Exceptional minimal sets of codimension two on flows

点击翻译按钮获取中文摘要

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.