As previously, the system has an attractor only when every
| ri | are smaller than 1.
However, in this case, the rotations associated to each function allows the system to take its value in the whole complex plane and so the system have now two degrees of freedom. The system has thus two dimensions and now, the attractor can be a fractal whose dimension is between 0 and 2.
It is also important to remark that the fractal dimension
of the attractor is independent of the angle of rotation associated to
each function and is still independent of the pi. The shape
of the attractor depends of course of these angles but not the final dimension
of the system.
The pdf of the attractor is also dependent of the angles of rotations and has still a multi-fractal stucture.
This notion which is not obvious can simply be understood as the dimension of the intersection of the plane of a given probability and the pdf.
As it is not easy to tell more in few (and understandable...) words, I prefer to give several exemples of attractors that I obtained during my experiments.