the potential function φ(z) is defined by φ(z) = logz. The equipotential lines for this function are concentric circles. As we have
where is the sequence of iteration generated by z. For the more general iteration , it has been proved that if the Julia set is connected (that is, if c belongs to the (usual) Mandelbrot set), then there exist a biholomorphic map ψ between the outer Fatou domain and the outer of the unit circle such that .^{[29]} This means that the potential function on the outer Fatou domain defined by this correspondence is given by:
This formula has meaning also if the Julia set is not connected, so that we for all c can define the potential function on the Fatou domain containing ∞ by this formula. For a general rational function f(z) such that ∞ is a critical point and a fixed point, that is, such that the degree m of the numerator is at least two larger than the degree n of the denominator, we define the potential function on the Fatou domain containing ∞ by:
where d = m − n is the degree of the rational function.^{[30]}
If N is a very large number (e.g. 10^{100}), and if k is the first iteration number such that , we have that
for some real number , which should be regarded as the real iteration number, and we have that:
where the last number is in the interval [0, 1).
For iteration towards a finite attracting cycle of order r, we have that if z* is a point of the cycle, then (the rfold composition), and the number
is the attraction of the cycle. If w is a point very near z* and w' is w iterated r times, we have that
Therefore the number is almost independent of k. We define the potential function on the Fatou domain by:
If ε is a very small number and k is the first iteration number such that , we have that
for some real number , which should be regarded as the real iteration number, and we have that:
If the attraction is ∞, meaning that the cycle is superattracting, meaning again that one of the points of the cycle is a critical point, we must replace α by
where w' is w iterated r times and the formula for φ(z) by:
And now the real iteration number is given by:
For the colouring we must have a cyclic scale of colours (constructed mathematically, for instance) and containing H colours numbered from 0 to H−1 (H = 500, for instance). We multiply the real number by a fixed real number determining the density of the colours in the picture, and take the integral part of this number modulo H.
The definition of the potential function and our way of colouring presuppose that the cycle is attracting, that is, not neutral. If the cycle is neutral, we cannot colour the Fatou domain in a natural way. As the terminus of the iteration is a revolving movement, we can, for instance, colour by the minimum distance from the cycle left fixed by the iteration.
CPM/J
editNote that potential inside Kc is zero so :
Pseudocode version :
if (LastIteration==IterationMax) then potential=0 /* inside Filledin Julia set */ else potential= GiveLogPhi(z0,c,ER,nMax); /* outside */
It also removes potential error for log(0).
Full version
editMath (full) notation :^{[31]}
Maxima (full) function :
GiveLogPhi(z0,c,ER,nMax):= block( [z:z0, logphi:log(cabs(z)), fac:1/2, n:0], while n<nMax and abs(z)<ER do (z:z*z+c, logphi:logphi+fac*log(cabs(1+c/(z*z))), n:n+1 ), return(float(logphi)) )$
Simplified version
editThe escape rate function of a polynomial f is defined by :
where :
"The function Gp is continous on C and harmonic on the complement of the Julia set. It vanishes identically on K(f) and as it has a logarithmic pole at infinity, it is a it is the Green's function for C/ K(f)." ( Laura G. DeMarco) ^{[32]}
Math simplified formula :
Maxima function :
GiveSLogPhi(z0,c,e_r,i_max):= block( [z:z0, logphi, fac:1/2, i:0 ], while i<i_max and cabs(z)<e_r do (z:z*z+c, fac:fac/2, i:i+1 ), logphi:fac*log(cabs(z)), return(float(logphi)) )$
If you don't check if orbit is not bounded ( escapes, bailout test) then use this Maxima function :
GiveSLogPhi(z0,c,e_r,i_max):= block( [z:z0, logphi, fac:1/2, i:0], while i<i_max and cabs(z)<e_r do (z:z*z+c, fac:fac/2, i:i+1 ), if i=i_max then logphi:0 else logphi:fac*log(cabs(z)), float(logphi) )$
C version :
double jlogphi(double zx0, double zy0, double cx, double cy) /* this function is based on function by W Jung http://mndynamics.com */ { int j; double zx=zx0, zy=zy0, s = 0.5, zx2=zx*zx, zy2=zy*zy, t; for (j = 1; j < 400; j++) { s *= 0.5; zy = 2 * zx * zy + cy; zx = zx2  zy2 + cx; zx2 = zx*zx; zy2 = zy*zy; t = fabs(zx2 + zy2); // abs(z) if ( t > 1e24) break; } return s*log2(t); // log(zn)* 2^(n) }//jlogphi
Euler version by R. Grothmann ( with small change : from z^2c to z^2+c) :^{[33]}
function iter (z,c,n=100) ... h=z; loop 1 to n; h=h^2+c; if totalmax(abs(h))>1e20; m=#; break; endif; end; return {h,m}; endfunction x=2:0.05:2; y=x'; z=x+I*y; {w,n}=iter(z,c); wr=max(0,log(abs(w)))/2^n;
Level Sets of potential = pLSM/J
editHere is Delphi function which gives level of potential :
Function GiveLevelOfPotential(potential:extended):integer; var r:extended; begin r:= log2(abs(potential)); result:=ceil(r); end;
Level Curves of potential = equipotential lines = pLCM/J
editThe continuous potential formula doesn't align properly with iteration bands
ArgPhi  External angle (angular component of complex potential) and external ray
editOne can start with binary decomposition of basin of attraction of infinity.
