Fractals/Rational

Iteration of complex rational functions

• analysis of rational map
• examples

Herman ring - image with c++ src code

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 $f(z)={\frac {\left(z+c\right)^{2}}{z^{3}+c}}$

Gallery

• Rational functions
• 
  ${\frac {z}{z-1}}$
• 
  ${\frac {z^{2}}{z^{2}-1}}$
• 
  ${\frac {z^{3}}{z^{3}-1}}$
• 
  ${\frac {z^{4}}{z^{4}-1}}$
• 
  ${\frac {z^{5}}{z^{5}-1}}$
• 
  ${\frac {z^{7}}{z^{7}-1}}$
• 
• 
• 
• 
• 
• 

• Rational functions
• 
  ${\frac {z^{4}+0.05}{z^{4}-1}}$
• 
• 
• 
• 
• 
• 
• 
• 
• 

• 
  ${\frac {z^{3}}{z^{5}-1}}$
• 
  ${\frac {z^{4}}{z^{5}-1}}$
• 
  ${\frac {z^{5}}{z^{5}-1}}$
• 
  ${\frac {z^{6}}{z^{5}-1}}$
• 
  ${\frac {z^{7}}{z^{5}-1}}$

• 
  ${\frac {(z-4i+3)(z-2-3i)z}{(z+2i+4)(z+3i-2)}}$

Blaschke fraction

Fatou sets for Blaschke fraction f(z) = rho * z^2 * (z-3) over (1-3z)

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 $B''(z0)\neq 0.$

• boudaries of BDM
• 
• 
• 

critial orbit

• 

McMullen maps

singularly perturbed maps, also called McMullen maps

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

• 
• 

degree 2

• 
  reversed Basilica Julia set
• 

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 2 by Michael Becker

1 period 4 basin

Julia set for f(z)=(z2+a) over (z2+b) a=-0.2+0.7i , b=0.917

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.

degree 3 by Michael Becker

1 period 2 basin

Julia set f(z)=1 over z3+z*(-3-3*I)

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

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

• 
  only Julia set
• 
  Julia set, basins of attractions, critical orbits, attracting cycles

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

degree 5

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

Julia set of rational function f(z)=z^2(3 − z^4 ) over 2.png

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

Julia set for f(z)=z2 over (z9-z+0.025)

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