Lipshitz matchbox manifolds

A matchbox manifold is a connected, compact foliated space with totally disconnected transversals; or in other notation, a generalized lamination. It is said to be Lipschitz if there exists a metric on its transversals for which the holonomy maps are Lipschitz. Examples of Lipschitz matchbox manifolds include the exceptional minimal sets for $C^1$-foliations of compact manifolds, tiling spaces, the classical solenoids, and the weak solenoids of McCord and Schori, among others. We address the question: When does a Lipschitz matchbox manifold admit an embedding as a minimal set for a smooth dynamical system, or more generally for as an exceptional minimal set for a $C^1$-foliation of a smooth manifold? We gives examples which do embed, and develop criteria for showing when they do not embed, and give examples. We also discuss the classification theory for Lipschitz weak solenoids.Comment: The paper has been significantly revised, with several proofs correcte

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

The dynamics of generic Kuperberg flows

In this work, we study the dynamical properties of Krystyna Kuperberg's aperiodic flows on $3$-manifolds. We introduce the notion of a ``zippered lamination'', and with suitable generic hypotheses, show that the unique minimal set for such a flow is an invariant zippered lamination. We obtain a precise description of the topology and dynamical properties of the minimal set, including the presence of non-zero entropy-type invariants and chaotic behavior. Moreover, we show that the minimal set does not have stable shape, yet satisfies the Mittag-Leffler condition for homology groups.Comment: This is the final version of the manuscript. Section 23 has been extended with many more details of the proof that the unique minimal set does not have stable shape, but does satisfy the Mittag-Leffler condition on homology group

Dynamics and the Godbillon-Vey Class of C^1 Foliations

Let F be a codimension-one, C^2-foliation on a manifold M without boundary. In this work we show that if the Godbillon--Vey class GV(F) \in H^3(M) is non-zero, then F has a hyperbolic resilient leaf. Our approach is based on methods of C^1-dynamical systems, and does not use the classification theory of C^2-foliations. We first prove that for a codimension--one C^1-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points E(F) has positive Lebesgue measure. We then prove that if E(F) has positive measure for a C^1-foliation F, then F must have a hyperbolic resilient leaf, and hence its geometric entropy must be positive. The proof of this uses a pseudogroup version of the Pliss Lemma. The theorem then follows, as a C^2-foliation with non-zero Godbillon-Vey class has non-trivial Godbillon measure. These results apply for both the case when M is compact, and when M is an open manifold.Comment: This manuscript is a revision of the section 3 material from the previous version, and includes edits to the pictures in the tex

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