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The link between the shape of the irrational Aubry-Mather sets and their Lyapunov exponents. (English) Zbl 1257.37028

After the fundamental work of Poincaré on the restricted 3-body problem, the twist maps of the two-dimensional annulus \(\mathbb A=\mathbb T\times\mathbb R\) are one of the central motivating examples in Hamiltonian dynamics. In this article the author study the relationship between the geometric shape of an Aubry-Mather set of an exact symplectic twist map and the hyperbolicity of its dynamics and prove the following interesting results:
(1) a Mather measure has zero Lyapunov exponents if and only if its support is \(C^1\)-regular almost everywhere;
(2) a Mather measure has nonzero Lyapunov exponents if and only if its support is \(C^1\)-irregular almost everywhere;
(3) an Aubry-Mather set is uniformly hyperbolic if and only if it is irregular everywhere;
(4) the Aubry-Mather sets which are close to the KAM invariant curves, even if they may be \(C^1\)-irregular, are not too irregular, i.e., have small paratingent cones.
A notion of \(C^1\)-regularity is introduced by the author as follows. A subset \(M\) is \(C^1\)-regular at \(x\in M\) if there exists a line \(D\) of \(T_x\mathbb A\) such that the paratingent cone \(P_M(x)\) is a subset of \(D\), otherwise \(M\) is \(C^1\)-irregular at \(x\). Here, \(P_M(x)\) is the cone of \(T_x\mathbb A\) whose elements are the limits \(v =\lim_{n\to\infty} (x_n-y_n)/\tau_n\); where \((x_n)\) and \((y_n)\) are sequences of elements of \(M\) converging to \(x\), \((\tau_n)\) is a sequence of elements of \(\mathbb R^*_+\) converging to \(0\), and \(x_n-y_n\) is the unique lift of this element of \(\mathbb A\) that belongs to \((-1/2,1/2)\times(-1/2,1/2)\).

MSC:

37E40 Dynamical aspects of twist maps
37J50 Action-minimizing orbits and measures (MSC2010)
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References:

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