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Minimal invariant tori in the resonant regions for nearly integrable Hamiltonian systems

Author: Chong-Qing Cheng
Journal: Trans. Amer. Math. Soc. 357 (2005), 5067-5095
MSC (2000): Primary 37J40, 37J50
Published electronically: March 31, 2005
MathSciNet review: 2165398
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Abstract: Consider a real analytical Hamiltonian system of KAM type $H(p,q)$ $=N(p)+P(p,q)$ that has $n$ degrees of freedom ($n>2$) and is positive definite in $p$. Let $\Omega =\{\omega\in \mathbb R^n \vert\langle \bar k,\omega\rangle =0, \bar k\in\mathbb Z^n\}$. In this paper we show that for most rotation vectors in $\Omega$, in the sense of ($n-1$)-dimensional Lebesgue measure, there is at least one ($n-1$)-dimensional invariant torus. These tori are the support of corresponding minimal measures. The Lebesgue measure estimate on this set is uniformly valid for any perturbation.

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Additional Information

Chong-Qing Cheng
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China – and – The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong, China

Keywords: KAM method, invariant torus, minimal measure
Received by editor(s): March 19, 2002
Received by editor(s) in revised form: March 22, 2004
Published electronically: March 31, 2005
Additional Notes: The author was supported by the state basic research project of China “Nonlinear Sciences" (G2000077303)
Article copyright: © Copyright 2005 American Mathematical Society

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