Exceptional minimal sets of codimension two on flows

1. 1995年，McSwiggen构造了一个具有局部连通例外极小集的曲面仿射。他插入了一个圆柱孔沿着一片叶子的强烈不稳定叶理的三维双曲toral自同构。我们从三维射影Anosov同态中构造了一个新的例子。2.曲面上群作用的例外极小集特别地，我们研究了球面上群作用的例外极小集。在双曲群的情况下，众所周知，存在许多例外极小集的形状。然后我们考虑了什么是作用在具有复杂极小集的曲面上的最简单群的问题。证明了幂零群作用的例外极小集不同胚于Siepinski地毯。3.三维流形的最小流在最小流上，我们研究了它们的无穷小流和射影流，得到了如下动力学性质：（1）利用Zimmer理论得到了射影流的不变纤维测度，并推导了原流的若干拓扑性质。(2)We研究了投影流恰好有两个极小集的极小流，证明了如果它的投影流有三个以上的极小集，它与无理流是拓扑等价的。此外，如果恰好有两个极小集，则存在分离极小集的丛截面。（3）Contreras得到了弱部分双曲流的重要结果。我们把他的理论应用到射影流上，证明了当射影流恰好有两个极小集时，射影流的“上”轨道和“下”轨道的存在性。

1.Construction of surface diffeomorphisms with exceptional minimal setsIn 1995, McSwiggen constructed a surface diffeomorphism with a locally connected exceptional minimal set. He inserted a cylindrical hole along a leaf of the strong unstable foliation of a 3-dimensional hyperbolic toral automorphism. We constructed a new example from a 3-dimensional projectively Anosov diffeomorphism.2.Exceptional minimal sets for group actions on surfacesIn particular, we studied exceptional minimal sets of group actions on the sphere. In the case of hyperbolic groups, it is well known that there are many shapes of exceptional minimal sets. Then we consider the problem what is the easiest group which acts on a surface with a complicated minimal sets. We showed that the exceptional minimal set of an action of a nilpotent group is not homeomorphic to the Siepinski carpet. Furthermore, in the case of the fundamental groups of torus bundles over the circle, we obtain some partial results.3.Minimal flows of 3-dimensional manifoldsOn minimal flows, we investigated their infinitesimal flows and projective flows, we obtain the following dynamical properties :(1)We used Zimmer's theory in order to obtain invariant fiber measures of projective flows, and deduced several topologaical properties of the original flows.(2)We investigated minimal flows whose projective flows have exactly two minimal sets, and we showed that it is topologically equivalent to an irrational flow if its projective flow has more than three minimal sets. Furthermore, there is a bundle section separating the minimal sets if there are exactly two minimal sets.(3)Contreras obtained important results on weakly partially hyperbolic flows. We used his theory to the projective flows and proved the existence of the "upper" orbits and the "lower" orbits of the projective flow if it has exactly two minimal sets.