On the tangent cones of Aubry sets (Q782418)

This paper considers the Tonelli Hamiltonian on \(\mathbb T^n \times \mathbb R^n\) and the Aubry set \(A\) in \(\mathbb T^n \times \mathbb R^n\). The main result describes cones of the Aubry set. A cone is a subset \(K\) of \(\mathbb R^{2n}\) such that \(0 \in K\) and \(\lambda K \subset K\) for all \(\lambda > 0\). A cone is determined uniquely by its intersection with the unit sphere. The author considers three different definitions of tangent cones: \begin{itemize} \item[1)] the contingent cone \( C_z (A)\) at \(z \in A\) is the set of all limit points \(\lim_{n \to +\infty} t_n (z_n - z)\) for \(t_n > 0, z_n \in A\) and \(z_n \rightarrow z\); \item[2)] the limit contingent cone \(\tilde C_z (A)\) is the set of all possible limit points of vectors \(v_n \in C_{z_n}\) with \(z_n \in A\) and \(z_n \rightarrow z\); \item[3)] the paratingent cone \( P_z (A)\) consists of the limit points of \(\lim_{n \to +\infty} t_n (z_n - w_n)\) where \(t_n > 0, z_n, w_n \in A\) and \(w_n \rightarrow z\). \end{itemize} Then \(C_z(A) \subset \tilde C_z(A) \subset P_z (A)\). Critical elements here are the Green bundles \(\mathcal {G}_\pm(x, p) \subset \mathbb R^n \times \mathbb R^n\), a family of invariant Lagrange subspaces transversal to \(\{0\} \times \mathbb R^n\). They are the graphs of symmetric matrices \(\mathcal {G}_\pm\) \(= \{(h, G_\pm) : h \in \mathbb R^n \}\). The author's main theorem is that the paratingent cone of the Aubry set of a Tonelli Hamiltonian is contained in a cone bounded by the Green bundles. More specifically \(P_z (A) \subset C(\mathcal G_-(z), \mathcal G_+(z))\) for all \(z \in A\). Here \(C(\mathcal G_-(z), \mathcal G_+(z))\) is defined as \(\{(h, Sh) : \mathcal G_-(z) \leq S \leq \mathcal G_+(z) \), for \(h \in \mathbb R^n\}\), and \(S_1 < S_2\) means that \(S_2 - S_1\) is positive definite for invariant Lagrange subspaces \(S_1\) and \(S_2\).