Existence and non-existence of length averages for foliations

TL;DR

This paper extends the concept of time averages to foliations, showing that length averages exist everywhere for certain singular foliations on compact surfaces, contrasting with the behavior of time averages in surface flows.

Contribution

It introduces the concept of length averages for singular foliations and demonstrates their universal existence under specific conditions, contrasting with time averages in dynamical systems.

Findings

Length averages exist everywhere for certain codimension one singular foliations.

Contrast between length averages in foliations and time averages in surface flows.

Provides examples of foliations with and without length averages.

Abstract

Since the pioneering work of Ghys, Langevin and Walczak among others, 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.