Finitary Analysis in Homogeneous Dynamics and Applications

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Dynamical systems study rules which predict the long-term behavior of a point in a space. The origin of dynamical systems may be traced back to Newtonian mechanics. Examples of mathematical models of dynamical systems include the study of fluid dynamics, airflow dynamics, and many others. Consider, for instance, gas particles moving in a space according to a certain rule. One is interested in understanding various states of the system over long-time intervals. A curious phenomenon is the following: if one assumes that initially all gas particles are in one half of the space, then infinitely often in the future, the gas molecules will collect in the initial portion of the space. This conclusion, which is predicted by mathematical aspects of the system, seems paradoxical. However, the mystery may be revealed if one observes that the number of the degree of freedom in this system is very large, thus in order to return to the original state, one needs to observe the system for very long (practically impossible) time intervals. This proposal aims at the study of finitary aspects of dynamical systems where one is interested in approximating various states of the system within an error. This project also include the training of a graduate student.The Principal Investigator seeks extensions and strengthening of certain rigidity results in dynamics and their applications in number theory and geometry. Special attention will be given to finitary and effective aspects of the analysis. The following will be the main objectives: (i) Dynamical systems have become a major player in modern mathematics. However, arguments relying on techniques from ergodic theory and dynamics are often non-quantitative. Providing finitary versions of these arguments are challenging and much sought after, especially in view of various applications. The principal investigator seeks results in this direction with two main goals in mind: provide finitary arguments which yield polynomial rates for strong rigidity results in homogeneous dynamics in some specific examples with interesting applications to number theory and geometry; provide finitary versions of these celebrated results in great generality, even though one may not obtain a polynomial rate in general. (ii) Geodesic planes in hyperbolic 3-manifolds have been studied from different angles. These investigations have made it clear that the behavior of geodesic planes is intimately related to the geometric, topological, and arithmetic properties of the ambient manifold. This proposal seeks to refine and strengthen results in this direction using tools from homogeneous dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.