Limit Theorems in Dynamical Systems

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This mathematical research project deals with probabilistic properties (or stochasticity) in deterministic systems. Many deterministic systems exhibit random behavior due to sensitive dependence on initial conditions. The goal of the present project is to develop methods of predicting the behavior of such systems using a statistical approach. Particular attention will be paid to describing the dynamics on short spatial and long temporal scales. The methods developed during this project will have applications to several fields, including probability theory, number theory, and statistical physics. The investigator also will continue his synergistic activities including organizing conferences, giving lectures and minicourses on recent advances in dynamics, writing survey papers, and mentoring graduate students.The investigator will work on limit theorems for dynamical systems. The main directions of research will concern (1) local limit theorems and related asymptotic expansions and (2) limit theorems for toral translations and other zero entropy systems. The methods will be drawn from dynamical systems and probability theory. In particular the investigations will rely on a spectral approach and renormalization techniques. The connection between the two parts of the project is that the limit theorems for renormalization dynamics developed in the first part of this project will play important role in the second part.