Markov set-valued functions and their inverse limits

In this paper we introduce the definition of a Markov set-valued function and show that the inverse limits of two similar Markov set-valued functions are homeomorphic. This generalizes results of S. Holte, I. Banič, M. Črepnjak, and T. Lunder. The definition we present differs from previous definitions of Markov interval functions in that we allow for points outside of the Markov partition to have non-degenerate images. Additionally, our definition focuses on the structure of the inverse of our function; we require that the inverse is a union of continuous mappings with specified restrictions on the domains, ranges, and points of intersection.