Embedding solenoids in foliations

TL;DR

This paper demonstrates the existence of smooth embeddings of solenoids within foliations, showing how certain conditions lead to complex minimal sets and complementing known stability results in foliation theory.

Contribution

It introduces a method to embed solenoids in smooth foliations under specific conditions, expanding understanding of minimal sets and foliation stability.

Findings

Existence of solenoidal minimal sets in certain foliations

Construction of foliations with uncountably many non-homeomorphic solenoids

Complementary to stability results by Reeb, Thurston, Langevin, and Rosenberg

Abstract

In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H^1(L;R)= 0 and if the foliation is a product foliation in some saturated open neighbourhood U of L, then there exists a foliation F' on M which is C^1-close to F, and F' has an uncountable set of solenoidal minimal sets contained in U that are pair wise non-homeomorphic. If H^1(L;R) is not 0, then it is known that any sufficiently small perturbation of F contains a saturated product neighbourhood. Thus, our result can be thought of as an instability result complementing the stability results of Reeb, Thurston and Langevin and Rosenberg.