We know that the Mandelbrot fractal contains a countable number of copies of itself.
See :
Does the Mandelbrot fractal contain countably or uncountably many copies of itself?
Where that is explained.
Notice that polynomials have a finite amount of zero's and entire functions have a countable amount of zero's.
So I started to wonder :
Is there a Julia fractal that contains uncountable many copies of itself ?
And if so, can they be iterations of entire functions ?
What are typical examples ?