Mathematical Sciences: Ergodic Theory, Differential Dynamicsand Combinatorial Number Theory

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9501383 Ornstein Part of the research involves a completely new approach to stability results of the KAM type. This approach fills the gaps in earlier work on diffeomorphisms of the circle and are optimal. The method gives a new proof of the Moser twist theorem; gives optimal smoothness for invariant curves whose rotation numbers belong to a "good" set of full measure; gives invariant curves for twist maps which are not necessarily close to a rigid twist. The project will involve completing this work and extending to higher dimensions. In other directions the investigators will continue their work on a statistical version of structural stability and applications of ergodic theory to combinatorics and number theory. Some of the research is focused on the question: What can we say about the long-term behavior of a dynamical system, if we have only approximate information about the equations that govern them? This research will focus on two simple examples involving maps of the circle or an annulus. A study will also be made of chaotic systems. Applications of ergodic theory and dynamical systems to combinatorial and number theoretic problems as well as statistics and data compression will be investigated. ***