Study of foliations and discrete group actions

假设平面的同胚允许一个紧致极小集是连通的，而不是一个点。我们证明了补码恰好有两个不变的连通分支。我们还证明了旋转数是唯一确定的。给定紧致流形上的叶层和叶状黎曼度量，定义了该流形的调和测度。如果度量是双曲的，并且调和度量是遍历的，我们证明了度量的二分性。在沿着平面的里布面理的流动中，有一种被称为标准的流动。我们用它的动力学性质来描述它。

Assume that a homeomorphism of the plane admits a compact minimal set which is connected and not a point. We showed that there are exactly two invariant connected components of the complement. We also showed that the rotation number is uniquely determined. Given a foliation on a compact manifold and a leafwise Riemannian metric, there is defined a harmonic measure of the manifold. In case the metric is hyperbolic and the harmonic measure is ergodic, we showed that there is a dichotomy of the measures. Among flows along the Reeb foliation of the plane, there is one which is called standard. We characterized it by its dynamical property.