Effective Ergodic Theory: Parabolic and Hyperbolic

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The project is devoted to the study of long-term behavior of a class of dynamical systems, called parabolic, which display sub-exponential (polynomial) divergence of nearby orbits. In contrast with hyperbolic dynamical systems, which display exponential divergence of orbits, parabolic systems are much less understood. Hyperbolic systems will also be studied insofar they appear as auxiliary tools in the study of parabolic ones, which are the main focus. The research will emphasize quantitative aspects that are relevant for applications. The project aims to advance our fundamental knowledge of dynamical systems and to develop new ideas and new methods of mathematical investigation with potential applications to geometry and number theory and thus indirectly to other scientific subjects. Also, since parabolic behavior appears in Hamiltonian mechanics, the project can directly advance our understanding of simple mathematical models of classical physical systems (billiards in polygons, statistical mechanics, planetary motions). Research training and mentoring of students and postdocs is an important goal of the project, with particular attention to members of underrepresented groups.The investigator will carry out research in several directions (effective weak mixing of translation flows, Ruelle asymptotics, dynamics on character varieties) and will continue work on longstanding open questions such as on effective ergodicity for non-horospherical unipotent flows (effective Ratner theory), optimal bounds on Weyl sums for higher degree polynomials, the ergodic theory of geodesic flows on flat surfaces and of billiards in non-rational polygons, as well as the ergodic and spectral theory of smooth parabolic flows. An important approach to parabolic systems, often called renormalization approach, replaces the direct studies of the parabolic dynamics with that of an auxiliary hyperbolic system, which is easier to understand by methods of hyperbolic theory (invariant manifolds, Lyapunov exponents). Motivated by recent interest in the relation between parabolic and hyperbolic dynamics via renormalization, the investigator has broadened his research with work on exponential decay of correlations and Ruelle asymptotics for hyperbolic systems from the point of view of the effective (polynomial) equidistribution of unstable foliations, and will further pursue research on the interplay between effective equidistribution in parabolic dynamics and effective mixing in hyperbolic 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.