The second step can be using
period of angle under doubling map
editHow to find period of external angle measured in turns under doubling map :
Here is Common Lisp code :
(defun giveperiod (ratioangle)
"gives period of angle in turns (ratio) under doubling map"
(let* ((n (numerator ratioangle))
(d (denominator ratioangle))
(temp n)) ; temporary numerator
(loop for p from 1 to 100 do
(setq temp (mod (* temp 2) d)) ; (2 x n) modulo d = doubling)
when ( or (= temp n) (= temp 0)) return p )))
Maxima CAS code :
doubling_map(n,d):=mod(2*n,d);
/* catchthrow version by Stavros Macrakis, works */ GivePeriodOfAngle(n0,d):= catch( block([ni:n0], for i thru 200 do if (ni:doubling_map(ni,d))=n0 then throw(i), 0 ) )$ /* goloop version, works */ GiveP(n0,d):=block( [ni:n0,i:0], block( loop, ni:doubling_map(ni,d), i:i+1, if i<100 and not (n0=ni) then go(loop) ), if (n0=ni) then i else 0 ); /* Barton Willis while version without for loop , works */ GivePeriod(n0,d):=block([ni : n0,k : 1], while (ni : doubling_map(ni,d)) # n0 and k < 100 do ( k : k + 1), if k = 100 then 0 else k)$
Computing external angle
External angle (argument) is argument of Boettcher coordinate
Because Boettcher coordinate is a product of complex numbers
so argument of product is :
dynamic external rays
editDynamic external rays description
Lamination of dynamical plane
editHere is long description
See also
editReferences
edit Ricardo A. Oliva: On the Combinatorics of External Rays in the Dynamics of the Complex Henon Map.
 A ThompsonLike Group for the Bubble Bath Julia Set by Jasper WeinrichBurd Boettcher map and internal rays
 Fixed curves near fixed points by Alastair Fletcher
 Mandelson set area via the Böttcher series by Geoffrey Irving and colab doc description
 ↑ ON THE NOTIONS OF MATING by CARSTEN LUNDE PETERSEN AND DANIEL MEYER
 ↑ Neighbourhood in wikipedia
 ↑ The work of George Szekeres on functional equations by Keith Briggs
 ↑ Topological conjugacy in wikipedia
 ↑ How to draw external rays by Wolf Jung
 ↑ wolfram : MandelbrotSetBoettcher function
 ↑ Böttcher equation at Hyperoperations Wiki
 ↑ wikipedia : Böttcher's equation
 ↑ Lucjan Emil Böttcher and his mathematical legacy by Stanisław Domoradzki, Małgorzata Stawiska
 ↑ wolfram language : JuliaSetBoettcher
 ↑ How to draw external rays by Wolf Jung
 ↑ math.stackexchange question: explicitriemannmappings
 ↑ A Thompson Group for the Basilica by James Belk, Bradley Forrest
 ↑ "Quasisymmetry group of the basilica Julia set", Sergiy Merenkov, March 26, 2020 NYGT Seminar talk
 ↑ Böttcher's equation in wikipedia
 ↑ A Thompson Group for the Basilica by James Belk, Bradley Forrest
 ↑ A ThompsonLike Group for the Bubble Bath Julia Set by Jasper WeinrichBurd, 2013
 ↑ Böttcher's equation in wikipedia
 ↑ The Beauty of Fractals  book by HeinzOtto Peitgen and Peter Richter, page 63
 ↑ wikipedia : Closedform expression
 ↑ Joukowsky transformation by John D. Cook
 ↑ wikipedia : Lucjan_Böttcher
 ↑ Schröder equation in wikipedia
 ↑ Lucjan Emil Böttcher and his mathematical legacy by Stanislaw Domoradzki, Malgorzata Stawiska
 ↑ L. E. Boettcher, The principal laws of convergence of iterates and their aplication to analysis (Russian), Izv. Kazan. fiz.Mat. Obshch. 14) (1904), 155234.
 ↑ wikipedia : Böttchers_equation
 ↑ Mathoverflow : Growth of the size of iterated polynomials
 ↑ da Silva, V., Novello, T., Lopes, H., Velho, L. (2021). RealTime Rendering of Complex Fractals. In: Marrs, A., Shirley, P., Wald, I. (eds) Ray Tracing Gems II. Apress, Berkeley, CA. https://doi.org/10.1007/9781484271858_33
 ↑ Adrien Douady and John H. Hubbard, Etude dynamique des polynômes complexes, Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)
 ↑ Peitgen, HeinzOtto; Richter Peter (1986). The Beauty of Fractals. Heidelberg: SpringerVerlag. ISBN 0387158510.
 ↑ The Beauty of Fractals, page 65
 ↑ Holomorphic families of rational maps: dynamics, geometry, and potential theory. A thesis presented by Laura G. DeMarco
 ↑ Euler examples by R. Grothmann