Given an Iterated System of Functions, say $f_1,...,f_n$, with associated transition probabilities $\sigma_{ij}$ for each $f_{i}$ to each $f_{j}$, how do we go about finding an estimate to the fractal dimension?
Motivation: For other methods of generating fractals, like recursive maps where we can define the n-th value $X_n$ in terms of the value $X_{n-1}$, we have many different methods to estimate the fractal dimension.For example, using the box counting technique on different values of N, and taking that limit as it approaches infinity. With techniques like these, we can also sometimes yield exact values of the fractal dimension.
My attempt: I've tried modelling the transitions of the IFS as a Markov Chain and finding the orbit of each function. If the orbit is finite, I'm thinking there might be a way to use its encoding to generate a number which could be used in the estimate of the dimension of the resulting fractal.
I am not sure if this is the right direction to take and if it is, how to proceed from there.