Aubry-Mather Theory Research Articles

Difference approximation to Aubry–Mather sets of the forced Burgers equation

Jauslin–Kreiss–Moser and W E made clear the connection between the Aubry–Mather theory and the inviscid forced Burgers equation with a -periodic forcing term and established the smooth approximation of -periodic entropy solutions of the PDE. This paper presents results of a difference approximation to the Aubry–Mather sets. We prove the convergence of the Lax–Friedrichs scheme for the -periodic entropy solutions. This result leads to difference approximations of the corresponding effective Hamiltonian and -periodic viscosity solutions of the Hamilton–Jacobi equation. We numerically construct the Aubry–Mather sets through the approximate entropy solutions, based on the dynamical properties of the Aubry–Mather sets.

Percival Lagrangian approach to the Aubry–Mather theory

We present some streamlined proofs of some of the basic results in the Aubry–Mather theory (existence of quasi-periodic minimizers, multiplicity results when there are gaps among minimizers) based on the study of hull functions. We present results in arbitrary number of dimensions.We also compare the proofs and results with those obtained in other formalisms.

Selection of Ground States in the Zero Temperature Limit for a One-Parameter Family of Potentials

For the subshift of nite type = f0; 1; 2g N we study the convergence and the selection at temperature zero of the Gibbs measure associated to a non-locally constant Holder potential which admits exactly two maximizing ergodic measures. These measures are Dirac measures at two dierent xed points and the potential is atter at one of these two xed points. We prove that there always is convergence but not necessarily to the Dirac measure at the point where the potential is the attest. This is contrary to what was expected in the light of the analogous problem in Aubry-Mather theory (1). This is also contrary to the nite range case where the equilibrium state converges to the equi-barycentre of the two Dirac measures. Moreover we emphasize the strange behavior of the Gibbs measure: the eigenmeasure selects one Dirac measure ( at the point where the potential is the attest) and the eigen- function selects the other one (at the point where the potential is the sharpest).

Ghost circles in lattice Aubry–Mather theory

Monotone lattice recurrence relations such as the Frenkel–Kontorova lattice, arise in Hamiltonian lattice mechanics, as models for ferromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multi-dimensional counterpart of monotone twist maps. Such recurrence relations often admit a variational structure, so that the solutions x : Z d → R are the stationary points of a formal action function W ( x ) . Given any rotation vector ω ∈ R d , classical Aubry–Mather theory establishes the existence of a large collection of solutions of ∇ W ( x ) = 0 of rotation vector ω. For irrational ω, this is the well-known Aubry–Mather set. It consists of global minimizers and it may have gaps. In this paper, we study the parabolic gradient flow d x d t = − ∇ W ( x ) and we will prove that every Aubry–Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called ‘ghost circle’. The existence of these ghost circles is known in dimension d = 1 , for rational rotation vectors and Morse action functions. The main technical result of this paper is therefore a compactness theorem for lattice ghost circles, based on a parabolic Harnack inequality for the gradient flow. This implies the existence of lattice ghost circles of arbitrary rotation vectors and for arbitrary actions. As a consequence, we can give a simple proof of the fact that when an Aubry–Mather set has a gap, then this gap must be filled with minimizers, or contain a non-minimizing solution.

Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

In this paper we are mainly concerned with existence and modulation of uniform sliding states for particle chains with damping γ and external driving force F. If the on-site potential vanishes, then for each F > 0 there exist trivial uniform sliding states xn(t) = nω + νt + α for which the particles are uniformly spaced with spacing ω > 0, the sliding velocity of each particle is ν = F/γ, and the phase α is arbitrary. If the particle chain with convex interaction potential is placed in a periodic on-site potential, we show under some conditions the existence of modulated uniform sliding states of the form $$x_n(t)=n\omega+\nu t+\alpha+u(n\omega+\nu t+\alpha),$$ where the modulation function u is periodic and unique up to phase. The conditions are that the system is overdamped and the driving force F exceeds some critical value Fd(ω) ≥ 0 depending on mean spacing ω. If \({F\in [0,F_d(\omega)]}\) , the system possesses a set of rotationally ordered equilibrium states for irrational ω, which can be described by a non-decreasing hull function, just as the case γ = F = 0, where Aubry-Mather theory applies to ground states. Meanwhile, we prove that Fd(ω) = 0, which was argued physically much earlier, if the hull function of ground states with irrational rotation number ω for F = 0 is continuous.

