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In this paper we define and study the topological entropy of a setvalued dynamical system. Actually, we obtain two entropies based on separated and spanning sets. Some properties of these entropies resembling the singlevalued case will be obtained.

We prove that every sectional-hyperbolic Lyapunov stable set contains a nontrivial homoclinic class.Comment: 5 page

We propose a definition of topological stability for set-valued maps. We prove that a single-valued map which is topologically stable in the setvalued sense is topologically stable in the classical sense [14]. Next, we prove that every upper semicontinuous closed-valued map which is positively expansive [15] and satisfies the positive pseudo-orbit tracing property [9] is topologically stable. Finally, we prove that every topologically stable set-valued map of a compact metric space has the positive pseudo-orbit tracing property and the periodic points are dense in the nonwandering set. These results extend the classical single-valued ones in [1] and [14].

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