Mather Sets Research Articles

On 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.

Mather measures selected by an approximation scheme

In this note, we will identify Mather measures selected by Evans’s variational approach in 1-d. Motivated by the low dimension case, we conjecture that Evans’s approximation scheme might catch the whole Mather set in all dimensions. We also discuss the connection with another approximation scheme in the works of Anantharaman, Evans and Gomes.

On the boundedness of solutions to a nonlinear singular oscillator

We study a second order scalar equation of the form x′′ + V′(x) = p(t), where p is a π-perodic function and V is a singular potential. We give sufficient conditions on V, p ensuring that all solutions are bounded; we prove the existence of Aubry–Mather sets as well.

The experimental localization of Aubry–Mather sets using regularization techniques inspired by viscosity theory

We provide a new method for the localization of Aubry-Mather sets in quasi-integrable two-dimensional twist maps. Inspired by viscosity theories, we introduce regularization techniques based on the new concept of "relative viscosity and friction," which allows one to obtain regularized parametrizations of invariant sets with irrational rotation number. Such regularized parametrizations allow one to compute a curve in the phase-space that passes near the Aubry-Mather set, and an invariant measure whose density allows one to locate the gaps on the curve. We show applications to the "golden" cantorus of the standard map as well as to a more general case.

Hyperbolic periodic orbits and Mather sets in certain symmetric cases

We consider a C∞ Lagrangian function L : T ∗M → R which is superlinear and convex in the fibers and has one antisymplectic symmetry. We prove that: • in every energy level strictly above the critical one, there exists a Mather set which is the union of some periodic orbits; • for L generic, these orbits are hyperbolic; • on the torus, these orbits have one homoclinic orbit.

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.

Existence of infinitely many homoclinic orbits to Aubry sets for positive definite Lagrangian systems

In this paper, we study the problem of homoclinic orbits to Aubry sets for time-periodic positive definite Lagrangian systems. We show that there are infinitely many homoclinic orbits to some Aubry set under the conditions that the associated Mather set is uniquely ergodic and the first relative homology group of the projection of this Aubry set is nonzero.

On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics

We study the zero-temperature limit for Gibbs measures associated to Frenkel-Kontorova models on (R d) Z/Z d . We prove that equilibrium states concentrate on configurations of minimal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lyapunov exponents, a bit like in the Ruelle-Pesin inequality. Then we transpose the result to certain continuous-time stationary stochastic processes associated to the viscous Hamilton-Jacobi equation. As the viscosity vanishes, the invariant measure of the process concentrates on the so-called Mather set of classical mechanics, and must, in addition, minimize the gap in the Ruelle-Pesin inequality.

Dynamics Around Minimal Hyperbolic Torus in Hamiltonian Systems

By investigating the topological properties of action minimal sets near a minimal hyperbolic torus in nearly integrable Hamiltonian system, a sufficient condition of existence of connections between different Mather sets is obtained.

Aubry-Mather Theory and Hamilton-Jacobi Equations

We review and extend some results of Jauslin, Kreiss, Moser and Weinan E on the smooth approximation of Mather sets.

Large time behavior of fronts governed by eikonal equations

Motivated by a model of solid combustion in heterogeneous media, we investigate the time-asymptotic behaviour of flame fronts evolving with a periodic space-dependent normal velocity; using the so-called \`\`level set approach'' we are led to study the large time behaviour of solutions of eikonal equations. We first provide a general approach which shows that the asymptotic normal velocity of such a flame front depends only on its normal direction and is given by the homogenized Hamiltonian of the eikonal equation. Then we turn to a more precise study of the asymptotic behaviour of the flame front when the initial front is a graph of a periodic function: in this case, the front moves asymptotically with a constant normal velocity and we are able to prove that, in coordinates moving with this constant velocity, the front has a time-periodic asymptotic behaviour in the following two cases: (i) when there is a straight line of maximal speed, and (ii) when the space dimension is 2. These results are obtained by using homogenization, control theory and dynamical systems methods (Aubry--Mather sets).

