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.