Class A spacetimes

We introduce class A spacetimes, i.e. compact vicious spacetimes (M, g) such that the Abelian cover \({(\overline{M}, \overline{g})}\) is globally hyperbolic. We study the main properties of class A spacetimes using methods similar to those introduced in Sullivan (Invent Math 36:225–255, 1976) and Burago (Adv Sov Math 9:205–210, 1992). As a consequence we are able to characterize manifolds admitting class A metrics completely as mapping tori. Further we show that the notion of class A spacetime is equivalent to that of SCTP (spacially compact time-periodic) spacetimes as introduced in Galloway (Comm Math Phys 96:423–429, 1984). The set of class A spacetimes is shown to be open in the C 0-topology on the set of Lorentzian metrics. As an application we prove a coarse Lipschitz property for the time separation of the Abelian cover. This coarse Lipschitz property is an essential part in the study of Aubry-Mather theory in Lorentzian geometry.

Destruction of invariant curves in the restricted circular planar three-body problem by using comparison of action

The classical principle of least action says that orbits of mechanical systems extremize action; an important subclass are those orbits that minimize action. In this paper we utilize this principle along with Aubry-Mather theory to construct (Birkhoff) regions of instability for a certain three-body problem, given by a Hamiltonian system of 2 degrees of freedom. We believe that these methods can be applied to construct instability regions for a variety of Hamiltonian systems with 2 degrees of freedom. The Hamiltonian model we consider describes dynamics of a Sun-Jupiter-comet system, and under some simplifying assumptions, we show the existence of instabilities for the orbit of the comet. In particular, we show that a comet which starts close to an orbit in the shape of an ellipse of eccentricity e=0.66 can increase in eccentricity up to e=0.96. In the sequels to this paper, we extend the result to beyond e=1 and show the existence of ejection orbits. Such orbits are initially well within the range of our solar system. This might give an indication of why most objects rotating around the Sun in our solar system have relatively low eccentricity.

Regions of applicability of Aubry-Mather Theory for non-convex Hamiltonian

Herman constructed an autonomous system of two degrees of freedom which says that in non-convex situations, oscillations do happen and Aubry-Mather Theory cannot apply (see the results due to W. F. Chen in 1992). In this paper, it is shown that although the orbits could visit a region far away from the initial point in phase space, they can only exist in some fixed regions in I = (I1, I2) plane. Moreover, Aubry-Mather Theory can be applied outside the regions.

Mañé's conjectures in codimension 1

AbstractWe prove Mañé's conjectures [9] in the context of codimension 1 Aubry‐Mather theory. © 2011 Wiley Periodicals, Inc.

A comparison principle for a Sobolev gradient semi-flow

We consider gradient descent equations for energy functionals of the type $S(u) = \frac{1}{2} _{L^2} + \int_{\Omega} V(x,u) dx$, where $A$ is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration. We consider the steepest descent equation for $S$ where the gradient is an element of the Sobolev space $H^{\beta}$, $\beta \in (0,1)$, with a metric that depends on $A$ and a positive number $\gamma >$sup$|V_{2 2}|$. We prove a weak comparison principle for such a gradient flow. We extend our methods to the case where $A$ is a fractional power of an elliptic operator, and provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional.

An introduction to the Aubry-Mather theory

This paper is a self-contained introduction to the Aubry-Mather theory and its connections with the theory of viscosity solutionsof Hamilton-Jacobi equations. Our starting point is Ma~ne's variationalapproach using holonomic measures [Mn96]. We present the Legendre-Fenchel-Rockafellar theorem from convex analysis and discuss the basictheory of viscosity solutions of rst order Hamilton-Jacobi equations.We apply these tools to study the Aubry-Mather problem following theideas in [EG01]. Finally, in the last section, we present a new proof ofthe invariance under the Euler-Lagrange ow of the Mather measuresusing ideas from calculus of variations.

Poncelet Pairs and the Twist Map Associated to the Poncelet Billiard

Consider a fixed differentiable curve $K$ (which is the boundary of a convex domain) and a family indexed by $\lambda\in [0,1]$ of differentiable curves $L = L_\lambda$ such that they are the boundary of convex (connected) domains $A_\lambda$. Suppose that for $\lambda_1<\lambda_2$ we have $A_{\lambda_1}\subset A_{\lambda_2}$. Then, the number of $n$-Poncelet pairs is given by $\frac{e (n)}{2}$, where $e(n)$ is the number of natural numbers $m$ smaller than $n$ and which satisfies mcd$(m,n)=1$. The curve $K$ does not have to be part of the family. In order to show this result we consider an associated billiard transformation and a twist map which preserves area. We use Aubry-Mather theory and the rotation number of invariant curves to obtain our main result. In the last section we estimate the derivative of the rotation number of a general twist map using some properties of the continued fraction expansion.

