Fractals/Rational



Iteration of complex rational functions
 * analysis of rational map
 * examples

=Examples=
 * Newton fractals
 * commons:Category:Complex rational maps
 * The Boundedness Locus and baby Mandelbrot sets for some generalized McMullen maps by Suzanne Boyd, Alexander J. Mitchell 2023
 * Mark McClure : a-julia-set-on-the-riemann-sphere
 * f(z)=z2/(z9-z+0,025)
 * f(z)=(z3-z)/(dz2+1) where d=-0,003+0,995i
 * f(z)=(z3-z)/(dz2+1) where d=1,001· e2Pi/30
 * Multibrot sets by Xender
 * $$f_{\lambda}(z) = z^n + \frac{\lambda}{z^d}$$
 * Rational Julia Sets by Marc McClure
 * f(z) = (z^n+c)/(c^n+z), for n = -2
 * Jasper Weinrich Burd: A Thompson-Like Group for the Bubble Bath Julia Set
 * Abalone Fractals by Anders Sandberg, 2004
 * (z2-1)/(z2+1). It has simple zeros at ±1 and simple poles at ±i.
 * Shigehiro Ushiki = ComplexExplorer Page
 * Rational maps with a superattracting 2-cycle by Wolf Jung
 * Julia Sets of Cubic Rational Maps with Escaping Critical Points by Jun Hu, Arkady Etkin
 * ON HYPERBOLIC RATIONAL MAPS WITH FINITELY CONNECTED FATOU SETS by YUSHENG LUO, 2021
 * On geometrically finite degenerations I: boundaries of main hyperbolic components by Yusheng Luo, 2021
 * On geometrically finite degenerations II: convergence and divergence by Yusheng Luo, 2021
 * math.stackexchange question: how-to-compute-a-negative-multibrot-set
 * Exotische Juliamengen - Jürgen Meier
 * There are rational functions whose Julia set is the whole plane. The first example was given by Lattès: $$R(z)=\frac{(z^2+1)^2 }{ 4z(z^2-1)}$$
 * Rational maps whose Julia sets are Cantor circles by Weiyuan Qiu, Fei Yang, Yongcheng Yin
 * Petersen, C. L., and S. Zakeri. “On the Julia Set of a Typical Quadratic Polynomial with a Siegel Disk.” Annals of Mathematics, vol. 159, no. 1, 2004, pp. 1–52.
 * A Note on a Quadratic Rational Map with Two Siegel Disks by Liang SHEN, Sheng Jian WU. Acta Mathematica Sinica, English Series. Jul., 2010, Vol. 26, No. 7, pp. 1393–1402 Published online: June 15, 2010 DOI: 10.1007/s10114-010-6611-3 Http://www.ActaMath.com
 * MandelbrotMaps by Chris King
 * fractalforums.org : z-plus-c2-z3-plus-c $$f(z) = \frac{\left( z + c \right)^2}{z^3 + c}$$

Blaschke fraction


Analysis of critical points: kill(all); remvalue(all); display2d:false; ratprint : false; /* remove "rat :replaced " */

rho : -0.6170144002709304 +0.7869518599370003*%i;

define(f(z), rho * z^2 * (z-3)/(1-3*z));

/* first derivativa wrt z */ define( d(z), diff(f(z),z,1));

/* hipow does not expand expr, so hipow (expr, x) and hipow (expand (expr, x)) may yield different results */ n : hipow(num(expand(f(z))),z); m : hipow(denom(expand(f(z))),z);

/* check if infinity is a fixed point */ limit(f(z),z,infinity);

/* finite critical points */

s:solve(d(z)=0)$ s : map(rhs,s)$ s : map('float,s)$ s : map('rectform,s)$

So there are 3 critical points :
 * 2 finite critical points : z=1.0 i z= 0.0
 * infinity

Dynamical plane consist of 3 basins
 * basin of attraction of fixed point z = infinity ( superattracting) with inf many componnets
 * basin of attraction of fixed point z = 0 ( superattracting) with inf many componnets
 * basin of parabolic period 3 cycle ( with z= 1 critical point)

