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 -
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Y1 - 1992/11/09
UR - http://rspa.royalsocietypublishing.org/content/439/1906/359.abstract
N2 - Given a bounded domain Ω ⊂ ℝm and a Lipschitz map Φ : Ω ⟼ ℝn, we determine the Hausdorff dimension of sets of points ω ∈ Ω for which the inequality |k·ω — l·Φ(ω)| < Ψ (|k| + |l|) has infinitely many distinct integer solutions (k, l)∈ℤm x 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 Ψ 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.
ER -