Smooth Dynamical Systems

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This research is primarily concerned with the dynamics of smooth diffeomorphisms of low- dimensional manifolds. Relevant questions deal with the existence of strange attractors (sometimes called "ergodic attractors"), the abundance of homoclinic tangencies, and the existence of symbolic extensions. By definition, an ergodic attractor is the support of a nonatomic ergodic SRB measure. Due to the work of Benedicks, Carleson, Wang, Young, Mora, and Viana it is known that there are smooth parametrized families of area-decreasing diffeomorphisms for which ergodic attractors exist for positive measure sets of parameters. A natural question is whether "positive measure" can be replaced with "open." Previous work of the principal investigator has shown that, in the presence of homoclinic tangencies, there are NO open sets of area-decreasing planar diffeomorphisms with ergodic attractors. Part of the present research investigates whether this lack of ergodic attractors is a "generic" property. In addition, attempts will be made to prove the existence of ergodic attractors in systems that are area-preserving in some parts of the phase space--thus removing the current dependence on sharp contraction of area in all known techniques. Other problems in the current project focus on the existence of symbolic extensions for maps with varying levels of smoothness. This research involves the dynamics of low-dimensional mappings. It is intimately related to understanding the solutions (or orbits) of nonlinear differential equations. Such equations arise in many scientific disciplines. To cite particular examples, they provide models for systems in biology such as circadian rhythms, blood flow, and the electrodynamics of nerve impulses; in astronomy in connection with the design of orbiting satellites; and even in weather prediction. In some specific models related to the foregoing general systems, it is known that the relevant equations cannot be solved explicitly. Instead, in the study of these systems one must rely on certain geometric information that is made available through numerical exploration with computers. This project attempts to provide a mathematically rigorous treatment of important properties related to equations that arise in such situations. The knowledge of these properties helps to provide tools that the scientist or engineer can use to describe, test, and refine models for a variety of things that occur in nature. As such, the present research has the potential to provide foundations for substantial scientific advancement.