Finite Blaschke product
$$ B_n(z) $$ is a finite Blaschke product of degree n. It is:
 * a rational function
 * an analytic function on the open unit disc such that f can be extended to a continuous function on the closed unit disc that maps the unit circle to itself
 * have no poles in the open unit disc
 * In particular, if ƒ satisfies the condition above and has no zeros inside the unit circle, then ƒ is constant (this fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function log(|ƒ(z)|)).
 * the Blaschke products B are rational perturbations of the doubling map of the circle R(z) = z^2 (equivalently given by θ → 2θ (mod 1)).
 * a finite Blaschke product may be uniquely described by the set of its critical points
 * rational map which fix a disc =  which takes the closed unit disc D to itself
 * their iteration theory can be analyzed from the point of view of Fuchsian groups.
 * polynomials’ in the hyperbolic plane = hyperbolic polynomial
 * A finite Blaschke product, restricted to the unit circle, is a smooth covering map
 * the unit disk D, the unit circle ∂D and the complement of the closed unit disk C\D are all completely invariant sets for B


 * $$ B_n(z)=\zeta\prod_{i=1}^n\left({{z-a_i}\over {1-\overline{a_i}z}}\right)^{m_i}

$$

where
 * $$\zeta$$ is a unimodular constant. It is a point which lies on the unit circle: $$ |\zeta| = 1$$
 * $$m_i$$ is the multiplicity of the zero $$a_i$$
 * $$(a_i)_{i=1}^n$$ is a finite sequence of n points in the open unit disc $$|a_i| < 1 $$


 * $$(a_i)_{i=1}^n = ( a_1, a_2, \ldots, a_n).$$

The building blocks of Blaschke products are Mobius transformations of the form

$$ b_k(z) = e^{i\theta k} \frac{a_k - z }{1 - \overline{a_k} z } $$

where
 * ak ∈ D := {z ∈ C|, |z| < 1}
 * θk ∈ R.

A finite (infinite) Blaschke product has the form

$$ w = B(z) = \prod_{k=1}^n b_k(z) $$

Examples:
 * Quadratic rational maps by Curtis T McMullen

classification
There is a classification of finite Blaschke products in analogy with Möbius transformations.
 * B is elliptic if the Denjoy-Wolff point z0 of B lies in D. |B' (z0)| < 1.
 * B is hyperbolic if the Denjoy-Wolff point z0 of B lies on ∂D and B'(z0) < 1,
 * B is parabolic if the Denjoy-Wolff point z0 of B lies on ∂D and B'(z0) = 1,

The Denjoy-Wolff point of B is a unique z0 ∈ D such that $$B^n(z) \to z_0$$ for every z ∈ D

Julia set
Let B be a finite Blaschke product of degree d > 1. Julia set $$J_B$$, the set on which the iterates $$B^n$$ fail to be normal on any neighbourhood is either the unit circle $$S^1$$ or a Cantor subset
 * if B is elliptic, J(B) = ∂D,
 * if B is hyperbolic, J(B) is a Cantor subset of D,
 * if B is parabolic and z0 ∈ ∂D is the Denjoy-Wolff point of B,
 * J(B) = ∂D if B''(z0) = 0
 * J(B) is a Cantor subset of ∂D if $$B''(z0) \ne 0.$$

McMullen maps
singularly perturbed maps, also called McMullen maps

$$ R_{\lambda}(z) = z^m + \lambda/ z^d$$

degree 2
Function: $$f(z) = \frac{z^2}{z^2-1}$$

maxima

Maxima 5.41.0 http://maxima.sourceforge.net using Lisp GNU Common Lisp (GCL) GCL 2.6.12 Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. The function bug_report provides bug reporting information. (%i1) display2d:false;

(%o1) false (%i2) f:z^2/(z^2-1);

(%o2) z^2/(z^2-1) (%i3) dz:diff(f,z,1);

(%o3) (2*z)/(z^2-1)-(2*z^3)/(z^2-1)^2 (%i4) s:solve(f=z);

(%o4) [z = -(sqrt(5)-1)/2,z = (sqrt(5)+1)/2,z = 0] (%i5) s:map('float,s);

(%o5) [z = -0.6180339887498949,z = 1.618033988749895,z = 0.0] (%i6)

So fixed points $$ z : f(z) = z $$:
 * z = -0.6180339887498949
 * z = 1.618033988749895
 * z = 0.0

The Bubble bath Julia set
The quadratic rational function f:
 * $$f(z) = \frac{1-z^2}{z^2} $$

The derivative wrt z is


 * $$f'(z) = \frac{-2}{z^3} $$

The Julia set for f is called the Bubble Bath Julia set. It is called the Bubble Bath for its visual similarity to a tub of bubbles.

