Henon Renormalization

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

The Palis Conjecture describes the behavior of smooth dynamical systems. It asserts that the chaos observed in typical systems can be successfully described with probabilistic methods. Only for one-dimensional systems has this conjecture been proved. The long-term goal of this project is to prove the conjecture for the Henon family, which consists of two-dimensional systems. The essential part of a system is concentrated in its attractor. In particular, the probabilistic behavior of a system is closely related to the microscopic geometric properties of the attractor. Renormalization is a method to study this microscopic geometry. A central theme in dynamics is the exploration of renormalization beyond the theory of one-dimensional systems, where it has been instrumental in making progress. This project concentrates on the development of a renormalization theory for Henon maps. Renormalization has already been successful in improving the understanding of Henon maps that are at transition to chaos, dissipative maps at the accumulation of period-doubling-type. The immediate goals of the proposed research are the following: first, to extend the period-doubling renormalization theory to more general combinatorial types in the Henon family; second, to refine the theory for the dynamics of Henon maps at the accumulation of period doubling; and finally, to use the renormalization results to establish the Palis Conjecture at the accumulation of period doubling. Dynamics is the study of processes. This can mean processes generated by mechanical systems (e.g., the solar system), but also chemical, biological, or even sociological processes. An underlying idea is that the chaos observed in such systems can be understood qualitatively with the aid of a limited number of mechanisms. One goal of dynamics is to help engineers and scientists by explaining the mechanisms that are at play in their chaotic systems. Indeed, dynamics has led to many industrial applications. The processes dynamics studies are deterministic. Thus, if the positions and speeds of all the planets in the solar system were known now, dynamical theory could predict with great precision what those quantities will be next month. The future is, in principle, determined. Unfortunately, when a system is chaotic it becomes practically impossible to predict its future precisely. During the last fifty years it has become clear that, in order to describe deterministic chaos, science needs probabilistic methods. It has also become clear that chaos is in a quite real sense "very well organized." This organization is reflected in its probabilistic behavior. We all have seen the "bell curve" being used in many applications. It is always the same bell curve, an aspect of the organization within chaos. The relevant probabilistic laws are the phenomena observed in chaos. The explanation of these laws is intrinsically related to the microscopic properties of what scientists know as "attractors," which contain the essential aspects of the observed behavior of systems. Renormalization is a method to study this microscopic geometry. The organization of chaos becomes clearly visible exactly at this microscopic scale. Dynamics is very far from a complete understanding of chaos, but renormalization has been instrumental in the most sophisticated theories available at the present time. This project will explore further applications of renormalization. In particular, it will concentrate on systems that are related to the creation of chaos, the so-called Henon systems.