Perturbation Theory for Viscosity Solutions of Hamilton--Jacobi Equations and Stability of Aubry--Mather Sets

In this paper we study the stability of integrable Hamiltonian systems under small perturbations, proving a weak form of the KAM/Nekhoroshev theory for viscosity solutions of Hamilton--Jacobi equations. The main advantage of our approach is that only a finite number of terms in an asymptotic expansion are needed in order to obtain uniform control. Therefore there are no convergence issues involved. An application of these results is to show that Diophantine invariant tori and Aubry--Mather sets are stable under small perturbations.

Regularity theory for Hamilton–Jacobi equations

The objective of this paper is to discuss the regularity of viscosity solutions of time independent Hamilton–Jacobi Equations. We prove analogs of the KAM theorem, show stability of the viscosity solutions and Mather sets under small perturbations of the Hamiltonian.

On the existence of a new type of periodic and quasi-periodic orbits for twist maps of the torus

We prove that for a large and important class of C1 twist maps of the torus periodic and quasi-periodic orbits of a new type exist, provided that there are no rotational invariant circles (RICs). These orbits have a non-zero `vertical rotation number' (VRN), in contrast to what happens to Birkhoff periodic orbits and Aubry–Mather sets. The VRN is rational for a periodic orbit and irrational for a quasi-periodic. We also prove that the existence of an orbit with a VRN = a>0, implies the existence of orbits with VRN = b, for all 0<b<a. In this way, related to a generalized definition of rotation number, we characterize all kinds of periodic and quasi-periodic orbits a twist map of the torus can have. As a consequence of the previous results we obtain that a twist map of the torus with no RICs has positive topological entropy, which is a very classical result. At the end of the paper we present some examples, like the standard map, such that our results apply.

Linear programming interpretations of Mather's variational principle

We discuss some implications of linear programming for Mather theory [13-15] and its finite dimensional approximations. We find that the complementary slackness condition of duality theory formally implies that the Mather set lies in an n -dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [5-8].

On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces

Given a rational homology classh in a two dimensional torusT2, we show that the set of Riemannian metrics inT2 with no geodesic foliations having rotation numberh isCk dense for everyk ∈ N. We also show that, generically in theC2 topology, there are no geodesic foliations with rational rotation number. We apply these results and Mather's theory to show the following: let (M, g) be a compact, differentiable Riemannian manifold with nonpositive curvature, if (M, g) satisfies the shadowing property, then (M, g) has no flat, totally geodesic, immersed tori. In particular,M has rank one and the Pesin set of the geodesic flow has positive Lebesgue measure. Moreover, if (M, g) is analytic, the universal covering ofM is a Gromov hyperbolic space.

On the minimal action function of autonomous lagrangians associated to magnetic fields

In this paper we show the existence of a plateau for the minimal action function associated with a model for a particle under the influence of a magnetic field (Hall effect). We describe the structure of the Mather sets, that is, sets that are supports of minimizing measures for the corresponding autonomous Lagrangian.This description is obtained by constructing a twist map induced by the first return map defined on a certain transversal section in a fixed level of energy.

On a theorem by Mather and Aubry-Mather sets for planar Hamiltonian systems

A result due to Mather on the existence of Aubry-Mather sets for superlinear positive definite Lagrangian systems is generalized in one-dimensional case. Applications to existence of Aubry-Mather sets of planar Hamiltonian systems are given.

A Variational Principle Associated to Positive Tilt Maps

In this paper we establish a variational principle for positive tilt maps. This implies that the Aubry–Mather set exists for positive tilt maps and most of the theory developed by Mather for twist maps now applies for positive tilt maps.

Homoclinic trajectories of invariant sets of Hamiltonian systems

We use variational methods for proving the existence of homoclinic trajectories for Mather sets of positive definite time-periodic Hamiltonian systems.