The PI will continue his work on limit theorem for dynamical systems. The main directions of research will be the following: convergence to stable laws, limit theorems for actions of higher rank abelian groups, limit theorems for random translations and diffusion limit theorems.
Examples of systems which evolve according to deterministic laws but exhibit stochasticity when the initial conditions are not known exactly appear in natural phenomena ranging from atomic scales (Brownian Motion of particles in fluids) to astronomical scales (motion of comets and asteroids). In cases where exact calculations are impossible due to stochasticity a statistical description has to be used. The research supported by the current proposal will develop methods of statistical description of deterministic system by identifying the regions in the phase space responsable for the stochasticity and analyzing the behaviour of the system in those regions. PI will also continue his synergistic activities including organizing conferences, giving lectures and minicourses on recent advances in dynamics, writing survey papers and mentoring graduate students and postdocs.
The research supported by the present grant deals with the origin of stochastic behavior in deterministic dynamical system. A particular emphasis ws made onsystems with several time scale where the randomness comes from the sensetive dependence on initial conditions: a small change in the fast variable can make a significant impact on the slow variables. The examples of the systems covered by the methods developed during the grant period are the following: (1) n body problem of celestial mechanics. Historically, the system of particles interacting by a gravitational potential was one of the first systems investigated by differential equations. Surprisingly, the question of completness of the corresponding equations, that is, the question when the solution can be continued for all times is still poorely understood. The research supported by this grant included the PhD thesis of Jinxin Xue where he showed that the four body problem admits solutions which has no collisions yet cannot be continued for all times. (2) Fermi-Ulam systems. Fermi acceleration-the stochastic acceleration of particles as a resultof interaction with moving environment plays important role in several branches of physics-including construction of particles accelerators. In 1950s it was suggested by Ulam that a similar phenomenon can appear in systems with a small numbers of degrees of freedom. In a joint paper with PIs former student, Jacopo de Simoi, the PI developed a theory which allowed to explain the numerical experiments performed by Ulam and Wells to test this hypothesis. Another direction of research funded by the present grant deals with applications of dynamical systems to probability theory-in particular to random walks in random environment which is a common model for transport in heterogenuous media. During the grant period the PI directed 2 PhD thesises. He also coorganized a semester long thematic program on Hyperbolic Dynamics, Large Deviations and Fluctuations which included two schools for graduate students and recent PhDs and coedited a volume containing the lectures from the schools.