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Boundaries of instability zones for symplectic twist maps. (English) Zbl 1290.37024
Let $$f: {\mathbb A } \rightarrow {\mathbb A}$$ be a symplectic twist map of the annulus. Recall that an “instability region” is an open invariant subset $$U \subset {\mathbb A}$$ which is homeomorphic to the annulus and contains a simple closed curve that is not homotopic to a point (so-called “essential curve”) and is maximal (for the inclusion) among such subsets that satisfy the following property:
The invariant curves contained in the closure $${\bar U}$$ of $$U$$ in $${\mathbb A}$$ that are not homotopic to a point are contained in the boundary $$\partial U$$.
The existence of instability zones was proven by G. D. Birkhoff [Ann. Inst. Henri Poincaré 2, 369–386 (1932; JFM 58.1274.02)] and M. R. Herman [Sur les courbes invariantes par les difféomorphismes de l’anneau. Vol. 1. Astérisque, 103–104. Paris: Société Mathématique de France. (1983; Zbl 0532.58011)] proved that, generically, the boundaries of the instability zones have irrational rotation numbers. P. Le Calvez [Ann. Sci. Éc. Norm. Supér. (4) 20, No. 3, 443–464 (1987; Zbl 0653.58014)] proved many interesting topological dynamical properties inside $$U$$, specially about the stable and unstable sets of the boundary of an instability zone. Using variational methods, J. N. Mather [J. Am. Math. Soc. 4, No. 2, 207–263 (1991; Zbl 0737.58029)] showed that if $$\Gamma$$ is at the boundary of an instability zone, then $$W^{s}(\Gamma) - \Gamma \neq \emptyset$$ and $$W^{u}(\Gamma) - \Gamma \neq \emptyset$$, where $$W^{s}(\Gamma)$$ (resp. $$W^{u}(\Gamma)$$) denotes the stable (resp. unstable) set of $$\Gamma$$. The main result of the paper is the following:
Theorem 1: Let $$\omega$$ be an irrational number. In any neighborhood of the map $$(\omega,r) \mapsto (\theta+r,r)$$ in the $$C^2$$ topology, there is a symplectic $$C^2$$ twist map $$f$$ in the annulus that has a homotopically non-trivial invariant curve $$\Gamma$$ such that:
(a) $$f_{| \Gamma}$$ is $$C^{0}$$ conjugate to a Denjoy counter-example;
(b) $$\Gamma$$ is of class $$C^1$$, except along a wandering orbit. In other words, $$\Gamma$$ is the graph of a function $$\gamma :{\mathbb T } \rightarrow {\mathbb R }$$ which is $$C^1$$ except along the projection of a wandering orbit $$x_n$$ where the left and right derivatives are disctinct.
The author concentrates the non-regularity of $$\Gamma$$ along a nonwandering orbit and gives a description in neighborhoods of the orbit in terms of its stable/unstable branches. In these neighborhoods, part of $$\Gamma$$ is made of the union of a stable branch with an unstable one, while the other two branches belong to $$U$$.
The starting point is a construction by Herman which considers a family of maps $$f :{\mathbb T} \times {\mathbb R} \rightarrow {\mathbb T} \times {\mathbb R}$$ of the form $$f(\theta,r)=(\theta+r,r+\varphi(\theta+r))$$, where $$\varphi: {\mathbb T} \rightarrow {\mathbb R}$$ is a $$C^1$$ function with $$\int_{\mathbb T} \varphi(\theta) d\theta=0$$. The function $$\varphi$$ obtained by Herman is of class $$C^2$$ and has the form $$\tilde{g}-{\tilde{g}}^{-1}-2Id$$, where $$\tilde{g}$$ is a lift of a Denjoy diffeomorphism. The proof of the theorem involves careful improvements of Herman’s estimates and the description of the stable and unstable sets of the wandering orbit.
The author also proves a theorem, in the $$C^1$$ class, that shows that in any neighborhood of a symplectic twist map $$f$$ with an essential invariant curve $$\Gamma$$ of irrational rotation number, there is a $$C^1$$ symplectic twist map $$g$$ such that $$\Gamma$$ is in the boundary of an instability zone of $$g$$ such that $$g_{|\Gamma}=f_{|\Gamma}$$.
The author also raises some interesting open questions about the regularity of the boundary of instability zones and their stable/unstable sets.

##### MSC:
 3.7e+41 Dynamical aspects of twist maps 3.7e+11 Dynamical systems involving maps of the circle
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##### References:
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