Lipshitz matchbox manifolds

zbMATH Open1385.57024arXiv1309.1512MaRDI QIDQ4604213

Publication date: 23 February 2018

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.

Full work available at URL: https://arxiv.org/abs/1309.1512

zbMATH Keywords

minimal set holonomy map matchbox manifold generalized lamination Lipschitz matchbox manifold Lipschitz weak solenoid

Mathematics Subject Classification ID

Continua and generalizations (54F15) Foliations (differential geometric aspects) (53C12) Continua theory in dynamics (37B45) Triangulating (57R05) Microbundles and block bundles (57N55)