Aubry-Mather Theory Research Articles

Ground states and critical points for generalized Frenkel–Kontorova models in

We consider a multidimensional model of the Frenkel–Kontorova type but we allow non-nearest-neighbour interactions, which satisfy some weak version of ferromagnetism.For every possible frequency vector, we show that there are quasiperiodic ground states which enjoy further geometric properties. The ground states we produce are either bigger or smaller than their integer translates. They are at a bounded distance from the plane wave with the given frequency.The comparison property above implies that the ground states and the translations are organized into laminations. If these leave a gap, we show that there are critical points inside the gap which also satisfy the comparison properties.In particular, given any frequency, we show that either there is a continuous parameter of ground states or there is a ground state and another critical point which is not a ground state. This is a higher dimensional analogue of the criterion of the non-existence of invariant circles if and only if there is a positive Peierls–Nabarro barrier.All the above results are higher dimensional extensions of similar results in Aubry–Mather theory.

Viscosity solutions for partial differential equations with Neumann type boundary conditions and some aspects of Aubry–Mather theory

We study partial differential inequalities (PDI) of the type c + H ( x , ∂ v ∂ x ) − N K ( x ) ∋ 0 where N K ( ⋅ ) is the normal cone to the set K. We prove existence of a constant c : = c ¯ such that the PDI of Hamilton–Jacobi type has a unique (global) Lipschitz viscosity solution. We provide a formula to calculate this constant. Moreover, we define a subset K ˜ of K such that any two solutions of the previous PDI which coincide on K ˜ will coincide on K. Our paper generalizes results of the case without boundary conditions for convex Hamiltonians obtained by L.C. Evans and A. Fathi.

PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians

We propose a PDE approach to the Aubry-Mather theory using viscosity solutions. This allows to treat Hamiltonians (on the flat torus \(\mathbb{T}^N\)) just coercive, continuous and quasiconvex, for which a Hamiltonian flow cannot necessarily be defined. The analysis is focused on the family of Hamilton-Jacobi equations \(H(x,Du) = a\) with a real parameter, and in particular on the unique equation of the family, corresponding to the so-called critical value a = c, for which there is a viscosity solution on \(\mathbb{T}^N\). We define generalized projected Aubry and Mather sets and recover several properties of these sets holding for regular Hamiltonians.

On diffusion in high-dimensional Hamiltonian systems

The purpose of this paper is to construct examples of diffusion for ε -Hamiltonian perturbations of completely integrable Hamiltonian systems in 2 d -dimensional phase space, with d large. In the first part of the paper, simple and explicit examples are constructed illustrating absence of ‘long-time’ stability for size ε Hamiltonian perturbations of quasi-convex integrable systems already when the dimension 2 d of phase space becomes as large as log 1 ε . We first produce the example in Gevrey class and then a real analytic one, with some additional work. In the second part, we consider again ε -Hamiltonian perturbations of completely integrable Hamiltonian system in 2 d -dimensional space with ε -small but not too small, | ε | > exp ( - d ) , with d the number of degrees of freedom assumed large. It is shown that for a class of analytic time-periodic perturbations, there exist linearly diffusing trajectories. The underlying idea for both examples is similar and consists in coupling a fixed degree of freedom with a large number of them. The procedure and analytical details are however significantly different. As mentioned, the construction in Part I is totally elementary while Part II is more involved, relying in particular on the theory of normally hyperbolic invariant manifolds, methods of generating functions, Aubry–Mather theory, and Mather's variational methods. Part I is due to Bourgain and Part II due to Kaloshin.

Counting geodesics which are optimal in homology

On a surface of negative curvature, we study the distribution of closed geodesics which are not far from minimizing length in their homology classes. These geodesics do not have the statistical behaviour obtained by Margulis for ‘generic’ geodesics and later extended by Lalley to geodesics satisfying certain homological constraints. We cannot use the usual techniques of ‘thermodynamic formalism for Anosov flows’, but we study the problem using ideas from Aubry–Mather theory, as well as some facts of low-dimensional topology.

Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory

We consider Lagrangian submanifolds lying on a fiberwise strictly convex hypersurface in some cotangent bundle or, respectively, in the domain bounded by such a hypersurface. We establish a new boundary rigidity phenomenon, saying that certain Lagrangians on the hypersurface cannot be deformed (via Lagrangians having the same Liouville class) into the interior domain. Moreover, we study the non-removable intersection between the Lagrangian and the hypersurface, and show that it contains a set with specific dynamical behavior, known as Aubry set in Aubry-Mather theory.

Boundary Rigidity for Lagrangian Submanifolds, Non-Removable Intersections, and Aubry—Mather Theory

We consider Lagrangian submanifolds lying on a fiberwise strictly convex hypersurface in some cotangent bundle or, respectively, in the domain bounded by such a hypersurface. We establish a new boundary rigidity phenomenon, saying that certain Lagrangians on the hypersurface cannot be deformed (via Lagrangians having the same Liouville class) into the interior domain. Moreover, we study the non-removable intersection between the Lagrangian and the hypersurface, and show that it contains a set with specific dynamical behavior, known as Aubry set in Aubry-Mather theory.

Existence of infinite non-Birkoff periodic orbits for area-preserving monotone twist maps of cylinders

The exact monotone twist map of infinite cylinders in the Birkhoff region of instability is studied. A variational method based on Aubry-Mather theory is used to discover infinitely many non-Birkhoff periodic orbits of fixed rotation number sufficiently close to some irrational number for which the angular invariant circle does not exist.

Optimal scheduling in a periodic environment

We consider the problem of scheduling a sequence of actions when the benefit obtained from an action depends on only the current time modulo a period and the time since the previous action. A simple example is a model of a shuttle bus service where the profit from running a bus depends on the time of day and the time since the previous bus. Mathematically, such problems can be formulated as maximizing V(tn-1,tn) over schedules (tn)n0 as tN goes to infinity, for functions V(t,t´) satisfying the periodicity condition V(t+T,t´+T) = V(t,t´), for some T>0. The problem is related to Aubry-Mather theory in the dynamics of area-preserving maps. We extend this theory in order to prove the existence of optimizing schedules for each initial condition t0, to characterize the properties of such schedules and to analyse their dependence on t0 and on the parameters of the model.

Aubry-mather theory and the inverse spectral problem for planar convex domains

The inverse spectral problem is concerned with the question to what extent the spectrum of a domain determines its geometry. We find that, associated to a convex domain Ω in ℝ2, there is a convexfunction which is a length spectrum invariant under continuous deformations. It includes several geometric quantities, such as the lengths and Lazutkin parameters of caustics, as well as the asymptotic invariants discovered by Marvizi and Melrose. Via a Poisson relation, we also find invariants determined by the Laplace spectrum of Ω.

AUBRY–MATHER THEORY AND IDEMPOTENT EIGENFUNCTIONS OF BELLMAN OPERATOR

We study the connection between Aubry–Mather theory of invariant sets of a dynamical system described by a Lagrangian L(t,x,v) = L0(v) - U(t,x) with periodic potential U(t,x), on the one hand, and idempotent spectral theory of Bellman operator of the corresponding optimization problem, on the other hand. This connection is applied to obtain a uniqueness result for an eigenfunction of Bellman operator in the case of irrational rotation number.

Periodic solutions of the Hamilton-Jacobi equation with a periodic non-homogeneous term and Aubry-Mather theory

It is proved that the one-dimensional Hamilton-Jacobi equation with a periodic non-homogeneous term admits a family of generalized solutions, each of which can be represented as the sum of a linear and a periodic function; a condition for the uniqueness of such a solution is given in terms of Aubry-Mather theory.

Lagrangian systems on hyperbolic manifolds

This paper gives two results that show that the dynamics of a time-periodic Lagrangian system on a hyperbolic manifold are at least as complicated as the geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the Poincaré ball, Theorem A asserts that there are minimizers of the lift of the Lagrangian system that are a bounded distance away and have a variety of approximate speeds. Theorem B gives the existence of a collection of compact invariant sets of the Euler–Lagrange flow that are semiconjugate to the geodesic flow of a hyperbolic metric. These results can be viewed as a generalization of the Aubry–Mather theory of twist maps and the Hedlund–Morse–Gromov theory of minimal geodesics on closed surfaces and hyperbolic manifolds.

