99-105 Alex Haro

The Primitive Function of an Exact Symplectomorphism. Variational principles, converse KAM theory and the problems of determination and interpolation. (20554K, gzipped ps) Apr 12, 99

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Abstract. This thesis has been made under the direction of Prof. Carles Simo. The main contribution is the systematic use of the primitive function of an exact symplectomorphism. The analytical, geometrical and numerical tools used along this thesis take into account the properties of this primitive function. We have divided the thesis in four parts. PART I: Exact symplectic geometry (introduction of the problems). PART II: On the standard symplectic manifold (analytical part). PART III: On the cotangent bundle (geometrical part). PART IV: Applications (numerical part). In last part we generalize converse KAM theory by MacKay, Meiss and Stark and relate it with Lipschitz theory by Birkhoff, Herman and Mather. We also perform a Greene method to detect the breakdown of an invariant torus based upon variational principles. We apply them to a broad class of examples: standard map, exponential standard map, quadratic standard map, Froeschl\'e map (and its family), rotational standard map (a symplectic skew-product), ... We study numerically the Aubry-Mather sets in higher dimensions. We also talk about geometrical obstructions, normal forms, foliations, ...

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