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 MisiurewiczPoints

Source Notebook

Compute values in the Mandelbrot set for which the critical point is strictly pre-periodic

Contributed by: Arnoud Buzing

ResourceFunction["MisiurewiczPoints"][m,k]

returns the set of parameter values c for which the critical point 0 of f(z)=z2+c is strictly pre-periodic with pre-period m and period k.

Details

A point c is pre-periodic if it iterates eventually to a periodic point. For example, if c=-2 and starting with z=0 then f(z) has the sequence (0,-2,2,2,2,…). The values 0 and -2 are pre-periodic, and the value 2 is periodic.
All Misiurewicz points lie on the boundary of the Mandelbrot set but are specifically excluded from the centers of hyperbolic components.
At a Misiurewicz point c, the corresponding Julia set is connected, contains no interior, and admits no attracting periodic cycles, making the dynamics purely chaotic.
Misiurewicz points are dense in the boundary of the Mandelbrot set, often serving as the "tips" of filaments or junctions where branches of the set meet.
The function solves a polynomial equation zm+k(c)=zm(c) where the degree grows exponentially as 2m+k-1, leading to a significant increase in computational complexity for large values of m and k.
Michał Misiurewicz is a Polish-American mathematician renowned for his contributions to chaos theory and dynamical systems, specifically for defining the parameters in the Mandelbrot set where the critical point is strictly pre-periodic but not periodic.

Examples

Basic Examples (3) 

Compute the Misiurewicz points with a pre-period of 2 and a period of 2:

In[1]:=
ResourceFunction["MisiurewiczPoints"][2, 2]
Out[1]=

Compute the Misiurewicz points with no pre-period and a period of 3:

In[2]:=
ResourceFunction["MisiurewiczPoints"][0, 3]
Out[2]=

Compute the Misiurewicz points with a pre-period of 2 and a period of 5:

In[3]:=
ResourceFunction["MisiurewiczPoints"][2, 5]
Out[3]=

Scope (3) 

Use ComplexListPlot to plot Misiurewicz points in the complex plane:

In[4]:=
ComplexListPlot[ResourceFunction["MisiurewiczPoints"][2, 5]]
Out[4]=

Increasing the values of the arguments leads to an outline of the Mandelbrot set:

In[5]:=
ComplexListPlot[ResourceFunction["MisiurewiczPoints"][3, 6]]
Out[5]=

At any Misiurewicz point c, the corresponding Julia set is connected, contains no interior, and admits no attracting periodic cycles, making the dynamics purely chaotic:

In[6]:=
SeedRandom[0]; p = RandomChoice[ResourceFunction["MisiurewiczPoints"][4, 4]]
Out[7]=
In[8]:=
JuliaSetPlot[p, MaxIterations -> 300]
Out[8]=

Example of a non-Misiurewicz point that has an interior:

In[9]:=
JuliaSetPlot[-1]
Out[9]=

Example of a non-Misiurewicz point that is fully disconnected:

In[10]:=
JuliaSetPlot[1, MaxIterations -> 20]
Out[10]=

Applications (3) 

Pick a random Misiurewicz point:

In[11]:=
z = RandomChoice[ResourceFunction["MisiurewiczPoints"][3, 6]]
Out[11]=

Misiurewicz points lie on the boundary of the Mandelbrot set, so they generally yield visually interesting results:

In[12]:=
r = 1.0 + 1.0 I; MandelbrotSetPlot[{z - 0.01 r, z + 0.01 r}]
Out[13]=

Create a video that zooms in on a Misiurewicz point:

In[14]:=
AnimationVideo[ MandelbrotSetPlot[{z - r 2^-t, z + r 2^-t}, Sequence[ Frame -> False, Axes -> False, PlotRangePadding -> None]], {t, 0, 10}]
Out[14]=

Publisher

Arnoud Buzing

Version History

  • 1.0.0 – 01 April 2026

Source Metadata

Related Resources

Author Notes

MisiurewiczPoints was written with the help of AI tools.

License Information