Nonperiodic leaves of codimension one foliations

TL;DR

This work addresses the Realization Problem for leaves of codimension-one foliations by constructing $5$-manifolds $Z_r$ that are not homeomorphic to any leaf of any transversely $C^2$ codimension-one foliation on a compact manifold, yet are realizable as proper leaves of $C^r$ codimension-one foliations for $0\le r<2$ and as proper leaves of $C^\infty$ codimension-two foliations. The approach centers on Ghys manifolds $M_\omega$ determined by bi-infinite sequences $\omega:\mathbb{Z}\to\{0,1\}$, and their perturbed variants $M_{p,\omega}$, with end-periodicity and almost-periodicity playing key roles in realizability. A crucial construction provides a $C^\infty$ codimension-one foliation on a compact $6$-manifold with boundary that, via a Denjoy diffeomorphism, yields an exceptional minimal set whose Ghys leaves are nonperiodic and realizable as proper leaves in low-regularity codimension-one foliations. Moreover, the paper shows that these nonperiodic Ghys leaves can be realized as repetitive leaves and analyzes the combinatorial structure of their associated sequences (notably Sturmian), while presenting simple modifications to realize broader classes of repetitive Ghys leaves. The results illuminate a sharp boundary between $C^2$ codimension-one realizability and realizability in lower regularity or higher codimension, and they raise open questions about end-repetitiveness and the existence of codimension-one foliations with finitely many nonrepetitive ends.

Abstract

In this work we exhibit examples of $5$-manifolds that are not homeomorphic to any leaf of any $C^2$ codimension one foliation of any compact $6$-manifold but are homeomorphic to (proper) leaves of some $C^1$ codimension one foliations and also to (proper) leaves of some $C^\inf$ codimension $2$ foliations. As far as we know, this is the first example of this nature. In addition, it is shown examples of $C^{r}$ codimension one foliations, $r\in[0,2)$, with a minimal invariant set whose leaves are pairwise nonhomeomorphic.

Figures (2)

• Figure 1: Initial foliation (the $S^2$ factor is collapsed) and the turbulized one along a suitably oriented loop $\gamma$. Observe the transverse paths $\delta^{\pm}$ joining the new boundary leaf with the older ones.
• Figure 2: The elements of the foliation $\text{$\mathcal{F}$}_6$ (the vertical dimension must be interpreted as a $T^2\times S^2$ factor at this step). Bold circles represent closed leaves obtained by turbulization and tunneling. The dotted arcs join boundary components of the exterior compact leaves (external and internal cylinders) indentified by the homeomorphism $f$. The transverse circles $S_-$ and $S_+$ can be seen depicted to the right and to the left respectively (dashed lines).

Theorems & Definitions (36)

• Definition 1.1: Nonleaf
• Theorem 1
• Definition 2.1: Ghys manifold
• Definition 2.2: Periodic end
• Definition 2.3: End periodic sequences
• Proposition 2.4
• Remark 2.5
• Lemma 2.6
• proof
• proof : Proof of Proposition \ref{['p:nonperiodic_Ghys']}