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Communications in Mathematical Physics

, Volume 372, Issue 2, pp 367–383 | Cite as

Existence and Non-existence of Length Averages for Foliations

  • Yushi Nakano
  • Tomoo YokoyamaEmail author
Article
  • 74 Downloads

Abstract

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 one \(\mathscr {C}^1 \) orientable 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.

Notes

Acknowledgements

We are deeply grateful to anonymous reviewers for many suggestions, all of which substantially improved the paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsTokai UniversityHiratukaJapan
  2. 2.Department of MathematicsKyoto University of Education/JST PrestoFushimi-kuJapan

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