- 00-11 D. Dickinson, T. Gramchev, M. Yoshino
- Perturbations of vector fields on tori: resonant normal forms
and Diophantine phenomena
Jan 10, 00
(auto. generated ps),
of related papers
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.