  We study Markov multi-maps of the interval from the point of view of
topological dynamics. Specifically, we investigate whether they have various
properties, including topological transitivity, topological mixing, dense
periodic points, and specification. To each Markov multi-map, we associate a
shift of finite type (SFT), and then our main results relate the properties of
the SFT with those of the Markov multi-map. These results complement existing
work showing a relationship between the topological entropy of a Markov
multi-map and its associated SFT. We also characterize when the inverse limit
systems associated to the Markov multi-maps have the properties mentioned
above.
