Difference between revisions of "Julia set"
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The definition of the [[Julia set]] above implies that the function $f$ is iterated |
The definition of the [[Julia set]] above implies that the function $f$ is iterated |
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integer number of times. In principle, similar set can be considered, assuming, |
integer number of times. In principle, similar set can be considered, assuming, |
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| − | that the number of iterate $n$ can take also non–integer values. |
+ | that the number of [[iterate]] $n$ can take also non–integer values. |
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| − | In general, analysis the [[Julia set]]s |
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==References== |
==References== |
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Revision as of 12:58, 10 July 2013
Julia set is set of values inside of the range of holomorphism of some function, that fall out from the range of holomorphism of some integer iteration this function. 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. On this case, the number of iteration is supposed to be integer.
The complementary set to Julia set is called Fateu 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
$\mathbb J(f) \cup \mathbb F(f)=C$
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 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.
References
http://en.wikipedia.org/wiki/Julia_set
