Length averages for codimension one foliations

TL;DR

This paper explores the relationship between length averages and ball averages in codimension one foliations, introducing a new group-based mechanism to produce irregular average behavior and constructing novel examples with non-existent length averages.

Contribution

It introduces a new group-structure-based method to generate irregular length average behavior in codimension one foliations, with geometric realizations on compact manifolds.

Findings

Existence of codimension one foliations with non-existent length averages on open sets.

A new mechanism based on non-amenable groups causes irregular ball average behavior.

Construction of smooth foliations with prescribed average properties.

Abstract

In this paper we study geometrical and dynamical properties of codimension one foliations, by exploring a relation between length averages and ball averages of certain group actions. We introduce a new mechanism, which relies on the group structure itself, to obtain irregular behavior of ball averages for certain non-amenable group actions. Several geometric realization results show that any such groups can appear connected with the topology of leaves which are connected sums of plugs with a special geometry, namely nearly equidistant boundary components. This is used to produce the first examples of codimension one $\mathcal C^\infty$ regular foliations on a compact Riemannian manifold $M$ for which the length average of some continuous function does not exist on a non-empty open subset of $M$.