Markov set-valued functions on compact metric spaces

We generalize the notion of Markov functions on closed intervals \([a,b]\) to Markov set-valued functions on compact metric spaces. We also introduce when two such Markov set-valued functions follow the same pattern and show that if the Markov set-valued functions \(F:X X\) and \(G:Y Y\) follow the same pattern, then the inverse limits \(lim_{{}}(X,F)\) and \(lim_{}(Y,G)\) are homeomorphic.