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.
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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.
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Mackay, J.M., Tyson, J.T. & Wildrick, K. Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets. Geom. Funct. Anal. 23, 985–1034 (2013). https://doi.org/10.1007/s00039-013-0227-6
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DOI: https://doi.org/10.1007/s00039-013-0227-6