The dynamics of surface diffeomorphisms

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

Proposal Number: DMS-0203975 PI: Andre de Carvalho ABSTRACT The study of 2-dimensional dynamical systems is an active area of research. Families such as the Henon and Lozi families have been intensively studied in recent years and many beautiful results have been obtained. However, we are still far from having a complete understanding of them. In dimension 1, due to the efforts of many researchers during the past two decades, there is an almost complete theory. Parallels and differences between the dynamics of families in dimensions 1 and 2 have been exploited by several authors. The past work of the PI includes the development of a 2-dimensional analogue of Milnor and Thurston's kneading theory and the study of braids and braid forcing. The present project proposes to further the study of dynamics in dimension 2 as well as some aspects of braid theory and Teichmuller theory. The main topics to be studied are: Horseshoe braids, Generalized pseudo-Anosov maps and piecewise linear models for diffeomorphisms in dimension 2, Real and Complex Henon maps and the Pruning Front Conjecture, Renormalization in dimension 2, and some aspects of infinite-dimensional Teichmuller Theory. Dynamical systems is the branch of mathematics that describes systems that evolve in time. Typical examples are planetary motion, population patterns, weather prediction or the economy. Since Newton, it is believed that simple mathematical laws are capable of describing the long term behavior of many of such systems. The problem is that often such simple mathematical laws lead to extremely complicated long time behavior.