Boundaries of instability zones for symplectic twist maps.

*(English)*Zbl 1290.37024Let \(f: {\mathbb A } \rightarrow {\mathbb A}\) 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.

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.

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\textit{M. C. Arnaud}, J. Inst. Math. Jussieu 13, No. 1, 19--41 (2014; Zbl 1290.37024)

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