Julia set

Julia set, set of values in range of holomorphism of function, but out of holomorphism of some its integer iteration.

Julia set is often defined with symbol \(J\) or \(\mathbb J\). The name of the function can be indicated either as the subscript or in the parenthesis immediately after this symbol.

Let \(f\) be holomorphic function defined at some \(C\in \mathbb C\).

Then

\(\mathbb J(f) = \{ z \in C : \exists ~ n\in \mathbb N_+ ~:~ f^n(z) \bar \in C\}\)

where the upper subscript after the name of the function indicates the number of iteration. This notation had been used by Walter Bergweiler at least since 1993 . In this case, the number \(n\) of iteration is supposed to be integer.

Origin of the name
Identifier Jilia set is choosen after Gaston Julia, who had presented the results about invariants of the iteration of functions in 1918 http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1918_8_1_A2_0 GASTON JULIA. Mémoire sur l’itération des fonctions rationnelles. Journal de mathématiques pures et appliquées 8e série, tome 1 (1918), p. 47-246. .

Fatou set
The complementary set to Julia set is called Fatou set ; symbol \(\mathbb F\) is used to denote it:

\(\mathbb F(f) = \{ z \in C : \forall ~ n\in \mathbb N_+ ~,~ f^n(z) \in C\}\)

in such a way that

\(\mathbb J(f) \cup \mathbb F(f)=C\)

At least for positive integer \(n\), the following relations hold:


 * $$f^n(\mathbb J(f)) = \mathbb J(f)$$


 * $$ f^n(\mathbb F(f)) = \mathbb F(f)$$

Fractal behavior
Usually, the Julia set of any non–trivial function with at least one singularity shows complicated, fractal behaviour; the similar structures reproduce again and again, displaced and scaled.

Science about properties of these fractals, and, in particular, those of the Julia sets and the Fatou sets, is called Complex dynamics.

Non-integer iterates
The definition of the Julia set above implies that the function \(f\) is iterated integer number of times. In principle, similar set can be considered, assuming, that the number of iterate \(n\) can take also non-integer values.

Generalization and the non–holomorphic transforms
The requirements for the iterated function \(F\) may be not so strict, as it is assumed above. In particular, the requirement of holomorhizm can be substituted to requirement of smoothness or to requirement of continuity. Then, the point \(z\) belongs to the Julia set, if \(F^n(z)\) is continuous function for every \(n\).

In addition, the dependence on the parameter can be considered, so, \(F=F_c\); then, the initial value \(z_0\) is assumed to be fixed. Number \(c\) is declared to belong to the set \(\mathbb J\), if \(\displaystyle \lim_{n \rightarrow \infty}F_c^n(z_0)\) exists and is smooth function of \(c\).

In such a way, there exist many various definitions and meanings of term Julia set.

In many cases, the Julia sets, defined with different way, show very similar fractal behavior, and looking at part of the plots of the approximations of the set, it is difficult to guess, which definition had been used, and iteration of which function does the plot correspond to.

If them Julia set is used for some scientific (not artistic) application, then, before to use this term, it should be explicitly defined.

Keywords
Fractal, Iteration, Superfunction