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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Conformal dimension of the antenna set
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by Christopher J. Bishop and Jeremy T. Tyson PDF
Proc. Amer. Math. Soc. 129 (2001), 3631-3636 Request permission

Abstract:

We show that the self-similar set known as the “antenna set” has the property that $\inf _f \dim (f(X)) =1$ (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.
References
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  • J. Heinonen, Lectures on analysis on metric spaces, Univ. of Michigan (1996), Lecture notes.
  • P. W. Jones, On removable sets for Sobolev spaces in the plane, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991) (Princeton University Press, Princeton, NJ), 1995, pp. 250–267.
  • Pierre Pansu, Dimension conforme et sphère à l’infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), no. 2, 177–212 (French, with English summary). MR 1024425, DOI 10.5186/aasfm.1989.1424
  • J. T. Tyson, Sets of minimal Hausdorff dimension for quasiconformal maps, Proc. Amer. Math. Soc. (to appear).
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Additional Information
  • Christopher J. Bishop
  • Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
  • MR Author ID: 37290
  • Email: bishop@math.sunysb.edu
  • Jeremy T. Tyson
  • Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
  • MR Author ID: 625886
  • Email: tyson@math.sunysb.edu
  • Received by editor(s): November 15, 1999
  • Received by editor(s) in revised form: April 27, 2000
  • Published electronically: April 25, 2001
  • Additional Notes: The first author was partially supported by NSF Grant DMS 98-00924. The second author was partially supported by an NSF postdoctoral fellowship
  • Communicated by: Albert Baernstein II
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 3631-3636
  • MSC (2000): Primary 30C62; Secondary 28A78
  • DOI: https://doi.org/10.1090/S0002-9939-01-05982-2
  • MathSciNet review: 1860497