New notion of measure-theoretic entropy for multivalued maps satisfying variational principle

The main goal of this paper is to formulate a full variational principle for measure-theoretic entropy for multivalued upper semicontinuous maps in a compact metric space. To this end, we introduce a new variant of measure-theoretic entropy, called \emph{natural extension entropy}, which is consistent with the standard one in the single-valued case and satisfies an analogue of the classical variational principle formulated by Dinaburg and Goodman in 1971. In our principle, the key role is played by the topological entropy for multivalued maps invented by Gromov in 1977 and rediscovered 40 years later by Kelly and Tennant. Several natural properties of this new concept are established here. As far as we know, there is no such result dealing with a full variational principle for general upper semicontinuous maps. The only attempts made so far concern only the half-variational (only one inequality) principles for iterated function systems generated by single-valued continuous maps.