[Paper Review] Lipshitz matchbox manifolds

Steven Hurder|arXiv (Cornell University)|Sep 6, 2013

Mathematical Dynamics and Fractals94 references3 citations

TL;DR

This paper introduces the concept of Lipschitz matchbox manifolds—foliated spaces with totally disconnected transversals equipped with a metric making holonomy maps Lipschitz—and establishes a bounded tower equivalence condition as a necessary and sufficient criterion for Lipschitz equivalence of such manifolds. The key contribution is a classification invariant based on displacement of indexing functions in inverse systems, distinguishing homeomorphic but non-Lipschitz-equivalent solenoids.

ABSTRACT

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.

Motivation & Objective

• To determine when a Lipschitz matchbox manifold can be embedded as an exceptional minimal set in a C¹-foliation of a compact smooth manifold.
• To develop intrinsic geometric invariants that obstruct such embeddings, focusing on recurrence and metric properties.
• To classify Lipschitz weak solenoids using a new invariant based on displacement of inverse system indexing functions.
• To distinguish between homeomorphic but non-Lipschitz-equivalent matchbox manifolds, particularly in the context of Vietoris solenoids.
• To extend previous non-embedding results by incorporating metric structure and quasi-isometry invariants into the classification framework.

Proposed method

• Define Lipschitz matchbox manifolds as compact, connected foliated spaces with totally disconnected transversals and a metric on transversals for which holonomy maps are Lipschitz.
• Use inverse limits of covering systems with subgroups of the fundamental group to model matchbox manifolds, particularly weak solenoids.
• Introduce the displacement of indexing functions ℓ ↦ νₗ and ν ↦ ℓᵥ as a measure of how far the inverse systems are from being uniformly aligned.
• Define bounded tower equivalence via finite displacement: Disp(ℓᵥ, νₗ) < ∞, which ensures compatibility between the inverse systems.
• Construct fiber metrics using decaying weights aₗ = 3⁻ˡ to define metrics on Cantor sections of the matchbox manifolds.
• Prove that Lipschitz equivalence of the holonomy actions on fibers is equivalent to bounded tower equivalence under these metrics.

Experimental results

Research questions

• RQ1 Under what conditions can a Lipschitz matchbox manifold be realized as an exceptional minimal set in a C¹-foliation of a compact smooth manifold?
• RQ2 What geometric or dynamical invariants obstruct such an embedding, particularly when recurrence or metric structure is involved?
• RQ3 When are two Lipschitz matchbox manifolds, such as weak solenoids, Lipschitz equivalent despite being homeomorphic?
• RQ4 How does the displacement of indexing functions in inverse systems relate to the Lipschitz equivalence of holonomy actions on fibers?
• RQ5 Can bounded tower equivalence serve as a complete invariant for classifying Lipschitz weak solenoids up to Lipschitz equivalence?

Key findings

• Lipschitz equivalence of the holonomy actions on fibers of two matchbox manifolds is equivalent to bounded tower equivalence of their presentations.
• Two Vietoris solenoids with different covering degree sequences can be homeomorphic but not Lipschitz equivalent if their displacement is infinite.
• The displacement invariant Disp(ℓᵥ, νₗ) is finite if and only if the induced maps between fibers are Lipschitz under the metric defined by aₗ = 3⁻ˡ.
• The example with alternating degrees 2 and 3, followed by exponentially increasing sequences of degree-2 covers, yields a pair of homeomorphic but non-Lipschitz-equivalent solenoids.
• The classification of Lipschitz weak solenoids is not determined by homeomorphism type alone, but requires the bounded tower equivalence condition.
• The result generalizes earlier classification results for classical solenoids by Bing and McCord, now refined by metric structure and Lipschitz invariance.