Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

2012/01/31 by John M. Mackay, Jeremy T. Tyson, Kevin Wildrick · 16 citations
Mathematics · #Analytic and geometric function theory #Class (philosophy) #Euclidean geometry #Euclidean space #Geometry and complex manifolds #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Measure (data warehouse) #Metric (unit) #Metric space #Modulus #math.DG #math.MG #msc:28A80 #msc:30L99 #msc:31E05

paper · pdf · doi:10.1007/s00039-013-0227-6

published in Geometric and Functional Analysis 23(3), 985-1034 (Birkhäuser) · v1: 42 pages, 11 figures. v2: 42 pages, 10 figures. Improved exposition

arxiv created 2013/02/01 · openalex publication_date 2013/03/25 · arxiv updated 2013/11/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06

Abstract

A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincaré inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.

Citations

• A Course in Metric Geometry
• Modulus and the Poincar� inequality on metric measure spaces
• Planar relative Schottky sets and quasisymmetric maps
• Quasiconformal maps in metric spaces with controlled geometry
• The Poincaré inequality is an open ended condition

Cited by

• Densely defined non-closable curl on carpet-like metric measure spaces

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