Study of foliations and discrete group actions

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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.