Julia set

A Julia set

In complex dynamics, the Julia set Failed to parse (Missing texvc executable; please see math/README to configure.): J(f)\,

of a holomorphic function Failed to parse (Missing texvc executable; please see math/README to configure.): f\,
informally consists of those points whose long-time behavior under repeated iteration of Failed to parse (Missing texvc executable; please see math/README to configure.): f\,
can change drastically under arbitrarily small perturbations.

The Fatou set Failed to parse (Missing texvc executable; please see math/README to configure.): F(f)\,

of Failed to parse (Missing texvc executable; please see math/README to configure.): f\,
is the complement of the Julia set: that is, the set of points which exhibit 'stable' behavior.

Thus on Failed to parse (Missing texvc executable; please see math/README to configure.): F(f)\, , the behavior of Failed to parse (Missing texvc executable; please see math/README to configure.): f\,

is 'regular', while on Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): f:X\to X\,

be an analytic self-map of a Riemann surface Failed to parse (Missing texvc executable; please see math/README to configure.): X\, . We will assume that Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): f\,

as a discrete dynamical system on the phase space

Failed to parse (Missing texvc executable; please see math/README to configure.): X\, , so we are interested in the behavior of the iterates Failed to parse (Missing texvc executable; please see math/README to configure.): f^n\,

of Failed to parse (Missing texvc executable; please see math/README to configure.): f\,
(that is, the Failed to parse (Missing texvc executable; please see math/README to configure.): n\,

-fold compositions of Failed to parse (Missing texvc executable; please see math/README to configure.): f\,

with itself).

The Fatou set of Failed to parse (Missing texvc executable; please see math/README to configure.): f\,

consists of all points Failed to parse (Missing texvc executable; please see math/README to configure.): z\in X\,
such that the family of iterates

Failed to parse (Missing texvc executable; please see math/README to configure.): (f^n)_{n\in\mathbb{N}}

forms a normal family in the sense of Montel when restricted to some open neighborhood of Failed to parse (Missing texvc executable; please see math/README to configure.): z\, .

The Julia set of Failed to parse (Missing texvc executable; please see math/README to configure.): f\,

is the complement of the Fatou set in Failed to parse (Missing texvc executable; please see math/README to configure.): X\,

.

Equivalent descriptions of the Julia set

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

Properties of the Julia set and Fatou set

The Julia set and the Fatou set of Failed to parse (Missing texvc executable; please see math/README to configure.): f

are both completely invariant under Failed to parse (Missing texvc executable; please see math/README to configure.): f

, i.e.

Failed to parse (Missing texvc executable; please see math/README to configure.): \ f^{-1}(J(f)) = f(J(f)) = J(f)

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \ f^{-1}(F(f)) = f(F(f)) = F(f) . ^[1]

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.^[2] Each component of the Fatou set of a rational map can be classified into one of four different classes.^[3]

Quadratic polynomials

Wikibooks has a book on the topic of

Fractals

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

Failed to parse (Missing texvc executable; please see math/README to configure.): f_c(z) = z^2 + c\,

(where Failed to parse (Missing texvc executable; please see math/README to configure.): c\,

is a complex parameter). 

Filled Julia set for fc, c=1−φ where φ is the golden ratio	Julia set for fc, c=(φ−2)+(φ−1)i =-0.4+0.6i	Julia set for fc, c=0.285+0i	Julia set (highres 01).jpg Julia set for fc, c=0.285+0.01i
Julia set camp3.jpg Julia set for fc, c=0.45+0.1428i	Julia set camp1.jpg Julia set for fc, c=-0.70176-0.3842i	Julia set camp2.jpg Julia set for fc, c=-0.835-0.2321i	Julia set camp4 hi rez.png Julia set for fc, c=-0.8+0.156i

The parameter plane of quadratic polynomials - that is, the plane of possible Failed to parse (Missing texvc executable; please see math/README to configure.): c -values - gives rise to the famous Mandelbrot set. Indeed, the Mandelbrot set is defined as the set of all Failed to parse (Missing texvc executable; please see math/README to configure.): c

such that Failed to parse (Missing texvc executable; please see math/README to configure.): 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. 

Map of 121 Julia sets in position over the Mandelbrot set

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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

In other words the Julia sets Failed to parse (Missing texvc executable; please see math/README to configure.): J(f_c)\,

are locally similar around Misiurewicz points.[4]

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 (inverse) iteration (IIM )

 


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 Failed to parse (Missing texvc executable; please see math/README to configure.): z\,

we know to be in the Julia set, such as a repelling periodic point, and compute all pre-images of Failed to parse (Missing texvc executable; please see math/README to configure.): z\,
under some high iterate Failed to parse (Missing texvc executable; please see math/README to configure.): f^n\,
of Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): f\,
.


For example, for the quadratic polynomial Failed to parse (Missing texvc executable; please see math/README to configure.): f_c\,
, the backwards iteration is described by

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

using DEM/J

See also

• Limit set
• Stable and unstable sets
• No wandering domain theorem
• Fatou components

Wikimedia Commons has media related to:

Julia set

References

1. ^ Beardon, Iteration of Rational Functions, Theorem 3.2.4
2. ^ Beardon, Iteration of Rational Functions, Theorem 5.6.2
3. ^ Beardon, Theorem 7.1.1
4. ^ Lei.pdf Tan Lei, "Similarity between the Mandelbrot set and Julia Sets", Communications in Mathematical Physics 134 (1990), pp. 587-617.

• 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

Links

• http://mathmo.blogspot.com/2007/04/essay-backtrack-julia-sets.html
• http://mcgoodwin.net/julia/juliajewels.html

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