Topological dynamics of Markov multi-maps of the interval

We study Markov multi-maps of the interval from point view topological dynamics. Specifically, we define forward trajectory system a multi-map, and investigate whether it has properties such as transitivity, mixing, density periodic points, specification. To each multi-map associate shift finite type (SFT), our main results relate SFT with those system. Under general coding condition, establish necessary sufficient conditions for transitivity mixing These complement existing work showing relationship between entropy its associated SFT. also characterize when inverse limit systems to have mentioned above.