Smooth Dynamics, Dimension Theory, Geodesic Flows, and Mathematical Biology

Invariant sets of dynamical systems are not generally self-similar in the strict sense. However, in work with others, the PI has shown that in some important cases, these sets can be decomposed into subsets each possessing a type of scaling symmetry. Sets which admit such structure are called multifractals (MF). This proposal involves the continuing investigation of the fine structure of these multifractals and an attempt to use the MF analysis to give new insights into dynamical systems and possibly yield a new (physical) classification of dynamical systems. In addition, the proposed work involves various problems in dimension theory which arise in mathematical biology as well as research on the relations between non-negative curvature and complicated dynamics of the geodesic flow. Regarding the latter, the (in)famous (xy)^2 Hamiltonian system, a model for a classical Yang-Mills field, which is orbit equivalent to a geodesic flow on a non-negative curved surface will be studied. Many physical and biological systems (including turbulent fluids, root systems of plants, stressed pieces of metal, NMR images of the brain, clouds, and galaxies in the universe) seem to possess some type of complicated fractal structure. Mathematically such objects are called multifractals. In previous work, the principal investigator presented a rigorous mathematical foundation for the study of some important classes of multifractals. The plan is to extend this work to larger classes of systems and to use this mathematical analysis to help understand the underlying physical or biological systems. The PI is particularly interested in applications to plant biology. In a different area, a large class of physical systems which are central to celestial mechanics and plasma physics can be studied by first transforming them to a ''geometric system'' called a geodesic flow and then studying the geodesic flow. In previous work, the PI showed that a large class of these flows, which some thought were easily understood and mathematically and physically boring, have extremely complicated behavior and are in fact chaotic. The investigations into some specific examples including an example from gauge field dynamics, which is one of the central theoretical problems in particle physics will be continued.