Parabolic Dynamics

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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.