Henon-Devaney like maps

We prove a general theorem characterizing the transitivity of homeomorphisms with singularities in the plane. We provide examples where this theorem applies including the classical Henon-Devaney map (1981 Commun. Math. Phys. 80 465-476). We also prove some results about the plane homeomorphisms satisfying the hypothesis of our theorem namely the Henon-Devaney like maps. Indeed, we show that the maximal invariant set of a Henon-Devaney like map is topologically conjugated to a shift map. Furthermore, every Henon-Devaney like map is fixed point free with dense periodic orbits. This generalizes some constructions by Devaney (1981 Commun. Math. Phys. 80 465-476) and Lenarduzzi (2015 Discrete Continuous Dyn. Syst. 35 1163-1177).