Continuity of the Peierls Barrier and Robustness of Minimal Laminations

This chapter deals with finite range monotone variational recurrence relationsover Z. Such recurrence relations arise in solid state physics, conservative lattice dynamics and as Hamiltonian twist maps. An example is the Frenkel-Kontorova lattice. AubryMather theory guarantees the existence of action-minimizing solutions of every rotation number ω ∈ R. For every ω these minimizers form ordered families, which for irrational rotation numbers can be either connected minimal foliations, or disconnected minimal laminations. A recurrence relation does not admit a minimal foliation of irrational rotation number ω, when the corresponding Peierls barrier function is non vanishing. We first show that the Peierls barrier function is sequentially continuous at irrational rotation numbers, with the help of continued fraction expansions. This allows us to prove a robustness statement for minimal laminations, under perturbations of local potentials that generate the variational recurrence relation. More precisely, the property that a local action has a positive Peierls barrier corresponding to a specific irrational rotation number ω, is open in the C2 topology. 156 Chapter 5: Continuity of the Peierls Barrier and Robustness of Minimal Laminations 5.