Continuum-wise expansiveness and specification for set-valued functions and topological entropy

Abstract: We define the concept of continuum wise expansive for set-valued functions and prove that if a compact metric space admit a set-valued $cw$-expansive function then the topological entropy of $X$ is positive.} We also introduce the notion of pointwise specification property for set-valued functions and prove that set-valued functions with this property has positive entropy.