On the dynamics of functional envelopes of multi-valued maps

In this paper, we study the dynamics of multi-valued maps and their functional envelopes. We demonstrate that the functional envelopes of multi-valued maps are invariant under topological conjugacy and prove that the concepts of equicontinuity, distality and uniform rigidity for multi-valued maps are equivalent to the corresponding concepts for their functional envelopes. We deal with the shadowing property, expansive properties and minimality for multi-valued maps and their functional envelopes. Furthermore, we also show that several dynamical properties of the functional envelopes of multi-valued maps depend only on their ranges.