00-11 D. Dickinson, T. Gramchev, M. Yoshino

Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena (413K, PS) Jan 10, 00

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Abstract. This paper is concerned with the study of perturbations of smooth vector fields on $\T^n $ (constant if $n\geq 3$) with zero'th order $C^\infty $ and Gevrey $G^\sigma $, $\sigma\geq 1$ pseudodifferential operators. A notion of simultaneous resonance is introduced and simultaneous resonant normal forms are shown (via conjugation with an elliptic pseudodifferential operator) under optimal simultaneous Diophantine conditions outside the resonances. In the $C^\infty$ category the results are complete. In the Gevrey category the effect of the loss of the Gevrey regularity of the conjugating operators due to Diophantine conditions is investigated by means of techniques similar to methods for obtaining Nekhoroshev type (effective stability) estimates in hamiltonian dynamics. The normal forms are used to study the global hypoellipticity in $C^\infty $ and Gevrey $G^\sigma$. Finally, the exceptional sets associated with the simultaneous Diophantine conditions are studied. Generalized Hausdorff dimension is used to give precise estimates of the ``size'' of different exceptional sets, including some inhomogenous exceptional sets.

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