Symplectic and contact properties of the Ma ~n 'e critical value of the\n universal cover

2013/09/04 by Gabriel P. Paternain, Paternain, Gabriel P., Alfonso Sorrentino +1
Computer Science · Mathematics · Physics and Astronomy · #37J05 #37J50 #37J55 #53D05 #53D10 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Quantum chaos and dynamical systems #Symplectic Geometry (math.SG) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1309.1001

openalex publication_date 2013/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We discuss several symplectic aspects related to the Ma ~n 'e critical value\nc_u of the universal cover of a Tonelli Hamiltonian. In particular we show that\nthe critical energy level is never of virtual contact type for manifolds of\ndimension greater than or equal to three. We also show the symplectic\ninvariance of the finiteness of the Peierls barrier and the Aubry set of the\nuniversal cover. We also provide an example where c_u coincides with the\ninfimum of Mather's \α -function but the Aubry set of the universal cover\nis empty and the Peierls barrier is finite. A second example exhibits all the\nergodic invariant minimizing measures with zero homotopy, showing that, quite\nsurprisingly, the union of their supports is not a graph, in contrast with\nMather's celebrated graph theorem.\n

Citations

• Lecture notes on Mather's theory for Lagrangian systems
• Action minimizing invariant measures for positive definite Lagrangian systems

Related

• On the stability of Mañé critical hypersurfaces
• On commuting Tonelli Hamiltonians: Autonomous case
• Quantitative contact Hamiltonian dynamics
• Aubry-Mather theory for contact Hamiltonian systems II
• Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures