RUI: Invariant geometric structures and rigidity in hyperbolic dynamics.

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project lies in the area of hyperbolic dynamical systems. These are smooth systems characterized by exponential expansion in some directions and exponential contraction in the other directions. Stability, rigidity, and classification of hyperbolic systems are among the central issues in smooth dynamics. These problems have been extensively studied, and many of them were solved for systems with one-dimensional invariant foliations. The situation is much more complicated for systems with higher-dimensional foliations. The main part of the proposed research is to study higher-dimensional hyperbolic systems. The principal investigator is particularly interested in properties related to local rigidity (i.e., regularity of conjugacy under a small perturbation) and global rigidity (i.e., existence of a smooth conjugacy to an algebraic model). Investigating properties of invariant foliations and geometric structures plays an important part in the proposed research.The field of dynamical systems is a modern branch of mathematics that grew out of the study of physical and mechanical problems. Its main objective is to describe how various abstract and real-life systems evolve over time. For this reason the theory of dynamical systems has a very wide range of applicability, from celestial mechanics and hydrodynamics to meteorology and social sciences. So-called hyperbolic systems have been one of the main objects of study in what is known as "smooth" dynamics. The exponential contraction and expansion in these systems produce chaotic behavior with complex orbit structure. This results in a rich theory with applications in various areas of applied and pure mathematics, including geometry and number theory. The principal investigator works in a primarily undergraduate-serving institution that is the main university in the southern Alabama region. The project will positively affect the department?s undergraduate mathematics major and its Master's program, as well as the broader student population and the surrounding community.