Function f is defined for all z in the Riemann sphere $$ \hat{\mathbf{C}}$$ = it is defined on the whole Riemann sphere


 * $$f : \hat{\mathbf{C}} \to \hat{\mathbf{C}}$$


 * The Fatou set of f is the basin of attraction of the 3-cycle consisting of the points 0, −1, and infinity. It is the only one attracting cycle and it is superattracting
 * The Julia set J(f) is the set of points whose orbits are not attracted to the above 3-cycle
 * the only critical points of f are:
 * z = 0 because it is the pole of order 3 of d(z), the zero of 1/d(z)
 * z = infinity because it is the zero of function d(z)

Maxima CAS code : kill(all); remvalue(all); display2d:false;

define(f(z), (1 -z^2)/(z^2));

(%o3) f(z):=(1-z^2)/z^2 define( d(z), ratsimp(diff(f(z),z,1)));

(%o13) d(z):=-2/z^3 (%i14) limit(d(z),z,infinity);

(%o14) 0 (%i15) limit(d(z),z,0);

(%o15) infinity

(%i2) f(-1); (%o2)                                 0 (%i3) limit(d(z),z,0); (%o3)                            limit  d(z) z -> 0 (%i4) limit(f(z),z,0); (%o4)                                inf (%i5) limit(f(z),z,inf); (%o5)                                - 1

Satbility of periodic cycle: kill(all); display2d:false; ratprint : false; /* remove "rat :replaced " */

define(f(z), (1 -z^2)/(z^2));

F(z0):= block(	[z],	if is(z0 = 0) then 	z: limit(f(z),z,0)	elseif is(z0 = infinity) then z: limit(f(z),z,infinity)	elseif is(z0 = inf) then z: limit(f(z),z,inf)	else z:f(z0),	return(z) )$

define( dz(z), ratsimp(diff(f(z),z,1)));

Dz(z0) := block(

[m,z], if is(z0 = 0) then m: limit(dz(z),z,0) elseif is(z0 = infinity) then m: limit(dz(z),z,infinity) elseif is(z0 = inf) then m: limit(dz(z),z,inf) else m:dz(z0), return(m)

)$

GiveStability(z0, p):=block(	[z,d],	/* initial values */	d : 1,	z : z0,	for i:1 thru p step 1 do ( d : Dz(z)*d, z: F(z) /*print("i = ", 0, " d =",d, "  z = ", z)*/ ),	return (cabs(d)) )$

GiveStability(-1,3);

See also :
 * Boettcher map

degree 3
The components of the map $$f(z) = z - (z^3-1)/3z^2$$ contain the attracting points that are the solutions to $$z^3=1$$. This is because the map is the one to use for finding solutions to the equation $$z^3=1$$ by Newton–Raphson formula. The solutions must naturally be attracting fixed points.

1 period 2 basin


$$f(z)=\frac{1}{z^3+z*(-3-3*I)}$$

2 critical points : { -0.4550898605622273*I -1.098684113467809, 0.4550898605622273*I+1.098684113467809}; Both critical points tend to the periodic cycle. There is only one attractive period cycle : period 2 cycle = {0, infinity}.

Whole plane ( sphere) is a basin of attraction of period 2 cycle ( which is divided into 2 components ). Julia set is a boundary.

2 period 2 basins
Function $$f(z)=\frac{1}{z^3+a*z+ b}$$ where
 * a = 2.099609375
 * b = 0.349609375

Derivative:

d(z):=-(3*z^2+2.099609375)/(z^3+2.099609375*z+0.349609375)^2

Critical points:

[-0.8365822085525526*%i,0.8365822085525526*%i] One can check it also using Wolfram Alpha

solve (3*z^2+2.099609375)/(z^3+2.099609375*z+0.349609375)^2=0 the result:

z = ± (5/16)* i* sqrt(43/6)) These are 2 finite critical points.

