Parabolic dynamics

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In physics and other sciences, relevant quantities can often be described as averages along the trajectories of a dynamical system. When the system has a property called ``ergodicity'', time averages for large times can be approximated by space averages, which are much easier to compute. For most applications of this fundamental idea, it is crucial to estimate the error involved in the approximation. Our goal is to develop methods for establishing ergodicity with error estimates for weakly chaotic, ``parabolic'' systems. Such systems are roughly characterized by subexponential, polynomial divergence of nearby orbits, in contrast with hyperbolic, strongly chaotic systems which exhibit exponential divergence. For parabolic systems there is no general theory comparable to the hyperbolic theory and the research has focused so far on key interesting examples. Some of the examples are now rather well understood, others are still beyond the reach of available methods of study. Parabolic systems are important in many applications to physics, to geometry and number theory. For instance, in simple models of Hamiltonian dynamics in the presence of singularities, of the type which occur in the motion of the Solar System, and in questions related to number theory, such as the number of ways a large integer can be written as a sum of a given number of squares, cubes and so on. A fundamental tool in the study of parabolic systems is the idea of a renormalization dynamics. The renormalization is like a powerful lense that allows us to explore the fine structure of a system (or family of systems) at smaller and smaller scales. Since the renormalization is typically a hyperbolic dynamical system, it can be studied by the tools of hyperbolic theory (invariant manifolds, Lyapunov exponents, etc.)