Existence and Non-existence of Length Averages for Foliations
Existence and Non-existence of Length Averages for Foliations
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DOI：
10.1007/s00220-019-03490-9
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发表时间：
2018-10
影响因子：
2.4
通讯作者：
Yushi Nakano;T. Yokoyama
中科院分区：
文献类型：
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作者：
Yushi Nakano;T. Yokoyama
自从 Ghys 等人的开创性工作以来，人们知道可以采用动力系统理论的几种方法来研究叶状结构。本文的目的是通过将动力系统理论中时间平均值的存在问题推广到叶状结构来研究叶状结构的复杂性：最近人们意识到，没有时间平均值的正勒贝格测度点集仅出现在复杂的动力系统中，例如具有异宿连接或同宿相切的动力系统。在本文中，我们将长度平均值的概念引入奇异叶状结构，并尝试收集具有/不具有长度平均值的有趣示例。特别是，我们证明，在准最小叶状集的温和条件下，对于任何余维可定向奇异叶状结构，在紧凑表面上没有退化奇点，长度平均值无处不在，这与表面流的时间平均值形成鲜明对比。
Since the pioneering work of Ghys et al., it has been known that several methods of dynamical systems theory can be adopted to study of foliations. Our aim in this paper is to investigate complexity of foliations, by generalising existence problem of time averages in dynamical systems theory to foliations: It has recently been realised that a positive Lebesgue measure set of points without time averages only appears for complicated dynamical systems, such as dynamical systems with heteroclinic connections or homoclinic tangencies. In this paper, we introduce the concept of length averages to singular foliations, and attempt to collect interesting examples with/without length averages. In particular, we demonstrate that length averages exist everywhere for any codimension oneorientable singular foliation without degenerate singularities on a compact surface under a mild condition on quasi-minimal sets of the foliation, which is in strong contrast to time averages of surface flows.