The Depinning Force For Monotone Recurrence Relations

Monotone recurrence relations determine a class of dynamical systems on high dimensional cylinder which generalizes the class of monotone twist maps on two-dimensional cylinder. A solution of a monotone recurrence relation corresponds to an equilibrium of the generalized Frenkel-Kontoroval(F-K) model. The Aubry-Mather theory for monotone recurrence relations says that for each ω ∈ R there is a Birkhoff minimizer with rotation number ω.Whether the set of all Birkhoff minimizers of rotation number ω form a foliation is a question like whether there is an invariant circle with rotation number ω for a monotone twist map. In this paper we give a criterion for the existence of minimal foliations and study its continuity.Depinning force Fd(ω), depending on rotation numbers, is a critical value of external driving force for the FK model, under which there are Birkhoff equilibria and hence the system is pinned, and above which there are no Birkhoff equilibria of rotation number ωand the system is sliding. We show that for irrational ω the set of all Birkhoff minimizers with rotation number ω forms a foliation if and only if Fd(ω) = 0. If ω = p/q is rational,then Fd( p/q) = 0 if and only if the set of( p, q)-periodic Birkhoff minimizers constitutes a foliation. Moreover, we show that Fd(ω) is continuous at irrational points and H?lder continuous at Diophantine points. Finally we will prove that the Depinning force depends continuously on parameters and hence the collection of local potentials that do not admit a foliation of a specific rotation number is open in the C2-topology.