On Aubry Sets and Mather’s Action Functional
نویسنده
چکیده
We study Lagrangian systems on a closed manifold M . We link the differentiability of Mather’s β -function with the topological complexity of the complement of the Aubry set. As a consequence, when dimM ≤ 3, the differentiability of the β -function at a given homology class is forced by the irrationality of the homology class. As an application we prove the two-dimensional case of two conjectures by Mañé.
منابع مشابه
Geometric Proofs of Mather’s Connecting and Accelerating Theorems
In this paper we present simplified proofs of two important theorems of J.Mather. The first (connecting) theorem [Ma2] is about wandering trajectories of exact area-preserving twist maps naturally arising for Hamiltonian systems with 2 degrees of freedom. The second (accelerating) theorem is about dynamics of generic time-periodic Hamiltonian systems on twotorus (2.5-degrees of freedom). Mather...
متن کاملDifferentiability of Mather’s average action and integrability on closed surfaces
In the study of Tonelli Lagrangian and Hamiltonian systems, a central role in understanding the dynamical and topological properties of the action-minimizing sets (also called Mather and Aubry sets), is played by the so-called Mather’s average action (sometimes referred to as β-function or effective Lagrangian), with particular attention to its differentiability and nondifferentiability propert...
متن کاملOn the Conley decomposition of Mather sets
In the context of Mather’s theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian. In the study of Lagrangian systems, John Mather introduced several invariant sets composed of globally minimizing ex...
متن کاملDeveloping a Semantic Similarity Judgment Test for Persian Action Verbs and Non-action Nouns in Patients With Brain Injury and Determining its Content Validity
Objective: Brain trauma evidences suggest that the two grammatical categories of noun and verb are processed in different regions of the brain due to differences in the complexity of grammatical and semantic information processing. Studies have shown that the verbs belonging to different semantic categories lead to neural activity in different areas of the brain, and action verb processing is r...
متن کاملA Minimax Selector for a Class of Hamiltonians on Cotangent Bundles
We construct a minimax selector for eventually quadratic hamiltonians on cotangent bundles. We use it to give a relation between Hofer’s energy and Mather’s action minimizing function. We also study the local flatness of the set of twist maps.
متن کامل