We exploit the techniques developed in [Le] to study N -expansive homeomorphisms on surfaces. We prove that when f is a 2-expansive homeomorphism defined on a compact boundaryless surface M with nonwandering set Ω(f ) being the whole of M then f is expansive. This condition on the nonwandering set cannot be relaxed: we present an example of a 2-expansive homeomorphisms on a surface with genus 2 with wandering points that is not expansive. * M. J.

In this note we prove that a homeomorphism is countably-expansive if and only if it is measure-expansive. This result is applied for showing that the C 1 -interior of the sets of expansive, measure-expansive and continuum-wise expansive C 1 -diffeomorphisms coincide.

We study expansive dynamical systems from the viewpoint of general topology. We introduce the notions of orbit and refinement expansivity on topological spaces extending expansivity in the compact metric setting. Examples are given on non-Hausdorff compact spaces. Topological properties are studied in relation to separability axioms, metrizability and uniform expansivity.

We show that every positive expansive flow on a compact metric space consists of a finite number of periodic orbits and fixed points. *

We prove that a flow on a compact surface is expansive if and only if the singularities are of saddle type and the union of their separatrices is dense. Moreover we show that such flows are obtained by surgery on the suspension of minimal interval exchange maps.Date: October 22, 2018.