This research project consists of three parts: Part I deals with billiards on non-compact tables, as models for hyperbolic dynamical systems with an infinite invariant measure. The plan is to extend to these dynamical systems certain important results available for compact billiards, including a suitable version of Pesin's Theory. This will require the rewriting of some notions of ergodic theory for the case of a non-probability measure. Part II concerns random billiards; more precisely, billiards with a random table (changing after each collision), a quenched random table (selected once and for all from a random ensemble), or a random law of reflection (in a fixed table). In the first two cases there is a physical invariant measure, and the PI will study its ergodic properties (including Lyapunov exponents and the like) for a typical realization of the random process. In the third case an equilibrium measure can be shown to exist, and its asymptotic properties will be investigated, together with the natural question of the stochastic stability at the zero-noise limit. Similar perturbative questions are considered for the other systems as well. Part III, in the realm of equilibrium statistical mechanics, considers the question of quantum large deviations. In a collaborative project, the PI sets out to initiate a theory of large deviations for a noncommutative quasi-local algebra on a d-dimensional lattice. Of primary interest is the convergence of the moment-generating function for an extensive observable, and the smoothness and physical significance of its limit.
Billiards are a class of dynamical systems that has been most extensively studied. This is so for a two-fold reason: On one hand, mathematically, they are relatively treatable, with their geometric features often giving hints on how to prove a certain sought result. On the other hand, from the point of view of physics, they are fairly realistic models that have been applied to a variety of fields, from statistical mechanics to optics to scattering theory. Therefore the drive is natural to expand this class to cover new areas where it could have a remarkable impact: the family of open systems (every system with unbounded dynamics belongs to this category) and of random systems (which tries to justify when, and why, deterministic predictions still make sense in a noisy world). As for Part III, the theory of large deviations in statistical mechanics gives a theoretical understanding of the fact that large systems, which are deeply random from the observer's standpoint, yield nonetheless very accurate deterministic measurements. Although large deviations for classical systems have been the subject of massive and very successful research, a counterpart of quantum system (which should rightfully be even more fundamental) is conspicuously missing at the moment.
