On Planar Piecewise and Two-Torus parabolic Maps

We investigate properties and examples of iterated planar piecewise parabolic (PWP) maps, including those induced by two-torus parabolic maps and their inverses. PWP maps are areapreserving maps that have constant linearizations but only one eigenvector with eigenvalue one. We obtain necessary and sufficient conditions for a two-torus parabolic map to be invertible. For noninvertible PWP maps, there are a number of questions about the existence and structure of attractors for such maps. We introduce the simplest example of a nontrivial PWP map, defined by a parabolic linear map with a translation that is a different constant on each side of a given line in the plane. For this “half plane parabolic map” we obtain sufficient conditions for a bounded attractor to exist, discuss their dynamical properties and give some examples of these and related maps.