We discuss a class of upper semi-continuous set-valued functions called Markov set-valued functions. Such set-valued functions have a corresponding symbolic dynamical system. Through examples, partial results, and open questions we explore the relationship between the topology of the set-valued function’s inverse limit and the topological entropy of the symbolic system. In particular, we establish some conditions where positive topological entropy is equivalent to the inverse limit containing a 1/2-indecomposable subcontinuum.