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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