Minimizing orbits in the discrete Aubry–Mather model

Abstract

We consider a generalization of the Frenkel-Kontorova model in higher dimension leading to a new theory of configurations with minimal energy, as in Aubry's theory or in Mather's twist approach in the periodic case. We consider a one dimensional chain of particles and their minimizing configurations and we allow the state of each particle to possess many degrees of freedom. We assume that the Hamiltonian of the system satisfies some twist condition. The usual "total ordering" of minimizing configurations does not exist any more and new tools need to be developed. The main mathematical tool is to cast the study the minimizing configurations into the framework of discrete Lagrangian theory. We introduce forward and backward Lax-Oleinik problems and interpret their solutions as discrete viscosity solutions as in Hamilton-Jacobi methods. We give a fairly complete description of a particular class of minimizing configurations: the calibrated class. These configurations may be thought of as "ground states" obtained in the thermodynamic limit at temperature zero. We obtain, in particular, Mather's graph property or the non-crossing property of two calibrated configurations and the existence of a rotation number for most of the calibrated configurations.

Lyapunov exponents for conservative twisting dynamics: a survey

“…For globally positive diffeomorphism in higher dimension, Garibaldi & Thieullen prove the existence of globally minimizing orbits and measures in [13]. The results that they obtain are very similar to the ones that we recall in subsection 1.3.3 for Tonelli Hamiltonians.…”

Lyapunov Exponents of Minimizing Measures for Globally Positive Diffeomorphisms in All Dimensions

“…The general reference for what is contained in this section is the article of Garibaldi & Thieullen [15] and the results that they obtain are very similar to the ones obtained by A. Fathi in the setting of the time-continuous weak K.A.M. theory (see [14]).…”

“…The dynamics that we study here are contained in the ones that they study and that are called "ferromagnetic". In [15], a big part of the article deals with a Lagrangian function L : R n × R n → R that is defined by L(x, v) = S(x, x + v) (let us recall that S is a generating function for F ) and the action F is denoted by L by them. They prove the existence of a unique L ∈ R such that there exists two Z n -periodic continuous functions u − , u + : R n → R such that:…”

Convergence of discrete Aubry–Mather model in the continuous limit

“…As in [DFIZ16b] we characterize the limit of the unique fixed point of T τ,δ in terms of minimizing plan, Mañé potential. We recall these two definitions, see [GT11] for more details. We consider here the projection on T d × T d of objects that should be defined on R d × R d if cohomology is needed.…”