This mathematics research project in the area of complex dynamics in higher dimension has three components. The first component investigates the properties of birational mappings in dimension 2 and higher. Bedford and his collaborators will use tools from complex analysis and algebraic geometry to analyze fundamental properties of these mappings. In particular, they will investigate how to determine the dynamical degrees in all dimensions. The second component of this project investigates the existence of automorphisms of positive entropy in complex surfaces and 3-folds. The third component is focused on the complex Henon family of dynamical systems, which has served as an important model family. These dynamical systems are represented by quite simple polynomials, but they exhibit complicated dynamical behaviors. They have been important because many observed phenomena can actually be proved mathematically.
This mathematics research project is related to the field of study of chaos theory, which has applications to several disciplines, including physics, engineering, economics and biology. An important phenomenon in chaos theory is the so-called butterfly effect, which illustrates the unstable dependence on initial conditions for a given dynamical system. The Henon functions studied in this project have been a fertile testing ground for transformations that have simple formulas but which exhibit the butterfly effect. The study of birational maps has applications to the field of Lattice Statistical Mechanics (LSM) which considers materials that have a lattice structure like crystals. LSM provides a formula for determining the energy and temperature of such a material in terms of the various interaction energies between the atoms in the lattice. This formula is very difficult to compute, but in many cases there are fundamental symmetries which lead to substantial simplifications. These symmetries are given by birational maps such as those studied in this project, and the amount of simplification is given by the degree complexity.
