We will study quantitative equidistribution and other ergodic properties, such as weak mixing/mixing, for several examples of parabolic flows, in particular billiard flows in polygons and reparametrizations of nilpotent flows. In the past several years, we have developed a method to investigate the ergodic theoretical properties of parabolic flows based on the study of invariant distributions (distributional obstructions to the existence of solutions of cohomological equations), on the Gottschalk-Hedlund theorem and on the construction of a renormalization dynamics. We have succesfully applied our ideas in several papers where we have proved bounds on the speed of ergodicity in a few fundamental cases. We intend to test our method further in other more challenging cases, which so far have been out of reach mainly beacuse no renormalization scheme is available. The problems that we intend to attack include longstanding open questions such as the question on weak mixing on invariant surfaces for flows in rational polygons and the question on the speed of (unique) ergodicity for nilflows.
Our long term goal is to contribute to develop a theory on a class of weakly chaotic dynamical systems, called parabolic, which, despite some recent progress, are not yet sufficiently well understood. Parabolic motion is characterized by a power-law divergence (for instance linear, quadratic, etc.) of nearby trajectories with time. It represents an intermediate situation between strongly chaotic motion (exponentially fast divergence) and regular motion (no or extremely slow divergence). Motions at the extreme ends of the spectrum are comparatevely much better understood than parabolic motion. We will study specific questions on the dynamics of specific classes of examples, chosen for their fundamental nature and for their relevance in applications to physics, to geometry and to number theory. For instance, certain parabolic systems are relevant in the study of celestial mechanics, or as a testing ground for conjectures on the relation between classical and quantum mechanics (quantum chaos), other systems have deep connections to questions in number theory. Advances in our understanding of these systems will improve our fundamental knowledge of dynamical phenomena which are relevant for the natural sciences and for technological applications.
The research has focused on the study of certain dynamical systems, often called parabolic, which display intermediate chaotic properties and are difficult to understand within the framework of existing theories. Parabolic dynamical systems appear in applications to other mathematical fields, in particular number theory and geometry, and to mathematical models of physical systems coming from celestial mechanics, solid state physics and statistical mechanics. In our investigations we have studied simple idealized mathematical models (interval exchange transformations, horocycle flows, nilflows) of parabolic behavior, which are however complicated enough to lead to fundamental theoretical questions in geometry and number theory. We have advanced the long term goal of developing an effective theory of studying important properties of parabolic systems, such as the rate of equidistribution of orbits, the existence and nature of limit distributions of normalized time averages of observables and the presence and rate of mixing. Some of the questions we have explored are related to the theory of Riemann surfaces and their moduli, which is a classical central subject in mathematics. Other questions are closely related to fundamental, longstanding open questions in number theory. In all cases, our main goal has been to introduce new ideas which hopefully will lead to far reaching developments. The results of the research have been detailed in ten papers (7 by the PI and collaborators and 3 by the PI's students, see below), published or accepted for publications on top peer reviewed mathematics journals. They have also been presented regularly in invited talks at international conferences in the US and abroad. A very important part of the project has consisted in the training of graduate students. Three graduate students have obtained their Ph. D.'s with top level research on questions related to the project. The award has supported them in many ways: it has reduced their teaching load and supported them over the summer, allowing them to devote more time to their research, and it has funded their participation to international conferences, where they have been exposed to current research and eventually presented their own. This support has been crucial in successfully completing their dissertation and starting their career as researchers and teachers. Another related activity has consisted in three mini-courses aimed at advanced undergraduates and beginning graduate students at international summer schools. These activities not only represent a direct contribution to the development of human resources in the mathematical sciences and to the dissemination of mathematical knowledge, but also faciltate the access of minorities to advanced mathematical research.