【作者】 王亚南；

【导师】 秦文新；

【作者基本信息】 苏州大学 ， 应用数学， 2014， 博士

【摘要】 我们考虑由r+1元生成函数S决定的单调回复关系。当r=1时,存在一个以S为生成函数的单调扭转映射。叶状结构(Foliation)对应于这个单调扭转映射的不变曲线。在本文中,我们将给出判断叶状结构是否存在的标志量：正向脱钉力和负向脱钉力。具有无理旋转数的叶状结构存在当且仅当相应的正向或者负向脱钉力消失。对于有理数ω=q/p,正向或者负向脱钉力消失当且仅当在(p,q)周期的Birkhoff构型集中存在叶状结构。如果某个单调扭转映射存在无理旋转数ω的不变曲线,则以ω为旋转数的Aubry-Mather集都在这条不变曲线上。我们证明在单调回复关系中也有类似的结果：如果存在某个无理旋转数的叶状结构,则在它之外没有同一旋转数的Denjoy极小集。实际上,无理旋转数的叶状结构上,Denjoy极小集最多只有一个。所以,不存在以ω为旋转数的叶状结构是存在多个ω旋转数的Denjoy极小集的必要条件。我们还将证明这个条件也是充分的。实际上,单调回复关系中无理旋转数的叶状结构是由Birkhoff的最小能量构型构成的。因此,上面的结论可以总结为：对无理数ω,存在多个ω旋转数的Denjoy极小集的充分必要条件是以ω为旋转数的最小能量构型不形成叶状结构。更多还原

【Abstract】 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.更多还原