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PERTURBATIONS OF VECTOR FIELDS ON TORI: RESONANT NORMAL FORMS AND DIOPHANTINE PHENOMENA

Published online by Cambridge University Press:  14 October 2002

Detta Dickinson
Affiliation:
Department of Mathematics, Logic House, NUI Maynooth, Co. Kildare, Ireland (ddickinson@maths.may.ie)
Todor Gramchev
Affiliation:
Dipartimento di Matematica, Università di Cagliari, 09124 Cagliari, Italy (todor@mat33.sc.unica.it)
Masafumi Yoshino
Affiliation:
Department of Mathematics, Chuo University, Tokyo 192-0393, Japan (yoshinom@tamacc.chuo-u.ac.jp)
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Abstract

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This paper concerns perturbations of smooth vector fields on $\mathbb{T}^n$ (constant if $n\geq3$) with zeroth-order $C^\infty$ and Gevrey $G^\sigma$, $\sigma\geq1$, 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 $C^\infty$ 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 $C^\infty$ and Gevrey $G^\sigma$. 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.

AMS 2000 Mathematics subject classification: Primary 37C15; 11J13. Secondary 58J40; 11J20; 35H05

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2002
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PERTURBATIONS OF VECTOR FIELDS ON TORI: RESONANT NORMAL FORMS AND DIOPHANTINE PHENOMENA
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