A Dichotomy in Area-Preserving Reversible Maps

Abstract

In this paper we studyR-reversible area-preserving mapsf : M! M on a two-dimensional Riemannian closed manifold M, i.e. dieomorphisms f such that R f = f 1 R where R: M! M is an isometric involution. We obtain a C 1 -residual subset where any map inside it is Anosov or else has a dense set of elliptic periodic orbits. As a consequence we obtain the proof of the stability conjecture for this class of maps. Along the paper we also derive the C 1 -closing lemma for reversible maps and other perturbation toolboxes.