Depinning Forces And Denjoy Minimal Sets Of Monotone Recurrence Relations

We investigate the monotone recurrence relation generated by a function S of r+1variables, which is called generating function. For the monotone recurrence relation, itis significant to know the existence or non-existence of foliations for they correspondto the invariant circles of a twist map with generating function S in case r=1.To give a criterion of existence or non-existence of foliations, we give two quantities:positive and negative depinning forces for each rotation number. We prove that thereis a foliation with irrational rotation number if and only if the corresponding positiveor negative depinning force vanishes. For a rational number ω=q/p, the positive ornegative depinning force vanishes if and only if there is a foliation contained in the(p, q)-periodic Birkhof configuration set.It is well known that all Aubry-Mather sets of a twist map with irrational number ωlie on an invariant circle with rotation number ω. We prove that the similar conclusionholds for the monotone recurrence relation: There is no Denjoy minimal set withirrational rotation number outside the foliation with the same rotation number. Weprove that the number of Denjoy minimal sets contained in foliation is no more thanone. Hence, a necessary condition for the existence of distinct Denjoy minimal sets ofirrational rotation number is the foliation of the same rotation number is disintegrating.We shall show the condition is also a sufcient one. The foliation with irrationalrotation number for the monotone recurrence relation consists of Birkhof minimizers.Hence, the conclusion can be summarized as: There are many Denjoy minimal sets ofirrational rotation number ω if and only if the Birkhof minimizers of rotation numberω can not form a foliation.