Continuous Complexity and Dynamics

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Abstract :The Principal Investigator is Michael Shub. Shub and co-workersPugh, Wilkinson and others have established the stable ergodicity of awide range of partially hyperbolic dynamical systems which preserve asmooth volume element, thus establishing the statistical robustness ofthese chaotic systems. The thrust of the work proposed is to extendthese results to include almost all partially hyperbolic systems and toeliminate the hypothesis of volume preservation. This is an ambitious goal,but the results obtained included the development of new tools, namelyjulienne quasi-conformality and julienne density point preservation of thestable holonomy maps, whose power has not been fully explored. A second theme is a study of the complexity of continuous problems centered around theanalysis and application of variants of Newton's method.Chaotic dynamical systems are pervasive in scientific, engineering andnumerical simulation applications. The proposed work would lay foundationsfor the statistical analysis of these systems, whose deterministic behavior isunpredictable due to the presence of sensitive dependence on initialconditions. Newton's method for solving systems of equations is one ofthe main algorithms of numerical analysis. A better understanding of itsproperties and those of variants can make the solution of problems ofscientific and engineering origin more efficient.