Stable rotation numbers and Birkhoff ordered circles for monotone recurrence relations

We study the rotation sets and dynamics of a class of twist maps on the high-dimensional cylinder generated by solutions of monotone recurrence relations. A rotation number is said to be stable if it lies in the rotation sets of all sufficiently close systems. We show that an irrational stable rotation number implies positive topological entropy. As a consequence, each non-wandering orbit has bounded deviations from the corresponding rigid rotation provided the system has zero topological entropy. Furthermore, by introducing Birkhoff ordered circles, we prove Boyland's conjecture in our setting, investigate the realization of rotation numbers defined by invariant measures, and obtain a result of uniformly bounded deviations from the rotation set.