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Destruction of invariant circles. (English) Zbl 0688.58024
Denote by \(T^ 1\) the set of all exact, area-preserving, monotone twist diffeomorphisms of infinite cylinder (\({\mathbb{R}}/{\mathbb{Z}})\times {\mathbb{R}}\). The precise conditions for a diffeomorphism \(\bar f\) of (\({\mathbb{R}}/{\mathbb{Z}})\times {\mathbb{R}}\), to be a member of \(T^ 1\) are the conditions imposed on \(\bar f\) by J. Mather [Comment. Math. Helv. 60, 508-557 (1985; Zbl 0587.58015)]. All elements of \(T^ 1\) which are \(C^{\infty}\) we denote by \(T^{\infty}\). A real number \(\omega\) is called a Liouville number if it is irrational but does not satisfy a Diophantine condition (for all \(C,N>0\) there exist \(p,q\in {\mathbb{Z}}\setminus \{0\}\) such that \(| q\omega -p| <c/q^ N)\). The main result of the paper is a partial converse of the Kolmogorov-Arnold- Moser result on persistence of invariant circles: if \(\omega\) is rational or Liouville, then any \(\bar f\in T^{\infty}\) can be arbitrarily well approximated in the \(C^{\infty}\)-topology by a \(g\in T^{\infty}\) which has no invariant circle of rotation number \(\omega\).
Reviewer: V.G.Angelov

MSC:
37A99 Ergodic theory
37C80 Symmetries, equivariant dynamical systems (MSC2010)
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[1] DOI: 10.1007/BF01209394 · Zbl 0617.58005 · doi:10.1007/BF01209394
[2] Herman, Astérisque 1 (1983)
[3] Birkhoff, Bull. Soc. Math. 60 pp 1– (1932)
[4] Rudin, Real and Complex Analysis (1966)
[5] DOI: 10.2307/2373361 · Zbl 0212.56502 · doi:10.2307/2373361
[6] Salamon, The Kolmogorov-Arnold-Moser theorem (1986) · Zbl 1136.37348
[7] Moser, Annals of Mathematical Studies 77 (1973)
[8] Moser, Nachr. Akad. Wiss. pp 1– (1962)
[9] Mather, Periodic Solutions of Hamiltonian Systems and Related Topics 209 pp 177– (1987) · doi:10.1007/978-94-009-3933-2_18
[10] Mather, Proc. ICM.
[11] Mather, Publ. IHES 63 pp 153– (1986) · Zbl 0603.58028 · doi:10.1007/BF02831625
[12] DOI: 10.1017/S0143385700002455 · Zbl 0557.58019 · doi:10.1017/S0143385700002455
[13] DOI: 10.1007/BF02567431 · Zbl 0597.58015 · doi:10.1007/BF02567431
[14] Herman, Astérisque 144 (1986)
[15] Collected Works II pp 418– (1950)
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