We investigate the dynamics of plane polynomial diffeomorphisms. These mappings define diffeomorphisms of the real plane as well as the complex plane. This project will investigate the dynamics of the complex mappings using the methods of complex analysis and complex geometry. An important part of our research will be to understand when and to what extent the complex results can be applied to the underlying real system. Our main focus will be on the Henon family of quadratic mappings of the plane. And within this family we study the horseshoe mappings. Among other things, we will describe the shape of the boundary of the horseshoe locus, and the behavior of the mappings on the boundary. We will also apply our methods to describe the dynamics of certain bifurcations.
One motivation is that an understanding of the polynomial diffeomorphisms of the plane should be a useful guide to the investigation of the dynamics of general diffeomorphisms of the plane. It should provide examples of mappings and bifurcations that can be understood in detail and provide the examples that serve as guideposts for the development of the theory for more general mappings. Closely related to this is the study of birational mappings of the plane. In particular, we will analyze birational maps that arise in renormalization questions in lattice statistical mechanics.
