98-487 Enrico Valdinoci

A remark on sharp estimates for high order nonresonant normal forms in Hamiltonian perturbation theory (24K, LaTeX 2.09) Jul 3, 98

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Abstract. Using the method of majorants, we give an estimate of the rest for the nonresonant action-angle normal forms and exhibit a simple example suggesting the ``optimality'' of this estimate. Given an integer $k$, calling $\g$ the size of the small denominators up to order $k$, we prove that the $k^{\mbox{th}}$ order remainder is approximatively bounded by $O(\e_0^{-k})$ with $\e_0=O(\g^2/k)$. Thus, if we disregard the dependence of $\g$ upon $k$, we obtain a rest bounded by $({\mbox{const}}\;k)^k$. These estimates are conjectured to be optimal: to support this idea we present a simplified model problem with no small denominators, formally related to the above calculations: this example exhibits a factorial divergence.

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