Infinity is a critical point too, as the 1st derivative's denominator degree is strictly greater than the numerator's. In numerical computations one can use the critical value (an image of critical point)

$$f( \infty) = 0 $$ There are two period 2 cycle:
 * { +0.4101296722285255 +0.5079485669960778*I,  +0.4101296722285255 -0.5079485669960778*I };
 * { +1.6890328811664648 +0.0000000000000000*I, +0.1147519899962205 +0.0000000000000000*I };

Both finite critical points fall into first cycle. Infinity ( or it's image zero) falls into the second cycle ( on the horizontal axis)

Infinity is not a fixed point remvalue(all); display2d:false; define(f(z), 1/(z^3+ 2.099609375*z + 0.349609375)); (%i5)limit(f(z),z,infinity); (%o5) 0 (%i6) limit(f(z),z,0); (%o6) 2.860335195530726

by L. Javier Hernandez Paricio
the rational map $$h(z) = \frac{1 + 4z^5}{5z^4}$$ has six fixed points:
 * ∞ ( repelling)
 * −0,809017 − 0,587785i
 * −0,809017 +0,587785i
 * 0,309017 − 0,951057i
 * 0,309017 + 0,951057i
 * 1

the basin of an end point associated to a fixed point (6= ∞) of f is the same that the attraction basin of the Newton-Raphson numerical method when it is applied to find the roots of the equation $$z^5 - 1 = 0$$

degree 6


The Julia set of the degree 6 function f :

$$f(z) = z^2\frac{3-z^4}{2} $$

There are 3 superattracting fixed points at :
 * z = 0
 * z = 1
 * z = ∞

All other critical points are in the backward orbit of 1.

How to compute iteration :

z:x+y*%i; z1:z^2*(3-z^4)/2; realpart(z1); ((x^2−y^2)*(−y^4+6*x^2*y^2−x^4+3)−2*x*y*(4*x*y^3−4*x^3*y))/2 imagpart(z1); (2*x*y*(−y^4+6*x^2*y^2−x^4+3)+(x^2−y^2)*(4*x*y^3−4*x^3*y))/2

Find fixed points using Maxima CAS :

z1:z^2*(3-z^4)/2; s:solve(z1=z); s:float(s);

result :

[z=−1.446857247913871,z=.7412709105660023,z=−1.357611535209976*%i−.1472068313260655,z=1.357611535209976*%i−.1472068313260655,z=1.0,z=0.0]

check multiplicities of the roots : multiplicities; [1,1,1,1,1,1]

z1:z^2*(3-z^4)/2; s:solve(z1=z)$ s:map(rhs,s)$ f:z1; k:diff(f,z,1); define(d(z),k); m:map(d,s)$ m:map(abs,m)$ s:float(s); m:float(m);

Result : there are 6 fixed point 2 of them are supperattracting ( m=0 ), rest are repelling ( m>1 ):

[−1.446857247913871,.7412709105660023,−1.357611535209976*%i−.1472068313260655,1.357611535209976*%i−.1472068313260655,1.0,0.0] [14.68114348748323,1.552374536603988,10.66447061028112,10.66447061028112,0.0,0.0]

Critical points :

[%i,−1.0,−1.0*%i,1.0,0.0]

degree 9 by Michael Becker


=References=
 * Describing Blaschke products by their critical points by Oleg Ivrii, Tel Aviv University: video, presentation
 * mathoverflow question: finding-the-critical-points-of-a-degree-5-blaschke-product

=See also=
 * On the dynamics of rational maps by Mañé, R. ; Sad, P.  ; Sullivan, D. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 16 (1983) no. 2, pp. 193-217.
 * Rational maps: the structure of Julia sets from accessible Mandelbrot sets by Fitzgibbon, Elizabeth Laura
 * Iteration of Rational Functions by Omar Antolín Camarena
 * math.stackexchange question: attracting-or-parabolic-cycles-other-than-fixed-points
 * 3D rational Julia sets by Algoristo