The Hausdorff Dimension of Small Divisors for LowerDimensional KAMTori
Abstract
Given a bounded domain [Note: See the image of page 359 for this formatted text] Ω subset R^{m} and a Lipschitz map [Note: See the image of page 359 for this formatted text] φ : Ω mapsto R^{n}, 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 Z^{m}× Z^{n} 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 lowerdimensional tori in nearly integrable hamiltonian systems (KAMtheory). 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 codimension, which is relevant in the KAMtheory of certain nonlinear partial differential equations.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 November 1992
 DOI:
 10.1098/rspa.1992.0155
 Bibcode:
 1992RSPSA.439..359D