Conformal dimension of the antenna set

Christopher J. Bishop; Jeremy T. Tyson

doi:10.1090/s0002-9939-01-05982-2

Conformal dimension of the antenna set

Christopher J. Bishop
, Jeremy T. Tyson

Mathematics

Stony Brook University

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

19 Scopus citations

Abstract

We show that the self-similar set known as the "antenna set" has the property that infy d\m(f(X)) = 1 (where the infimum is over all quasiconformal mappings of the plane), but that this infimum is not attained by any quasiconformal map; indeed, is not attained for any quasisymmetric map into any metric space.

Original language	English
Pages (from-to)	3631-3636
Number of pages	6
Journal	Proceedings of the American Mathematical Society
Volume	129
Issue number	12
DOIs	https://doi.org/10.1090/s0002-9939-01-05982-2
State	Published - 2001

Keywords

Conformai dimension
Hausdorff dimension
Quasiconformal map
Selfsimilar sets

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10.1090/s0002-9939-01-05982-2

Cite this

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title = "Conformal dimension of the antenna set",

abstract = "We show that the self-similar set known as the {"}antenna set{"} has the property that infy d\textbackslash{}m(f(X)) = 1 (where the infimum is over all quasiconformal mappings of the plane), but that this infimum is not attained by any quasiconformal map; indeed, is not attained for any quasisymmetric map into any metric space.",

keywords = "Conformai dimension, Hausdorff dimension, Quasiconformal map, Selfsimilar sets",

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AU - Bishop, Christopher J.

AU - Tyson, Jeremy T.

PY - 2001

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N2 - We show that the self-similar set known as the "antenna set" has the property that infy d\m(f(X)) = 1 (where the infimum is over all quasiconformal mappings of the plane), but that this infimum is not attained by any quasiconformal map; indeed, is not attained for any quasisymmetric map into any metric space.

AB - We show that the self-similar set known as the "antenna set" has the property that infy d\m(f(X)) = 1 (where the infimum is over all quasiconformal mappings of the plane), but that this infimum is not attained by any quasiconformal map; indeed, is not attained for any quasisymmetric map into any metric space.

KW - Conformai dimension

KW - Hausdorff dimension

KW - Quasiconformal map

KW - Selfsimilar sets

UR - https://www.scopus.com/pages/publications/23044531122

U2 - 10.1090/s0002-9939-01-05982-2

DO - 10.1090/s0002-9939-01-05982-2

M3 - Article

AN - SCOPUS:23044531122

SN - 0002-9939

VL - 129

SP - 3631

EP - 3636

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 12

ER -

Conformal dimension of the antenna set

Abstract

Keywords

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