The tiered Aubry set for autonomous Lagrangian functions (Q939596)

If \(M\) is a compact, connected manifold of dimension two or greater, a real-valued function \(L: TM\to\mathbb{R}\) on the tangent bundle \(TM\) is called a Tonelli Lagrangian function if it is \(C^\infty\) and the following conditions hold: (a) \(L\) is uniformly superlinear, so \(\lim_{\| v\|\to\infty} {L(x,v)\over\| v\|}= +\infty\) uniformly in \(x\in M\), and (b) \(L\) is strictly convex, so for all \((x,v)\in TM\), \({\partial^2L\over\partial v^2}(x,v)\) is positive definite. The tiered Aubry set for \(L\), \(A^T(L)\), is the union of the Aubry sets \(A(L+\lambda)\), where \(\lambda\) is a \(C^\infty\) closed \(t\)-form on \(M\). The tiered Mañé set \(N^T(L)\) is defined similary, via Mañé sets. The author's main results are as follows: (1) The set \(N^T(L)\) is closed and connected. If the dimension of the cohomology group \(H'(M)\) is at least two, then the intersection of \(N^T(L)\) with any constant energy surface is connected and chain-transitive; (2) If \(L\) is generic (in the sense of Mañé), then the closures of \(A^T(L)\) and \(N^T(L)\) have no interior; and (3) If the interior of the closure of \(A^T(L)\) is non-empty, then it contains a dense subset of periodic points. The author also provides two illuminating examples. The first constructs an explicit Tonelli Lagrangian function that satisfies (2) above. The second shows that when \(M\) is the 2-torus, the closure of the tiered Aubry set can be different from the closure of the union of the KAM tori.