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I think the proper way to think about these fractals is to see them as defined on the sphere, via the stereographic projection. See this blog post for more about how to see these fractals as living on the sphere:
http://algorithmic-worlds.net/blog/blog.php?Post=20120316
We are then studying the iterations of either conformal (for dense Julia sets) or almost conformal (for Ducks-type fractals) maps of the sphere to itself. By "almost conformal", I mean a map that fails to be conformal on a subspace of dimension 1 on the sphere. This is the case for all the implementations of Ducks-type fractals I know of. The subspaces of dimension 1 are then the axes of the mirrors symmetries. I am not sure if the Ansatz you propose can really account for all these almost conformal maps.

What happens to the orbits is the following. Basically there are two types of orbits. Either a neighborhood of the point is mapped smoothly along the orbit, the point is said to be "regular". Else, the map is chaotic in a neighborhood of the point and the point is "irregular". Irregular points belong to the Julia set, yielding interesting visual structures, while regular points form Fatou domains, that appear visually smooth. For some maps, it happens that the Julia set fills the whole sphere, i.e. all the orbits are chaotic. This is what yields dense fractals. You can check explicitly by plotting the orbits that they do become visually chaotic precisely when the pattern fills the plane.

Iterations of conformal maps have been studied extensively in the math literature, see for example this nice review if you have some background in mathematics:
http://arxiv.org/abs/math.DS/9201272
On the other hand, I haven't been able to find a single reference on the "almost conformal" maps that are responsible for Ducks/Kaliset fractals. This is something much less natural to study from the point of view of mathematicians.

I hope this helps,

Sam

The most fascinating and appealing I found "entangled trees". This pattern should be something mathematical as it appears from time to time with different formulas, even not a kaliset formulas. Some mix of parabolas or hyperbolas in complex plane???



Havent read jet, I will download and read at home.

In points where 4 tree branches meets there are some kind of spirals. Lines looks more like hyperbolic function, and not the curved second degree lines of mandelbrot, and in formula z=abs(z)/c+c there are just power 1. 
So these probably must be made by hyperbolic function.

I don't understand where you're going.
The fractal, if we stop at a finite iteration, is mathematically a finite progression of complex numbers from a starting complex... What we "see" is up to the coloring method...
It's kinda hard to define what we are talking about when we say "this is like an hyperbola".
Saying "it is an hyperbola" is weird because the coloring method is continuous. WHAt mathematically IS the hyperbola you're talking about ?
But i'm not saying you're wrong! You're maybe on something .

Anyway, when I'll have the time I'll post some pictures related to the differentiability of the the complex function associated with the last iteration in Julia mode. I think i'm on something interesting.

fractus