Let the graph of a function take a fractal form, such as the following representation of the Collatz conjecture topologically conjugated to the interval $[\frac12,1)\to[\frac12,1)$
In this example, the top and bottom of the map are equivalent - fine, we can think of the function acting on a hollow cylinder with no ends. Left and right are also equivalent - fine, we get a Torus.
And there is what I'm going to call a singularity at $\frac23$ where the function maps to $0$, which is off the graph. I mention the singularity, because one of the function's self-similarities which you can clearly see is that the interval $[\frac34,1)$ is equal to the interval one quarter its size and immediately to its left. That repeats and converges to the singularity.
Question
Are there theorems about functions represented by 2d fractals, such as e.g. Sierpinski's triangle, and under what circumstances they converge (when iterated)? In particular, I'm interested in relationships between the self-similarities of the fractal and convergence.
My Thoughts
Intuitively, I can see an obvious opportunity - mod/quotient out symmetries to find a structure which is well-founded, and show that function iteration descends the well-founded structure. Is this and/or other formulised somewhere so I don't have to reinvent the wheel?