Positively measure-expansive differentiable maps

Let C1(M)C1(M) be the space of differentiable maps of a closed C∞C∞ manifold M   endowed with the C1C1-topology, and let f∈C1(M)f∈C1(M). The purpose of this paper is to characterize the dynamics of positively measure-expansive differentiable maps from the measure theoretical view point. We show that (i) f   is in the C1C1-interior of the set of differentiable maps that is positively μ-expansive for any non-atomic Borel probability measure μ if and only if f   is expanding, and (ii) C1C1-generically, f is positively μ-expansive for any non-atomic Borel probability measure μ if and only if f is expanding.