Abstract

The N-expansive systems have been recently studied in the literature [6], [7], [9], [14]. Here we characterize them as those homeomorphisms for which every Borel probability measure is N-expansive. In particular, the strongly measure expansive homeomorphisms in the sense of [8] are precisely the homeomorphisms for which every invariant measure is 1-expansive. We also characterize the 1-expansive measures for equicontinuous homeomorphisms as the convex sum of finitely many Dirac measures supported on isolated points. In particular, such measures do not exist on metric spaces without isolated points. Furthermore, we consider N-expansive measure for flows and prove that a flow is N-expansive in the sense of [9] if and only if every Borel probability measure is N-expansive. Finally, we obtain a lower bound of the topological entropy of the N-expansive flows as the exponential growth rate of the number of periodic orbits. (C) 2019 Elsevier Inc. All rights reserved.