可重正化抛物流各态历经积分的渐近性
Asymptotic behaviour of ergodic integrals of `renormalizable' parabolic flows
摘要：
十年前，A。Zorich通过区间交换变换的计算机实验，发现了高亏格曲面上一般非精确Hamilton流的遍历积分的一些引人注目的新幂律。在Zorich后来的工作和M. Kontsevich，Zorich和Kontsevich能够解释绝大多数Zorich的发现有关的遍历论Teichmüller流的模空间的阿贝尔微分。在这篇文章中，我们概述了一个推广的Kontsevich-Zorich框架的一类“renormalizable”流的“伪齐次”空间。我们为这样的流构造一个“重整化动力学”的适当的“模空间”，推广Teichmüller流。如果一个流是可重整化的，并且光滑函数的空间是“稳定的”，在这个意义上，光滑函数上的李导数算子具有封闭的值域，那么遍历积分的行为至少在原则上可以根据流的轨道叶理的“基本流”丛上的“重整化上循环”的奥塞莱德克分解来分析。这种方法是由作者对Kontsevich-Zorich定理的证明所提出的，并在与L。Flaminio，证明了Zorich现象推广到几个经典的例子保体积，唯一遍历，抛物流，如horocycle流和幂零流齐次三维流形。
英文摘要：
Ten years ago A. Zorich discovered, by computer experiments on interval exchange transformations, some striking new power laws for the ergodic integrals of generic non-exact Hamiltonian flows on higher genus surfaces. In Zorich's later work and in a joint paper authored by M. Kontsevich, Zorich and Kontsevich were able to explain conjecturally most of Zorich's discoveries by relating them to the ergodic theory of Teichmüller flows on moduli spaces of Abelian differentials. In this article, we outline a generalization of the Kontsevich-Zorich framework to a class of `renormalizable' flows on `pseudo-homogeneous' spaces. We construct for such flows a `renormalization dynamics' on an appropriate `moduli space', which generalizes the Teichmüller flow. If a flow is renormalizable and the space of smooth functions is `stable', in the sense that the Lie derivative operator on smooth functions has closed range, the behaviour of ergodic integrals can be analyzed, at least in principle, in terms of an Oseledec's decomposition for a `renormalization cocycle' over the bundle of `basic currents' for the orbit foliation of the flow. This approach was suggested by the author's proof of the Kontsevich-Zorich conjectures and it has since been applied, in collaboration with L. Flaminio, to prove that the Zorich phenomenon generalizes to several classical examples of volume preserving, uniquely ergodic, parabolic flows such as horocycle flows and nilpotent flows on homogeneous 3-manifolds.
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