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Length averages for codimension one foliations

Masayuki Asaoka, Yushi Nakano, Paulo Varandas, Tomoo Yokoyama

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 $M$.

Length averages for codimension one foliations

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 regular foliations on a compact Riemannian manifold for which the length average of some continuous function does not exist on a non-empty open subset of .

Paper Structure

This paper contains 25 sections, 16 theorems, 120 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Irregular behavior of length averages
  4. Mechanism: length averages and ball averages
  5. Proof of Theorem \ref{['mainthm:1']}
  6. Suspension of an interval exchange transformation with a residual irregular set
  7. A basic foliated surface
  8. Removal of auxiliary boundary
  9. The foliated compact surface of Theorem \ref{['mainthm:1']}
  10. Non-existence of length averages
  11. Proof of Proposition \ref{['prop:2.2']} and Theorem \ref{['prop:pieces']}
  12. Proof of Proposition \ref{['prop:2.2']}
  13. A pair of pants with equidistant boundary components
  14. A compact surface with equidistant boundary components
  15. Thin legs
  16. ...and 10 more sections

Key Result

Theorem 1

There are a codimension one $\mathcal{C}^\infty$ non-degenerate singular foliation $\mathcal{F}$ on a compact surface $M$ and a continuous function on $M$ whose length averages for $\mathcal{F}$ do not exist on a residual subset of $M$.

Figures (13)

  • Figure 1: A leaf of the foliation formed as a suspension of pants with nearly equidistant boundaries over the group action by the free group $F_2$ (on the left); Cayley graph of $F_2$ (on the right)
  • Figure 2: (a) $\mathcal{Y}_0$, $\mathcal{Y}_1$; (b) $(M_c',\mathcal{F}_c')$; (c) $xy$-projection of $(M_{[a,b]},\mathcal{F}_{[a,c]})$
  • Figure 3: Smooth curves for a smooth doubling of $M_c'$
  • Figure 4: Compact surface $X_0$ with corners with equidistant property
  • Figure 5: Deformation of $X_0$ to obtain $X_0'$
  • ...and 8 more figures

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Corollary 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 3
  • Theorem 4
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • ...and 18 more