In complex dynamics, the Julia set [1] of a holomorphic function
informally consists of those points whose long-time behavior under repeated iteration of
can change drastically under arbitrarily small perturbations (bifurcation locus).
The Fatou set of
is the complement of the Julia set: that is, the set of points which exhibit 'stable' behavior.
Thus on , the behavior of
is 'regular', while on
, it is 'chaotic'.
These sets are named after the French mathematicians Gaston Julia,[2] and Pierre Fatou[3] who initiated the theory of complex dynamics in the early 20th century.
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Let
be a holomorphic map of a Riemann surface to itself. Assume that
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.)
Consider as a discrete dynamical system on the phase space
, and consider the behavior of the iterates
of
(that is, the
-fold compositions of
with itself).
The Fatou set of consists of all points
such that the family of iterates
forms a normal family in the sense of Montel when restricted to some open neighborhood of .
The Julia set of is the complement of the Fatou set in
.
The Julia set and the Fatou set of are both completely invariant under
, i.e.
and
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.[5] Each component of the Fatou set of a rational map can be classified into one of four different classes.[6]
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
where is a complex parameter.
The parameter plane of quadratic polynomials - that is, the plane of possible -values - gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all
such that
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 looks like the Mandelbrot set in sufficiently small neighborhoods of
. This is true, in particular, for so-called 'Misiurewicz' parameters, i.e. parameters
for which the critical point is pre-periodic. For instance:
In other words the Julia sets are locally similar around Misiurewicz points.[7]
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.
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 we know to be in the Julia set, such as a repelling periodic point, and compute all pre-images of
under some high iterate
of
.
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 .
For example, for the quadratic polynomial , the backwards iteration is described by
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.