Real and Complex Dynamical Systems

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We investigate the dynamics of plane polynomial diffeomorphisms.These mappings define diffeomorphisms of the real plane as well asthe complex plane. This project will investigate the dynamics ofthe complex mappings using the methods of complex analysis andcomplex geometry. An important part of our research will be tounderstand when and to what extent the complex results can beapplied to the underlying real system. Our main focus will be onthe Henon family of quadratic mappings of the plane. And withinthis family we study the horseshoe mappings. Among other things, wewill describe the shape of the boundary of the horseshoe locus, and the behavior of the mappings on the boundary. Wewill also apply our methods to describe the dynamics of certain bifurcations.One motivation is that an understanding of the polynomialdiffeomorphisms of the plane should be a useful guide to theinvestigation of the dynamics of general diffeomorphisms of theplane. It should provide examples of mappings and bifurcations thatcan be understood in detail and provide the examples thatserve as guideposts for the development of the theory for moregeneral mappings. Closely related to this is the study of birationalmappings of the plane. In particular, we will analyze birationalmaps that arise in renormalization questions in lattice statisticalmechanics.