Advanced Studies in Pure Mathematics

Lipschitz matchbox manifolds

Steven Hurder

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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 as an exceptional minimal set for a $C^{1}$-foliation of a smooth manifold? We give 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.

Article information

Source
Geometry, Dynamics, and Foliations 2013: In honor of Steven Hurder and Takashi Tsuboi on the occasion of their 60th birthdays, T. Asuke, S. Matsumoto and Y. Mitsumatsu, eds. (Tokyo: Mathematical Society of Japan, 2017), 71-115

Dates
Received: 27 August 2014
Revised: 17 October 2014
First available in Project Euclid: 4 October 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1538671763

Digital Object Identifier
doi:10.2969/aspm/07210071

Mathematical Reviews number (MathSciNet)
MR3726706

Zentralblatt MATH identifier
1385.57024

Subjects
Primary: 52C23: Quasicrystals, aperiodic tilings 57R05: Triangulating 54F15: Continua and generalizations 37B45: Continua theory in dynamics
Secondary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32] 57N55: Microbundles and block bundles [See also 55R60, 57Q50]

Citation

Hurder, Steven. Lipschitz matchbox manifolds. Geometry, Dynamics, and Foliations 2013: In honor of Steven Hurder and Takashi Tsuboi on the occasion of their 60th birthdays, 71--115, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07210071. https://projecteuclid.org/euclid.aspm/1538671763


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