Julia Sets Produced by Cubic Polynomials
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Consider the cubic polynomial 
, where 
and 
are complex numbers. It is known that if 
or 
is a root of unity, then 0 is in the Julia set of 
, while if 
, then 0 is in the Fatou set of 
. Thus, we focus on the remaining, more interesting case of 
with 
not a root of unity, since in this case 0 could be in either the Julia or the Fatou set. Write 
and 
for 
, 
. This Demonstration illustrates how the filled Julia set of 
changes around 0 as you vary the coefficients 
and 
and the variables 
, 
and 
.
Contributed by: Xiaojun Jia, Troy Yang, Sarah Zimmerman and Efstathios Konstantinos Chrontsios Garitsis
(Based on an undergraduate research project at the Illinois Geometry Lab in Spring 2021.) (August 2022)
Open content licensed under CC BY-NC-SA
Snapshots
Details
Let 
be a polynomial and denote its 
iterate by 
. The Julia set of 
is a fractal set of points where the dynamics of the polynomial exhibit chaotic behavior that is defined as follows.
Definition: The set of points 
for which the set of iterates 
is bounded is called a filled Julia set of 
. The boundary of this set is called a Julia set of 
and its complement is called a Fatou set of 
.
The fixed points of 
, that is, points 
for which 
, often provide information on how the dynamics of the polynomial behave around 
. Namely:
Theorem: let 
be a fixed point of 
.
• If 
, then 
is called attracting and lies in the Fatou set of 
.
• If 
, then 
is called repelling and lies in the Julia set of 
.
• If 
is a root of unity, then 
is called rationally indifferent and lies in the Julia set of 
.
• If 
with 
, then 
is called irrationally indifferent and can be in either the Julia or the Fatou set. Moreover, we can conjugate 
by a translation to get a new polynomial with an irrationally indifferent fixed point at 0 and conjugate by scaling to make it monic. By the following proposition, it is then sufficient to focus on polynomials of the form 
with 
, 
and 
.
Proposition: Let 
be a Möbius map. The Julia and Fatou sets of 
are equal to the image under 
of the Julia and the Fatou sets of 
, respectively.
For proofs and more details, see [1].
You can vary the graphics:
• there are sliders to choose any arbitrary argument 
and magnitude 
for the coefficient of the second-degree term,
• a few natural fixed options for the argument 
of said coefficient
• fixed options for the first-degree term, namely for specific values of 
• ways to zoom in, zoom out and change the color of the plot
• the circle on the lower left shows where the coefficient for the second degree is located in the plane, in case you want to fix 
and let 
vary
For additional theoretical background, see [2, 3].
References
[1] A. F. Beardon, Iteration of Rational Functions: Complex Analytic Dynamical Systems, New York: Springer-Verlag, 1991.
[2] R. Mañè, P. Sad and D. Sullivan, "On the Dynamics of Rational Maps," Annales scientifiques de l'École Normale Supérieure, 16(2), 1983 pp. 193–217. eudml.org/doc/82115.
[3] C. McMullen and D. Sullivan, "Quasiconformal Homeomorphisms and Dynamics III. The Teichmüller Space of a Holomorphic Dynamical System," Advances in Mathematics, 135(2), 1998 pp. 351–395. doi:10.1006/aima.1998.1726.
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