degree 2

• 
  reversed Basilica Julia set
• 

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

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 ${\displaystyle z:f(z)=z}$:

• z = -0.6180339887498949
• z = 1.618033988749895
• z = 0.0

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 :^[10]

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

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]