Difference between revisions of "Julia set"

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[[Julia set]], set of values in range of holomorphism of function, but out of holomorphism of some its integer iteration.
 
[[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.
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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 $C\in \mathbb C$.
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Let \(f\) be holomorphic function defined at some \(C\in \mathbb C\).
   
 
Then
 
Then
   
$\mathbb J(f) = \{ z \in C : \exists ~ n\in \mathbb N_+ ~:~ f^n(z) \bar \in C\}$
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\(\mathbb J(f) = \{ z \in C : \exists ~ n\in \mathbb N_+ ~:~ f^n(z) \bar \in C\}\)
   
 
where the upper subscript after the name of the function indicates the number of [[iteration]].
 
where the upper subscript after the name of the function indicates the number of [[iteration]].
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Bull. Amer. Math. Soc. 29 (1993), 151-188.
 
Bull. Amer. Math. Soc. 29 (1993), 151-188.
 
</ref>.
 
</ref>.
In this case, the number $n$ of iteration is supposed to be integer.
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In this case, the number \(n\) of iteration is supposed to be integer.
   
 
==Origin of the name==
 
==Origin of the name==
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P.Fatou. Sur les equations fonctionelles.
 
P.Fatou. Sur les equations fonctionelles.
 
Bulletin de la S.M.F., tome 48 (1920), p.208-314.
 
Bulletin de la S.M.F., tome 48 (1920), p.208-314.
</ref>; symbol $\mathbb F$ is used to denote it:
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</ref>; symbol \(\mathbb F\) is used to denote it:
   
$\mathbb F(f) = \{ z \in C : \forall ~ n\in \mathbb N_+ ~,~ f^n(z) \in C\}$
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\(\mathbb F(f) = \{ z \in C : \forall ~ n\in \mathbb N_+ ~,~ f^n(z) \in C\}\)
   
 
in such a way that
 
in such a way that
   
$\mathbb J(f) \cup \mathbb F(f)=C$
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\(\mathbb J(f) \cup \mathbb F(f)=C\)
   
At least for positive integer $n$, the following relations hold:
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At least for positive integer \(n\), the following relations hold:
   
 
:<math>f^n(\mathbb J(f)) = \mathbb J(f)</math>
 
:<math>f^n(\mathbb J(f)) = \mathbb J(f)</math>
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[[Science]] about properties of these fractals, and, in particular, those of the [[Julia set]]s and the [[Fatou set]]s, is called [[Complex dynamics]].
 
[[Science]] about properties of these fractals, and, in particular, those of the [[Julia set]]s and the [[Fatou set]]s, is called [[Complex dynamics]].
   
==Non–integer iterates==
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==Non-integer iterates==
The definition of the [[Julia set]] above implies that the function $f$ is iterated
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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,
 
integer number of times. In principle, similar set can be considered, assuming,
that the number of [[iterate]] $n$ can take also non–integer values.
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that the number of [[iterate]] \(n\) can take also non-integer values.
   
==Generalization and the Non–holomorphic transforms==
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==Generalization and the non–holomorphic transforms==
The requirements for the iterated function $F$ may be not so strict, as it is assumed above.
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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.
 
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 $n$.
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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.
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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
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Number \(c\) is declared to belong to the set \(\mathbb J\), if
$\displaystyle
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\(\displaystyle
\lim_{n \rightarrow \infty}F_c^n(z_0)$ exists and is smooth function of $c$.
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\lim_{n \rightarrow \infty}F_c^n(z_0)\) exists and is smooth function of \(c\).
   
 
In such a way, there exist many various definitions and meanings of term [[Julia set]].
 
In such a way, there exist many various definitions and meanings of term [[Julia set]].
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==Keywords==
 
==Keywords==
[[Fractal]] , [[Superfunction]] , [[Iteration]]
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[[Fractal]],
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[[Iteration]],
  +
[[Superfunction]]
  +
  +
[[Category:English]]
  +
[[Category:Fractal]]
  +
[[Category:Iteration]]
  +
[[Category:Superfunction]]

Latest revision as of 18:44, 30 July 2019

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 [1]. 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 </ref> 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. </ref>.

Fatou set

The complementary set to Julia set is called Fatou set [2]; 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.

References

  1. http://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00432-4/ Walter Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151-188.
  2. http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1920__48_/BSMF_1920__48__208_1/BSMF_1920__48__208_1.pdf P.Fatou. Sur les equations fonctionelles. Bulletin de la S.M.F., tome 48 (1920), p.208-314.

http://en.wikipedia.org/wiki/Julia_set

Keywords

Fractal, Iteration, Superfunction