Self-similarity of a dendrite fractal The Julia set of the map $z \mapsto z^2+i$ is a dendrite fractal.  I would like to know which affine maps (other than identity) map this region to a subset of itself.  I imagine there are two three generators, but maybe there are more.  Perhaps I am after an "iterated function system" which will generate the dendrite fractal (possibly as the limit of trees).
 (source: Wikipedia)
 A: What you are after is really the combinatorics of the branched cover $z\mapsto z^2+i$ on the Julia set $J$. Namely, you really want to consider $J$ as a dynamical system, and not just as a topological space.
As mentioned above, the map $z\mapsto z^2+i$ is not linear, but you could approximate it by something piecewise linear if you want: you would then get an equivalent dynamical system on the corresponding Julia set.
As a mere topological space, the set of homeomorphisms of $J$ is HUGE.
The space $J$ is actually uniquely characterized up to homeomorphism by the following properties:


*

*it is compact metrizable.

*it is one dimensional.

*it is locally connected and simply connected.

*all its points have valence 1, 2, or 3
(here, the valence is the number of connected components that you get after removing the point)

*the set of points of valence 3 is dense.


This is a little bit similar to the characterization of the Cantor set as the unique compact metrizable zero-dimensional space with no isolated points.
So you see that you have a lot of freedom, and that you can represent many, many dynamical systems on that same topological space $J$.
A: OK, as hinted in my comment.  Here is the fractal $J$:  

Now choose a branch of the squareroot so that $\sqrt{w-i}$ is continuous on this set.  Here is the image of $J$ under the map  $\sqrt{w-i}$ in green, and the image of $J$ under the map $-\sqrt{w-i}$ in red:
 
Thus $J$ is the attractor of a certain IFS (but not using affine maps).
