Proceedings of the American Mathematical Society

Author(s): Christopher J. Bishop; Jeremy T. Tyson
Journal: Proc. Amer. Math. Soc. 129 (2001), 3631-3636.
MSC (2000): Primary 30C62; Secondary 28A78
Posted: April 25, 2001
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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.

1.: C. J. Bishop, Quasiconformal mappings which increase dimension, Ann. Acad. Sci. Fenn. Ser. A I Math. 24 (1999), 397-407. MR 2000i:30044
2.: C. J. Bishop and P. W. Jones, Wiggly sets and limit sets, Ark. Mat. 35 (1997), no. 2, 201-224. MR 99f:30066
3.: C. J. Bishop and J. T. Tyson, Locally minimal sets for conformal dimension, Ann. Acad. Sci. Fenn. Ser. A I Math. (to appear).
4.: M. Bourdon, Au bord de certains polyèdres hyperboliques, Ann. Inst. Fourier (Grenoble) 45 (1995), 119-141. MR 96b:20045
5.: K. J. Falconer, Fractal geometry, Mathematical Foundations and Applications, John Wiley and Sons Ltd., Chichester, 1990. MR 92j:28008
6.: F. W. Gehring and J. Väisälä, Hausdorff dimension and quasiconformal mappings, J. London Math. Soc. (2) 6 (1973), 504-512.
7.: J. Heinonen, Lectures on analysis on metric spaces, Univ. of Michigan (1996), Lecture notes.
8.: 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.
9.: P. Pansu, Dimension conforme et sphère à l'infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 177-212. MR 90k:53079
10.: J. T. Tyson, Sets of minimal Hausdorff dimension for quasiconformal maps, Proc. Amer. Math. Soc. (to appear). CMP 99:09

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Christopher J. Bishop
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
Email: bishop@math.sunysb.edu

Jeremy T. Tyson
Affiliation: Department of Mathematics, SUNY at Stony Brook, Stony Brook, New York 11794-3651
Email: tyson@math.sunysb.edu

DOI: 10.1090/S0002-9939-01-05982-2
PII: S 0002-9939(01)05982-2
Keywords: Quasiconformal map, Hausdorff dimension, conformal dimension, self-similar sets
Received by editor(s): November 15, 1999
Received by editor(s) in revised form: April 27, 2000
Posted: 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 of article: Copyright 2001, American Mathematical Society

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