Gevrey smooth topology is proper to detect normalization under Siegel type small divisor conditions

Abstract

We shape the results on the formal Gevrey normalization. More precisely, we investigate the better expression ofα, which makes the formal Gevrey-α coordinates substitution turning the Gevrey-α smooth vector fields X into their normal forms in several cases. Such results show that the 'loss' of the Gevrey smoothness is not always necessary even under Siegel type small divisor conditions, which are different from others.

On the ultradifferentiable normalization

“…Especially, when we restrict our focus on the normalization with small divisors of the Diophantine type, the classical Gevrey smoothness is proper. In our category, the key lemma in [13] can be represented as follows.…”

“…In the view of our series results, out of the Poincaré domain, it is enough to apply formal Gevrey conjugacy to classify different resonant terms without any small divisors by Theorem 1.3. When Siegel type small divisors appear, the formal loss of smoothness for the normalization stops in the same formal Gevrey class but with different Gevrey indices by [13]. Now turning to (i) of Theorem 1.2, the formal loss of smoothness for the ultradifferentiable normalization stops in the class with different weight functions, while the slight loss must be allowed in (ii) of Theorem 1.2.…”

“…Recently, to bridge the gap between analytical and C ∞ conjugations, the ultradifferentiable topology, which is the refinement of C ∞ topology and was once introduced in [10] by Rudin, emerges before our eyes. In the Gevrey smooth case, it was proved in [11] that the Gevrey-α smooth vector fields can be changed into their normal forms by the Gevrey-(α + μ + 1) smooth coordinates substitutions at the origin for the hyperbolic liner part and the Siegel type small divisor condition with the index μ, which was improved into an accurate one in [13] for the diagonal linear part. Meanwhile, more degenerated formal Gevrey-α vector fields were studied in [6] and see also [2] for the Gevrey linearization.…”

“…The next is generalized from [13] to confirm the stop of the loss of smoothness for the 'larger' small divisor. Consider the following small divisor condition (1.2) fixed with μ (t) = e μt τ (1.6) for τ ∈ (0, ∞) and μ > 0.…”