Expansiveness, Lyapunov exponents and entropy for set valued maps

In this paper we introduce a notion of expansiveness for a set valued map defined on a topological space different from that given by Richard Williams at \cite{Wi, Wi2} and prove that the topological entropy of an expansive set valued map defined on a Peano space of positive dimension is greater than zero. We define Lyapunov exponent for set valued maps and prove that positiveness of its Lyapunov exponent implies positiveness for the topological entropy. Finally we introduce the definition of (Lyapunov) stable points for set valued maps and prove a dichotomy for the set of stable points for set valued maps defined on Peano spaces: either it is empty or the whole space.