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Julia set (Definition)

Let $ U$ be an open subset of the complex plane and let $ f \colon U \to U$ be analytic. Denote the $ n$-th iterate of $ f$ by $ f^n$, i.e. $ f^1 = f$ and $ f^{n+1} = f \circ f^n$. Then the Julia set of $ f$ is the subset $ J$ of $ U$ characterized by the following property: if $ z \in J$ then the restriction of $ \{f^n \mid n \in \mathbb{N}\}$ to any neighborhood of $ z$ is not a normal family.

It can also be shown that the Julia set of $ f$ is the closure of the set of repelling periodic points of $ f$. (Repelling periodic point means that, for some $ n$, we have $ f^n (z) = z$ and $ \vert f'(z)\vert > 1$.)

A simple example is afforded by the map $ f(z) = z^2$; in this case, the Julia set is the unit circle. In general, however, things are much more complicated and the Julia set is a fractal.

From the definition, it follows that the Julia set is closed under $ f$ and its inverse -- $ f(J) = J$ and $ f^{-1} (J) = J$. Topologically, Julia sets are perfect and have empty interior.



"Julia set" is owned by rspuzio.
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quadratic Julia set (Definition) by PrimeFan
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Cross-references: interior, perfect, inverse, closed under, fractal, unit circle, map, simple, repelling periodic points, closure, normal family, neighborhood, restriction, property, subset, iterate, analytic, complex plane, open subset
There are 2 references to this entry.

This is version 4 of Julia set, born on 2007-06-14, modified 2007-06-15.
Object id is 9593, canonical name is JuliaSet.
Accessed 584 times total.

Classification:
AMS MSC: 28A80 (Measure and integration :: Classical measure theory :: Fractals)
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