Rotation Sets For Monotone Recurrence Relations

We consider rotation sets for monotone recurrence relations.For the monotone recurrence relation determined by a potential function with certain conditions,the Aubry-Mather theory tells us that for each ??R,there is a Birkhoff solution with rotation number ?.Hence in this case,the rotation set is R.In this article,we inves-tigate generalized monotone recurrence relations.We have two main results:Firstly,if the monotone recurrence relation has a solution with bounded action,then it has a Birkhoff solution.Secondly,we will discuss some properties of the rotation set 7Z for the monotone recurrence relation.To obtain the first result,we need two steps.In the first step,we prove that if the monotone recurrence relation has a solution with bounded action,then it has a soluton with bounded action and rotation number.We prove in the second step that if the monotone recurrence relation has a solution with bounded action and rotation number,then it has a Birkhoff solution with the same rotation number.With respect to the second result,we show relying on the first result that R is closed and discuss the consequences of rational or irrational rotation numbers.