A study of minimal sets in differentiable flows and foliations

本文对三维流形上的可微流进行了测量理论和拓扑研究。通过努力更具体地理解齐默关于群体行为的遍历理论，我们为发现流动动力学的新方面奠定了基础。我们的方法是这样的：给定一个流，我们将它提升到它的投影法线束的总空间。然后，通过分解总空间上的一个不变测度，得到每个光纤上的一个测度。我们称由此得到的测度族为不变纤维测度。不变纤维测量为我们提供了一种测量流动扭转现象的装置。本文建立了一些关于不变光纤测度的基本定理，并给出了详细的证明。我们还利用一个不变量纤维测度得到了Ruelle不变量的一个新的表示公式，并且作为该公式的应用，我们得到了以下结果：考虑二维圆盘的保持微分同态的测度群和定义在这个群上的Ruelle映射。当我们将映射限制在一个可调整的子群中时，映射就成为同态映射。2000年，我们在华沙举行的一个关于叶子的国际研讨会上就这个结果做了一个小时的演讲。

In this research we made measure theoretic and topological investigation of differentiable flows on 3-dimensional manifolds. Through an effort to understand more concretely Zimmer's ergodic theory on group actions, we established a basis of finding new aspects of dynamics of flows. Our method is as follows : Given a flow we lift it to the total space of its projectivized normal bundle. Then, by disintegrating an invariant measure on the total space, we obtain a measure on each fiber. We Call the family of measures thus obtained an invariant fiber measure. An invariant fiber measure provides us with a setup for measuring the twisting phenomenon of the flow.In this research we established some fundamental theorems on invariant fiber measures, giving careful and accessible proofs for all of them. We also obtained a new representation formula of the Ruelle invariant by making use of an invariant fiber measure, Moreover, as an application of this formula, we showed the following result : Consider the group of measure preserving diffeomorphisms of the 2-dimensional disk and the Ruelle map defined on this group. Then the map becomes a homomorphism whenever we restrict it to an amenable subgroup.We gave a one-hour talk on this result at an international symposium on foliations held at Warsaw in 2000.