This project is devoted to a special class of chaotic dynamical systems, namely hyperbolic (and focusing, or `elliptic') mathematical billiards. In the hyperbolic case they are the prototype examples of non-uniformly hyperbolic dynamical systems with singularities. Such systems have been playing an important role in the rigorous mathematical foundation of statistical physics, so that the study and establishing their strong mixing properties is getting more and more physical relevance. The first part of the proposal focuses on a fundamental conjecture regarding this family of dynamical systems, namely the so called `Boltzmann-Sinai Ergodic Hypothesis', which states that any finite system of (totally elastic) hard spheres moving on a flat torus is fully hyperbolic and ergodic, of course, on the level set of its trivial first integrals. The proof of this conjecture (in its full generality) has been so far notoriously withstanding any attack against it. The first major part of the present proposal directly targets this conjecture. The second part is a blueprint for further research in this direction by generalizing the original Boltzmann-Sinai Hypothesis to cylindric billiards, and billiards in physically more relevant containers, i.e. rectangular boxes, convex domains, etc. The third part addresses a question posed by M. Herman, which asks if the scattering (hyperbolic) effect of the hard ball dynamics eventually prevails over the focusing effect of the convex boundary, if the motion takes place in a compact, convex domain. In the fourth part the fundamental complexity problems are targeted for the $n$-step singularity sets of higher dimensional, strictly dispersive, non-uniformly hyperbolic billiards. The answers to those questions are pivotal in further studies of the fine statistical properties of such systems. Part five poses some basic questions and problems concerning the ergodic properties of high-dimensional billiards, in which the hard core interaction potential is replaced by a smooth, rotational symmetric one. Finally, the closing part aims at some open problems in the theory of planar billiards.
The theory of dynamical systems studies the time-evolution of complicated,multi-component systems, like particle systems in statistical physics, reaction kinetics from chemistry, the behavior of the atmosphere (hence the relevance in weather forecasting), population dynamics, developments on the stock market, etc. By nature, this theory is closely related to - and is partly arising from - the theory of differential equations and stochastic processes. An interesting feature of the theory of dynamical systems is that it helps us better understand such crucial phenomena in the time evolution of the above mentioned systems, as the high sensitivity of the solution to the initial conditions, sometimes referred to as chaos, or chaotic behavior. My present proposal targets the investigation and better understanding of a popular and important class of mostly chaotically behaving mathematical models, namely the so called mathematical billiards. They got their name after the fact that they model the physical motion of ball shaped particles interacting with each other via elastic collisions.
