Embedding solenoids in foliations

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.