Monotone variational recurrence relations such as the Frenkel-Kontorova lattice, arise in solid state physics, conservative lattice dynamics and as Hamiltonian twist maps. For such recurrence relations, Aubry-Mather theory guarantees the existence of solutions of every rotation number. They are the action minimizers that constitute the Aubry-Mather set. When the rotation number is irrational, the Aubry-Mather set is either connected or a Cantor set. A connected Aubry-Mather set is called a minimal foliation. In the case of twist maps, it describes an invariant circle, while in solid state physics it corresponds to a continuum of ground states. A Cantor Aubry-Mather set is called a minimal lamination.
In this paper we prove that when the rotation number of a minimal foliation is either rational or easy to approximate by rational numbers, then the foliation can be destroyed into a lamination by an arbitrarily small smooth perturbation of the recurrence relation. This generalizes a theorem of Mather for twist maps to general recurrence relations.
Recurrence, pointwise almost periodicity and orbit closure relation for flows and foliations
本文给出了闭连通曲面$M$的连续向量场$w$的常返性的一个刻划。以下是等价的：1）$w$是逐点递归的。2)$w$是逐点概周期的。3)$w$是最小的或逐点周期的。此外，如果$w$是正则的，则以下等价：1）$w$是逐点递归的。2)$w$最小，或者轨道空间$M/w$为$[0,1]$或$S^1$。3)$R$是闭合的（其中$R := \{(x,y) \in M \times M \mid y \in \bar{O(x)} \}$是轨道闭合关系）。另一方面，我们证明了紧流形上余维为1的叶理$\mathcal{F}$的下列等价性：1）$\mathcal{F}$是点态概周期的. 2)$\mathcal{F}$是最小的或紧凑的。3)$\mathcal{F}$是$R$ -封闭的。证明了如果紧致可度量化空间上的叶状空间是极小的，或者是紧致的且没有无穷完整性的，则它是$R$ -闭的.
