Embedding solenoids in foliations

摘要

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) not equal to 0 and if the foliation is a product foliation in some saturated open neighborhood 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 pairwise non-homeomorphic. If H~1(L; R) is 0, then it is known that any sufficiently small perturbation of F contains a saturated product neighborhood. Thus, our result can be thought of as an instability result complementing the stability results of Reeb, Thurston and Langevin and Rosenberg.