Dynamics in Two Complex Variables

Cornell University, Ithaca NY

Investigators

Abstract

PI: John Smillie, Cornell University DMS-0302357 Abstract The study of mathematical models that describe how systems change with time continues to provide fundamental mathematical challenges. When the models are nonlinear the long-term behavior, or dynamics, of these models can be chaotic and the dependence of the dynamics on the parameters of the model can be extraordinarily delicate. One classic approach to understanding the fundamental mechanisms that explain nonlinear dynamics is to look for generic behavior in the collection of all maps or diffeomorphisms; another approach is to look carefully at well chosen special families. Recent advances in one variable dynamics show that these approaches can be complementary. In particular the study of the complex quadratic family has led to new results about analytic and smooth unimodal families of real maps. The PI will study a particular family of two-dimensional dynamical systems: polynomial diffeomorphisms in two complex dimensions. These will be studied from many points of view. The PI will use ideas from the study of smooth diffeomorphisms of surfaces, methods from potential theory, methods from dynamics in one complex variable and computer tools. The hope is to provide a pathway for the migration of ideas from one-dimensional complex dynamics to higher dimensional real and complex dynamics. The understanding of mathematical models that describe how systems evolve with time continues to provide essential insights in many areas of science. Such models are used to describe the rhythms of the heart, the pulsing of lasers and the spread of disease. Despite the progress that has been made there are still fundamental problems remaining in the study of such mathematical models. Even seemingly simple models can present formidable mathematical difficulties. In the past 20 years important progress has been made in using ideas connected with fractal objects such as Julia sets and the Mandelbrot set in understanding the dynamics of systems with one degree of freedom. The aim of the proposed research is to bring some of these new methods and concepts to bear on the problem of understanding systems with two degrees of freedom. If these ideas prove fruitful there could be a positive effect on the field of dynamical systems. This could in turn have an influence on a range of scientific fields in which mathematical models of time evolution play a role.