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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)
##### Keywords:
invariant circles
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##### References:
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