Ground States and Critical Points for Aubry–Mather Theory in Statistical Mechanics

We consider several models of networks of interacting particles and prove the existence of quasi-periodic equilibrium solutions. We assume (1) that the network and the interaction among particles are invariant under a group that satisfies some mild assumptions; (2) that the state of each particle is given by a real number; (3) that the interaction is invariant by adding an integer to the state of all the particles at the same time; (4) that the interaction is ferromagnetic and coercive (it favors local alignment and penalizes large local oscillations); and (5) some technical assumptions on the regularity speed of decay of the interaction with the distance. Note that the assumptions are mainly qualitative, so that they cover many of the models proposed in the literature. We conclude (A) that there are minimizing (ground states) quasi-periodic solutions of every frequency and that they satisfy several geometric properties; (B) if the minimizing solutions do not cover all possible values at a point, there is another equilibrium point which is not a ground state. These results generalize basic results of Aubry–Mather theory (take the network and the group to be ℤ). In particular, we provide with a generalization of the celebrated criterion of existence of invariant circles if and only iff the Peierls–Nabarro barrier vanishes.

Fast Transport Optimization for Monge Costs on the Circle

Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter. This theory is applied to a transportation problem arising in image processing: for two sets of point masses on the circle, both of which have the same total mass, find an optimal transport plan with respect to a given cost function satisfying the Monge condition. In the circular case the sorting strategy fails to provide a unique candidate solution and a naive approach requires a quadratic number of operations. For the case of $N$ real-valued point masses we present an O(N |log epsilon|) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations.

Randomly perturbed dynamical systems and Aubry-Mather theory

We give a new PDE proof of a Freidlin-Wentzell theorem about the exit points from a domain of a random process, obtained by perturbing a dynamical system through the addition of a small noise. The relevant part of the analysis concerns an Hamilton-Jacobi equation, coupled with a Neumann boundary condition, which does not possess any strict subsolution. A metric method based on the introduction of an intrinsic length is adopted, and a notion of Aubry set, adjusted to the setting, is given.

A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations

We discuss an Aubry–Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators.We show that for certain PDEs and ΨDEs with periodic coefficients and a variational structure it is possible to find quasi-periodic solutions for all frequencies. This results also hold under a generalized definition of periodicity that makes it possible to consider problems in covers of several manifolds, including manifolds with non-commutative fundamental groups.An abstract result will be provided, from which an Aubry–Mather-type theory for concrete models will be derived.

Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations

We provide a general framework of inequalities induced by theAubry-Mather theory of Hamilton-Jacobi equations. This frameworkdeals with a sufficient condition on functions $f\inC^1(\mathbb R^n)$ and $g\in C(\mathbb R^n)$ inorder that $f-g$ takes its minimum over $\mathbb R^n$ on the set{$x\in \mathbb R^n |Df(x)=0$}. As an application of thisframework, we provide proofs of the arithmetic mean-geometric meaninequality, Hölder's inequality and Hilbert's inequality in aunified way.

The large-time behavior of solutions of the Cauchy-Dirichlet problem for Hamilton-Jacobi equations

We investigate the large-time behavior of viscosity solutions of the Cauchy-Dirichlet problem (CD) for Hamilton-Jacobi equations on bounded domains. We establish general convergence results for viscosity solutions of (CD) by using the Aubry-Mather theory.

On the Aubry–Mather theory for symbolic dynamics

AbstractWe propose a new model of ergodic optimization for expanding dynamical systems: the holonomic setting. In fact, we introduce an extension of the standard model used in this theory. The formulation we consider here is quite natural if one wants a meaning for possible variations of a real trajectory under the forward shift. In other contexts (for twist maps, for instance), this property appears in a crucial way. A version of the Aubry–Mather theory for symbolic dynamics is introduced. We are mainly interested here in problems related to the properties of maximizing probabilities for the two-sided shift. Under the transitive hypothesis, we show the existence of sub-actions for Hölder potentials also in the holonomic setting. We analyze then connections between calibrated sub-actions and the Mañé potential. A representation formula for calibrated sub-actions is presented, which drives us naturally to a classification theorem for these sub-actions. We also investigate properties of the support of maximizing probabilities.

Minimizers of Dirichlet functionals on the n-torus and the weak KAM theory

Given a probability measure μ on the n-torus Tn and a rotation vector P∈Rn, we ask whether there exists a minimizer to the integral ∫Tn|∇ϕ+P|2dμ. This problem leads, naturally, to a class of elliptic PDE and to an optimal transportation (Monge–Kantorovich) class of problems on the torus. It is also related to higher dimensional Aubry–Mather theory, dealing with invariant sets of periodic Lagrangians, and is known as the “weak-KAM theory”.