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Im looking for simple examples of entire functions that have "fractal properties".

With "fractal properties" I mean that $|f(z)| < 1$ has a "fractal structure" in the complex plane.

With "fractal structure" I mean a nonperiodic fractal such as Mandelbrot.

So when we zoom out on $f(z)$ we begin to see fractal structure for $|f(z)| < 1$. This structure goes on forever.

This of course excludes polynomials.

How to achieve this ?

Edit : After some comments it appears that with fractal structure I mean more in the sense of " copies " and that was not clear enough. So I add it.

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There is a holomorphic bijection between the exterior of the Mandelbrot set and the exterior of the unit disc. But I think this is not enough for you? –  Hagen von Eitzen Oct 4 '12 at 15:09
    
If you really want $f$ to be entire, $\{z:|f(z)|=1\}$ will be far too nice a set to be anything I would call a fractal (or the boundary of a fractal). Think about the inverse function theorem... –  Micah Oct 4 '12 at 15:17
    
@ Micah : I do not know why you talk about the inverse function theorem. I do not see it as relevant ? –  mick Oct 4 '12 at 15:37
    
@HagenvonEitzen : Im not sure if you misunderstood or pretend that :). But anyway I did an edit and I think it is more clearly now. –  mick Oct 4 '12 at 15:38
    
What do you mean by "copies"? The reason why the inverse function theorem is relevant is that it means the boundary of your set must be a smooth curve (except for some the discrete set of points where $f'=0$), which means it cannot be fractal unless you are using a very strange definition of "fractal" (certainly it can look nothing like the Mandelbrot set!) –  Micah Oct 4 '12 at 15:57
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