Continuity of depinning force

The depinning force Fd(ω), depending on rotation numbers, is a critical value of the 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 ω and the system is sliding. We show that all Birkhoff minimizers with irrational rotation number ω form a foliation if and only if Fd(ω)=0. If ω=p/q is rational, then Fd(p/q)=0 if and only if all (p,q)-periodic Birkhoff minimizers constitute a foliation. Moreover, we show that Fd(ω) is continuous at irrational ω and Hölder continuous at Diophantine points, and it also depends continuously on system parameters.