Generalized Hénon maps: the cubic diffeomorphisms of the plane

In general a polynomial diffeomorphism of the plane can be transformed into a composition of generalized Hénon maps. These maps exhibit some of the familiar properties of the quadratic Hénon map, including a bounded set of bounded orbits and an anti-integrable limit. We investigate in particular the cubic, area-preserving case, which reduces to two, two-parameter families of maps. The bifurcations of low period orbits of these maps are discussed in detail.