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Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena

Dickinson, Detta and Gramchev, Todor and Yoshino, Masafumi (2002) Perturbations of vector fields on tori: resonant normal forms and Diophantine phenomena. Proceedings of the Edinburgh Mathematical Society, 45 (3). pp. 731-759. ISSN 1464-3839

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Abstract

This paper concerns perturbations of smooth vector fields on Tn (constant if n ≥ 3) with zeroth-order C∞ and Gevrey Gσ, σ ≥ 1, pseudodifferential operators. Simultaneous resonance is introduced and simultaneous resonant normal forms are exhibited (via conjugation with an elliptic pseudodifferential operator) under optimal simultaneous Diophantine conditions outside the resonances. In the C∞ category the results are complete, while in the Gevrey category the effect of the loss of the Gevrey regularity of the conjugating operators due to Diophantine conditions is encountered. The normal forms are used to study global hypoellipticity in C∞ and Gevrey Gσ. Finally, the exceptional sets associated with the simultaneous Diophantine conditions are studied. A generalized Hausdorff dimension is used to give precise estimates of the ‘size’ of different exceptional sets, including some inhomogeneous examples.

Item Type: Article
Additional Information: Copyright Notice. "Published online by Cambridge University. Proceedings of the Edinburgh Mathematical Society." http://journals.cambridge.org/download.php?file=%2FPEM%2FPEM2_45_03%2FS001309150000064Xa.pdf&code=a9baa4f865505a96cc1b3917051f9ad5
Keywords: Vector fields on tori; Simultaneous resonances; Diophantine conditions; Resonant normal forms; Global hypoellipticity; Generalized Hausdorff dimension.
Subjects: Science & Engineering > Mathematics & Statistics
Item ID: 1518
Identification Number: 10.1017/S001309150000064X
Depositing User: Dr. Detta Dickinson
Date Deposited: 25 Aug 2009 13:32
Journal or Publication Title: Proceedings of the Edinburgh Mathematical Society
Publisher: Cambridge University Press
Refereed: Yes
URI:

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