A study of minimal sets in differentiable flows and foliations

在本研究中，我们对三维流形上的可微流进行了测度论和拓扑学研究。通过努力更具体地理解齐默尔关于群作用的遍历理论，我们为发现流动动力学的新方面奠定了基础。我们的方法如下：给定一个流，我们将它提升到它的射影法丛的全空间。然后，通过分解全空间上的一个不变测度，我们得到了每个纤维上的一个测度。我们称该度量族为不变纤维测度。不变纤维测度为我们提供了一种测量流体扭转现象的方法。在这项研究中，我们建立了一些关于不变纤维测度的基本定理，并对它们进行了细致而易于理解的证明。我们还利用一个不变的纤维测度得到了Ruelle不变量的一个新的表示式，并且，作为这个公式的一个应用，我们证明了如下结果：考虑保持二维圆盘的微分同态射的测度群和定义在这个群上的Ruelle映射。然后，当我们把它限制在一个可服从的子群时，映射就变成了同态。2000年，我们在华沙举行的国际叶学研讨会上就这一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.