Aubry-Mather theory and periodic solutions of the forced Burgers equation

Consider a Hamiltonian system with Hamiltonian of the form H(x, t, p) where H is convex in p and periodic in x, and t and x ∈ ℝ1. It is well-known that its smooth invariant curves correspond to smooth Z2-periodic solutions of the PDE ut + H(x, t, u)x = 0. In this paper, we establish a connection between the Aubry-Mather theory of invariant sets of the Hamiltonian system and Z2-periodic weak solutions of this PDE by realizing the Aubry-Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry-Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry-Mather set, defined in (2.24). The graph itself is a backward-invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry-Mather theory into the characteristic fields of the above PDE. This is done by making use of one- and two-sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z2-periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two-sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc.

On the Aubry-Mather Theory in Statistical Mechanics

We generalize Aubry-Mather theory for configurations on the line to general sets with a group action. Cocycles on the group play the role of rotation numbers. The notion of Birkhoff configuration can be generalized to this setting. Under mild conditions on the group, we show how to find Birkhoff ground states for many-body interactions which are ferromagnetic, invariant under the group action and having periodic phase space.

Shadowing property of geodesics in Hedlund's metric

In this paper we show that the geodesic flow in a Hedlund-type metric on the 3-torus possesses the shadowing property. This implies, in particular, that any rotation vector is represented by a geodesic, a fact that in the two-dimensional case is given by the Aubry–Mather theory, while in the higher-dimensional case is still unknown.

Aubry-Mather theory for functions on lattices

We generalize the Aubry-Mather theorem on the existence of quasi-periodic solutions of one dimensional difference equations to situations in which the independent variable ranges over more complicated lattices. This is a natural generalization of Frenkel-Kontorova models to physical situations in a higher dimensional space. We also consider generalizations in which the interactions among the particles are not just nearest neighbor, and indeed do not have finite range.

Construction of invariant measures supported within the gaps of Aubry–Mather sets

AbstractThis paper represents a contribution to the variational approach to the understanding of the dynamics of exact area-preserving monotone twist maps of the annulus, currently known as the Aubry–Mather theory. The method introduced by Mather to construct invariant measures of Denjoy type is extended to produce almost-periodic measures, having arbitrary rationally independent frequencies, and positive entropy measures, supported within the gaps of Aubry–Mather sets which do not lie on invariant curves. This extension is based on a generalized version of the Percival's Lagrangian and on a new minimization procedure, which also gives a simplified proof of the basic existence theorem for the Aubry–Mather sets.

Geodesic rays, Busemann functions and monotone twist maps

We study the action-minimizing half-orbits of an area-preserving monotone twist map of an annulus. We show that these so-called rays are always asymptotic to action-minimizing orbits. In the spirit of Aubry-Mather theory which analyses the set of action-minimizing orbits we investigate existence and properties of rays. By analogy with the geometry of the geodesics on a Riemannian 2-torus we define a Busemann function for every ray. We use this concept to prove that the minimal average action A(α) is differentiable at irrational rotation numbersα while it is generically non-differentiable at rational rotation numbers (cf. also [18]). As an application of our results in the geometric framework we prove that a Riemannian 2-torus which has the same marked length spectrum as a flat 2-torus is actually isometric to this flat torus.

Mather Sets for Superlinear Dulling Equations

In this paper, we apply Aubry-Mather theory developed in recent years about monotone twist mappings to the study of the superlinear Duffing equation +g(x)=p(t), (0) where p(t)∈C 0 (R) is periodic with period 1 and g(x) satisfies the superlinearity condition Consequently, this gives descriptions of the global dynamical behavior, particularly periodic solutions and quasi-periodic solutions of a wide class of Eq. (0), not requiring high order smoothness assumption.