Julia set

In complex dynamics, the Julia set $J(f)\,$ of a holomorphic function $f\,$ informally consists of those points whose long-time behavior under repeated iteration of $f\,$ can change drastically under arbitrarily small perturbations.

The Fatou set $F(f)\,$ of $f\,$ is the complement of the Julia set: that is, the set of points which exhibit 'stable' behavior.

Thus on $F(f)\,$ , the behavior of $f\,$ is 'regular', while on $J(f)\,$ , it is 'chaotic'.

These sets are named in honor of the French mathematicians Gaston Julia and Pierre Fatou, who initiated the theory of complex dynamics in the early 20th century.

Formal definition

Let

f:X\to X\,

be an analytic self-map of a Riemann surface $X\,$ . We will assume that $X\,$ is either the Riemann sphere, the complex plane, or the once-punctured complex plane, as the other cases do not give rise to interesting dynamics. (Such maps are completely classified.)

We will be considering $f\,$ as a discrete dynamical system on the phase space $X\,$ , so we are interested in the behavior of the iterates $f^{n}\,$ of $f\,$ (that is, the $n\,$ -fold compositions of $f\,$ with itself).

The Fatou set of $f\,$ consists of all points $z\in X\,$ such that the family of iterates

(f^{n})_{n\in \mathbb {N} }

forms a normal family in the sense of Montel when restricted to some open neighborhood of $z\,$ .

The Julia set of $f\,$ is the complement of the Fatou set in $X\,$ .

Equivalent descriptions of the Julia set

$J(f)\,$ is the smallest closed set containing at least three points which is completely invariant under $f\,$ .
$J(f)\,$ is the closure of the set of repelling periodic points.
For all but at most two points $z\in X\,$ , the Julia set is the set of limit points of the full backwards orbit $\bigcup _{n}f^{-n}(z)$ . (This suggests a simple algorithm for plotting Julia sets, see below.)
If $f\,$ is an entire function - in particular, when $f\,$ is a polynomial, then $J(f)\,$ is the boundary of the set of points which converge to infinity under iteration.
If $f\,$ is a polynomial, then $J(f)\,$ is the boundary of the filled Julia set; that is, those points whose orbits under $f\,$ remain bounded.

Rational maps

There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational maps. For example, it is known that the Fatou set of a rational map has either 0,1,2 or infinitely many components.^[1] Each component of the Fatou set of a rational map can be classified into one of four different classes.^[2]

Quadratic polynomials

A very popular complex dynamical system is given by the family of quadratic polynomials, a special case of rational maps. The quadratic polynomials can be expressed as

f_{c}(z)=z^{2}+c\,

(where $c\,$ is a complex parameter).

Filled Julia set for f_c, c=φ−2 where φ is the golden ratio
Julia set for f_c, c=(φ−2)+(φ−1)i =-0.4+0.6i
Julia set for f_c, c=0.285+0i
Julia set for f_c, c=0.285+0.01i
Julia set for f_c, c=0.45+0.1428i
Julia set for f_c, c=-0.70176-0.3842i
Julia set for f_c, c=-0.835-0.2321i
Julia set for f_c, c=-0.8+0.156i

The parameter plane of quadratic polynomials - that is, the plane of possible $c$ -values - gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all $c$ such that $J(f_{c})\,$ is connected. For parameters outside the Mandelbrot set, the Julia set is a Cantor set: in this case it is sometimes referred to as Fatou dust.

In many cases, the Julia set of c looks like the Mandelbrot set in sufficiently small neighborhoods of c. This is true, in particular, for so-called 'Misiurewicz' parameters, i.e. parameters $c$ for which the critical point is pre-periodic. For instance:

At c= i, the shorter, front toe of the forefoot, the Julia set looks like a branched lightning bolt.
At c = −2, the tip of the long spiky tail, the Julia set is a straight line segment.

Generalizations

The definition of Julia and Fatou sets easily carries over to the case of certain maps whose image contains their domain; most notably transcendental meromorphic functions and Epstein's 'finite-type maps'.

Julia sets are also commonly defined in the study of dynamics in several complex variables.

Plotting the Julia set using backwards iteration

As mentioned above, the Julia set can be found as the set of limit points of the set of pre-images of (essentially) any given point. So we can try to plot the Julia set of a given function as follows. Start with any point $z\,$ we know to be in the Julia set, such as a repelling periodic point, and compute all pre-images of $z\,$ under some high iterate $f^{n}\,$ of $f\,$ .

Unfortunately, as the number of iterated pre-images grows exponentially, this is not computationally feasible. However, we can adjust this method, in a similar way as the "random game" method for iterated function systems. That is, in each step, we choose at random one of the inverse images of $f\,$ .

For example, for the quadratic polynomial $f_{c}\,$ , the backwards iteration is described by

z_{n+1}^{2}=z_{n}-c.

At each step, one of the two square roots is selected at random.

Note that certain parts of the Julia set are quite hard to reach with the reverse Julia algorithm. For this reason, other methods usually produce better images.

References

^ Beardon, Iteration of Rational Functions, Theorem 5.6.2
^ Beardon, Theorem 7.1.1

Lennart Carleson and Theodore W. Gamelin, Complex Dynamics, Springer 1993
Adrien Douady and John H. Hubbard, "Etude dynamique des polynômes complexes", Prépublications mathémathiques d'Orsay 2/4 (1984 / 1985)
John W. Milnor, Dynamics in One Complex Variable (Third Edition), Annals of Mathematics Studies 160, Princeton University Press 2006 (First appeared in 1990 as a Stony Brook IMS Preprint, available as arXiV:math.DS/9201272.)
Alexander Bogomolny, "Mandelbrot Set and Indexing of Julia Sets" at cut-the-knot.
Evgeny Demidov, "The Mandelbrot and Julia sets Anatomy" (2003)
Alan F. Beardon, Iteration of Rational Functions, Springer 1991, ISBN 0-387-95151-2

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[1] Beardon, Iteration of Rational Functions, Theorem 5.6.2

[2] Beardon, Theorem 7.1.1

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