The Hausdorff Dimension of Small Divisors for Lower-Dimensional KAM-Tori
Given a bounded domain [Note: See the image of page 359 for this formatted text] Ω subset Rm and a Lipschitz map [Note: See the image of page 359 for this formatted text] φ : Ω mapsto Rn, we determine the Hausdorff dimension of sets of points ω in Ω for which the inequality |k\cdot ω -l\cdot φ (ω )| < psi (|k|+|l|) has infinitely many distinct integer solutions [Note: See the image of page 359 for this formatted text] (k, l) in Zm× Zn satisfying |l|<= h, where h is a fixed integer. These sets `interpolate' between the cases h = 0 and h = ∞ , 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 = ∞ of infinite co-dimension, which is relevant in the KAM-theory of certain nonlinear partial differential equations.
Proceedings of the Royal Society of London Series A
- Pub Date:
- November 1992