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 $J$.