This project will investigate three aspects of complex dynamics in higher dimension: (1) We consider real, birational mappings of the plane, and we will show how complex-analytic and algebraic-geometric ideas can be brought to bear on the dynamics of these maps. (2) We study an "inversion" mapping on the space of matrices. We will determine the degree complexities of this map as it acts on various subspaces of matrices. (3) We will work with the complex Henon family, which has served as an important model family to exhibit complicated dynamical behaviors and has been important because many observed phenomena can actually be proved in the complex case. We will investigate certain parabolic bifurcation phenomena within this family.
This work involves the areas of Dynamical Systems and Complex Analysis. More specifically, the geometric shape of the dynamical set is connected with the dynamical behavior that takes place on the set, which is an interesting object of study within Complex Analysis. The mapping studied in (2) arises from certain fundamental symmetries in Lattice Statistical Mechanics. And part of the study of the complex Henon family in (3) should impact a "complex" approach to Quantum Mechanical Tunneling. Thus the work in (2) and (3) will have impact on Mathematical Physics.
