Positively expansive dynamical systems

We introduce the notions of weakly and strongly positively expansive (wPE and sPE, respectively) discrete dynamical systems. Both are topological generalizations of the well-known metric notion of positive expansiveness (PE). We prove that the three notions are identical on compact metrizable spaces, but not on noncompact spaces. We investigate properties of PE, wPE, and sPE dynamical systems and show how they are related. Finally, we show that the possible dynamics for wPE and sPE homeomorphisms are severely limited, and we classify sPE homeomorphisms.