In this talk, we introduce definitions of Markov set-valued functions on finite graphs and the same pattern between two Markov set-valued functions. These functions are defined using the framework of cell complexes. They allow for infinite Markov partitions and have graphs that may contain $2$-cells. We also show that two generalized inverse limits with bonding functions that are Markov set-valued functions following the same pattern are homeomorphic.

This is joint work with E. Matsuhashi and Y. Oshima.