Invariant measures and symbolic coding for dynamical systems with hyperbolic behavior

The Principal Investigator proposes three directions of continued and future research. The first project will study symbolic coding procedures for flows on homogeneous spaces. Based on previous joint work with S. Katok, the PI intends to continue the study of symbolic coding for noncompact surfaces of constant negative curvature with finite volume and their connections with continued fractions algorithms. Another objective is the existence of a feasible symbolic coding (multidimensional shift space of finite type) for dynamical systems obtained from higher rank abelian actions of Lie groups with the goal of obtaining a better understanding of the possible invariant measures for such actions. The novelty of this approach is the fact that the invariant measures will be obtained from the symbolic representation of the corresponding action. In recent years, questions from number theory, statistical mechanics and computer science have been solved using the above mathematical tools and results. A second direction of research is in the area of non-uniformly hyperbolic dynamical systems (or systems with nonzero Lyapunov exponents). The PI plans to study the existence of invariant measures of maximal entropy and to establish precise asymptotic estimates for the number of periodic orbits for such systems with additional dynamical properties (like invariant cone families). A third direction of research is on applications of the theory of chaotic dynamical systems to population models. In previous joint work with H. Weiss, complicated (chaotic) behaviors and several bifurcation routes have been documented for a class of density dependent population models with fertility rates depending exponentially or polynomially on the total population. The goal of the current project is to prove rigorously the existence of chaotic attractors and physical measures for such systems. A successful outcome of the first project will find important applications to number theory and statistical mechanics, while the second project will deepen the theoretical understanding of systems with nonzero Lyapunov exponents, and will be beneficial to those scientists that try to explain the existence of complicated behavior in physical models. The outcome of the third project will have an impact on population biologists and demographers, since it will rigorously demonstrate that chaotic behavior is a certainty in several classes of population models. The results will help developing better suitable mathematical models for population projections. The proposed research topics will also contribute to the mathematical education of undergraduate and Master's students at DePaul University, in part by lectures by the investigator, and in part by students working on some of these topics and related subjects.