Some amazing images have been made of fractal sets on the complex plane, but I don't think I've ever seen one which uses hyperbolic space in a clever way. I'm not counting hyperbolic tessellations here because the Euclidean analogue is not a fractal at all - it's just a repeated tiling.
The hyperbolic plane is particularly interesting because it is in some sense "bigger" than the Euclidean plane - you can tile the hyperbolic plane with regular heptagons for example. Now, you could just take a fractal defined in the complex plane and map it to the hyperbolic plane somehow, but that doesn't take advantage of any of the interesting structure that the hyperbolic plane has. It's also locally flat, so doesn't add anything new. If you use some orbit function that is more natural in the hyperbolic plane, I think something much more interesting could result. I may have to play about with this a bit.
Similarly, one could also do fractals on the surface of a sphere (a positively curved space - the hyperbolic plane is negatively curved and the Euclidean plane has zero curvature).
Hi my name is Cris, I study mathematics and I was just looking on the internet if there was some information about fractals on hyperbolic space and actually I had the illusion to find no information about it, but now I can see that it is not a new idea.
Curiously, i had very similar ideas that you mention in you entry and i've had the same problem about what kind of orbit to use because, unlike the Mandelbrot set, which uses every complex number, in the model of the Upper half-plane is not just to use the same method of squaring the number and adding a constant to get the orbit, so, I've been wondering how to do it.
I don't know if nowadays you have done something about it or if it was just an idea you had like me. In any case it was interesting to read what you wrote.
Success!
One possibility would be to think about the complex number operations involved in computing the Mandelbrot set as geometrical operations, and then performing those operations in the hyperbolic plane instead of the Euclidean one. Scalings, rotations and translations all have hyperbolic analogues. I guess the problem then is, given a point in the hyperbolic plane, how do you find the square of it? You can scale it and rotate it but that's an operation that takes a point and a complex number and yields a point. What you really want is an operation that takes two points and yields a point. Perhaps you can get a complex number from a point by finding the angle and distance from the origin (distance being computed using whatever metric it is that makes all the heptagons in the heptagonal tiling of hyperbolic plane the same size). I don't know offhand if that would give you a multiplication with nice properties like being commutative, being not too difficult to compute and being something different from normal complex multiplication.