TY - JOUR
T1 - The Hausdorff Dimension of Small Divisors for Lower-Dimensional KAM-Tori
JF - Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
SP - 359
LP - 371
M3 - 10.1098/rspa.1992.0155
VL - 439
IS - 1906
AU - Dodson, M. M.
AU - Poschel, J.
AU - Rynne, B. P.
AU - Vickers, J. A. G.
Y1 - 1992/11/09
UR - http://rspa.royalsocietypublishing.org/content/439/1906/359.abstract
N2 - Given a bounded domain [Note: See the image of page 359 for this formatted text] $\Omega \subset R^{m}$ and a Lipschitz map [Note: See the image of page 359 for this formatted text] $\phi $: $\Omega \mapsto R^{n}$, we determine the Hausdorff dimension of sets of points $\omega \in \Omega $ for which the inequality $|$k$\cdot \omega $-l$\cdot \phi $($\omega $)$|$ < $\psi $($|$k$|$+$|$l$|$) has infinitely many distinct integer solutions [Note: See the image of page 359 for this formatted text] (k, l) $\in Z^{m}\times Z^{n}$ satisfying $|$l$|\leq $ h, where h is a fixed integer. These sets `interpolate' between the cases h = 0 and h = $\infty $, which occur in the metric theory of Diophantine approximation of independent and dependent quantities, respectively. They arise, for example, in the perturbation theories of lower-dimensional tori in nearly integrable hamiltonian systems (KAM-theory). Among others, it turns out that their Hausdorff dimension is independent of h and n, it only depends on m and the lower order of $\psi $ at infinity. Part of this result even extends to the case n = $\infty $ of infinite co-dimension, which is relevant in the KAM-theory of certain nonlinear partial differential equations.
ER -