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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

37A99 Ergodic theory
37C80 Symmetries, equivariant dynamical systems (MSC2010)
Full Text: DOI
[1] DOI: 10.1007/BF01209394 · Zbl 0617.58005 · doi:10.1007/BF01209394
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