2,034 research outputs found
Homogeneous matchbox manifolds
We prove that a homogeneous matchbox manifold of any finite dimension is
homeomorphic to a McCord solenoid, thereby proving a strong version of a
conjecture of Fokkink and Oversteegen. The proof uses techniques from the
theory of foliations that involve making important connections between
homogeneity and equicontinuity. The results provide a framework for the study
of equicontinuous minimal sets of foliations that have the structure of a
matchbox manifold.Comment: This is a major revision of the original article. Theorem 1.4 has
been broadened, in that the assumption of no holonomy is no longer required,
only that the holonomy action is equicontinuous. Appendices A and B have been
removed, and the fundamental results from these Appendices are now contained
in the preprint, arXiv:1107.1910v
Embedding solenoids in foliations
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
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