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

John M. Mackay; Jeremy T. Tyson; Kevin Wildrick

doi:10.1007/s00039-013-0227-6

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

John M. Mackay
, Jeremy T. Tyson
, Kevin Wildrick

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

A carpet is a metric space homeomorphic to the Sierpiński 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.

Original language	English (US)
Pages (from-to)	985-1034
Number of pages	50
Journal	Geometric and Functional Analysis
Volume	23
Issue number	3
DOIs	https://doi.org/10.1007/s00039-013-0227-6
State	Published - Jun 2013

Keywords

Doubling measure
Gromov-Hausdorff tangent cone
Modulus
Poincaré inequality
Sierpiński carpet

ASJC Scopus subject areas

Analysis
Geometry and Topology

Online availability

10.1007/s00039-013-0227-6

Library availability

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@article{52ed3d49e4fe4cd2810cf7452945b708,

title = "Modulus and Poincar{\'e} Inequalities on Non-Self-Similar Sierpi{\'n}ski Carpets",

abstract = "A carpet is a metric space homeomorphic to the Sierpi{\'n}ski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincar{\'e} inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincar{\'e} inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.",

keywords = "Doubling measure, Gromov-Hausdorff tangent cone, Modulus, Poincar{\'e} inequality, Sierpi{\'n}ski carpet",

author = "Mackay, \{John M.\} and Tyson, \{Jeremy T.\} and Kevin Wildrick",

note = "Keywords and phrases: Sierpi{\'n}ski carpet, Doubling measure, Modulus, Poincar{\'e} inequality, Gromov{\textendash}Hausdorff tangent cone Mathematics Subject Classification (1991): 30L99; 31E05; 28A80 J. M. Mackay and J. T. Tyson were supported by US National Science Foundation Grant DMS-0901620. J. M. Mackay was supported by EPSRC grant {\textquotedblleft}Geometric and analytic aspects of infinite groups{\textquotedblright}. J. T. Tyson was supported by US National Science Foundation Grant DMS-1201875. K. Wildrick supported by Academy of Finland Grants 120972 and 128144, the Swiss National Science Foundation, ERC Project CG-DICE, and European Science Council Project HCAA.",

year = "2013",

month = jun,

doi = "10.1007/s00039-013-0227-6",

language = "English (US)",

volume = "23",

pages = "985--1034",

journal = "Geometric and Functional Analysis",

issn = "1016-443X",

publisher = "Springer International Publishing",

number = "3",

}

TY - JOUR

T1 - Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

AU - Mackay, John M.

AU - Tyson, Jeremy T.

AU - Wildrick, Kevin

N1 - Keywords and phrases: Sierpiński carpet, Doubling measure, Modulus, Poincaré inequality, Gromov–Hausdorff tangent cone Mathematics Subject Classification (1991): 30L99; 31E05; 28A80 J. M. Mackay and J. T. Tyson were supported by US National Science Foundation Grant DMS-0901620. J. M. Mackay was supported by EPSRC grant “Geometric and analytic aspects of infinite groups”. J. T. Tyson was supported by US National Science Foundation Grant DMS-1201875. K. Wildrick supported by Academy of Finland Grants 120972 and 128144, the Swiss National Science Foundation, ERC Project CG-DICE, and European Science Council Project HCAA.

PY - 2013/6

Y1 - 2013/6

N2 - A carpet is a metric space homeomorphic to the Sierpiński 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.

AB - A carpet is a metric space homeomorphic to the Sierpiński 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.

KW - Doubling measure

KW - Gromov-Hausdorff tangent cone

KW - Modulus

KW - Poincaré inequality

KW - Sierpiński carpet

UR - https://www.scopus.com/pages/publications/84878527834

UR - https://www.scopus.com/pages/publications/84878527834#tab=citedBy

U2 - 10.1007/s00039-013-0227-6

DO - 10.1007/s00039-013-0227-6

M3 - Article

AN - SCOPUS:84878527834

SN - 1016-443X

VL - 23

SP - 985

EP - 1034

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

IS - 3

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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