Wolfram Demonstrations Project

Consider the cubic polynomial

, where

and

are complex numbers. It is known that if

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.)

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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.