Expansive homeomorphisms on non-Hausdorff spaces

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 and dynamical properties are studied in relation to separability axioms, metrizability, uniform expansivity, asymptotic points and positive expansivity. We also prove that every refinement expansive homeomorphism on a T1T1 space, in particular every expansive homeomorphism in the usual sense, is conjugate to a subshift with a possibly non-Hausdorff topology on the symbols.