PERTURBATIONS OF VECTOR FIELDS ON TORI: RESONANT NORMAL FORMS AND DIOPHANTINE PHENOMENA
PERTURBATIONS OF VECTOR FIELDS ON TORI: RESONANT NORMAL FORMS AND DIOPHANTINE PHENOMENA
复制标题
环面矢量场的扰动：共振范式和丢番图现象
DOI：
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发表时间：
2002
影响因子：
0.7
通讯作者：
M. Yoshino
中科院分区：
文献类型：
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作者：
Detta Dickinson;T. Gramchev;M. Yoshino
本文讨论了具有零级拟微分算子的光滑向量场的扰动问题。引入了同时共振，并在共振之外的最佳同时丢番图条件下展示了同时共振范式(通过与椭圆拟微分算子的共轭)。在$C^\inty$范畴中，结果是完全的，而在Gevrey范畴中，由于丢番图条件的原因，共轭算子的Gevrey正则性的损失是有影响的。正规形被用来研究$C^inty$和Gevrey$G^\sigma$中的整体亚椭圆性。最后，研究了与同时丢番图条件相关的例外集。推广的Hausdorff维度被用来给出不同例外集的‘大小’的精确估计，包括一些非齐次的例子。AMS 2000数学科目分类：小学37C15；11J13。辅助58J40；11J20；35H05
Abstract 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