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Sets of minimal Hausdorff dimension for quasiconformal maps


Author: Jeremy T. Tyson
Journal: Proc. Amer. Math. Soc. 128 (2000), 3361-3367
MSC (2000): Primary 30C65; Secondary 28A78
DOI: https://doi.org/10.1090/S0002-9939-00-05433-2
Published electronically: May 18, 2000
MathSciNet review: 1676353
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Abstract: For any $1\le\alpha\le n$, there is a compact set $E\subset\mathbb{R}^n$ of (Hausdorff) dimension $\alpha$ whose dimension cannot be lowered by any quasiconformal map $f:\mathbb{R}^n\to\mathbb{R}^n$. We conjecture that no such set exists in the case $\alpha<1$. More generally, we identify a broad class of metric spaces whose Hausdorff dimension is minimal among quasisymmetric images.


References [Enhancements On Off] (What's this?)

  • 1. C. J. Bishop.
    Non-removable sets for quasiconformal and locally bi-Lipschitz mappings in ${R}\sp{\mbox{\unboldmath$3$ }}$.
    preprint.
  • 2. C. J. Bishop.
    Quasiconformal mappings which increase dimension.
    Ann. Acad. Sci. Fenn. Ser. A I Math.
    to appear.
  • 3. M. Bourdon.
    Au bord de certains polyèdres hyperboliques.
    Ann. Inst. Fourier (Grenoble), 45(1):119-141, 1995. MR 96b:20045
  • 4. H. Federer.
    Geometric measure theory, volume 153 of Die Grundlehren der mathematischen Wissenschaften.
    Springer-Verlag Inc., 1969. MR 41:1976
  • 5. F. W. Gehring and J. Väisälä.
    Hausdorff dimension and quasiconformal mappings.
    J. London Math. Soc. (2), 6:504-512, 1973. MR 48:2380
  • 6. J. Heinonen and P. Koskela.
    Quasiconformal maps in metric spaces with controlled geometry.
    Acta Math., 181:1-61, 1998. MR 99j:30025
  • 7. W. Hurewicz and H. Wallman.
    Dimension Theory.
    Princeton University Press, Princeton, N. J., 1941.
    Princeton Mathematical Series, v. 4. MR 3:312b
  • 8. P. Mattila.
    Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics.
    Cambridge University Press, 1995. MR 96h:28006
  • 9. P. Pansu.
    Dimension conforme et sphère à l'infini des variétés à courbure négative.
    Ann. Acad. Sci. Fenn. Ser. A I Math., 14(2):177-212, 1989. MR 90k:53079
  • 10. P. Pansu.
    Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un.
    Ann. Math. (2), 129(1):1-60, 1989. MR 90e:53058
  • 11. P. Tukia and J. Väisälä.
    Quasisymmetric embeddings of metric spaces.
    Ann. Acad. Sci. Fenn. Ser. A I Math., 5(1):97-114, 1980. MR 82g:30038
  • 12. J. Tyson.
    Assouad dimension and quasisymmetric mappings.
    in preparation.
  • 13. J. Tyson.
    Quasiconformality and quasisymmetry in metric measure spaces.
    Ann. Acad. Sci. Fenn. Ser. A I Math., 23(2):525-548, 1998. MR 99i:30038
  • 14. J. Väisälä.
    Lectures on $n$-dimensional quasiconformal mappings.
    Springer-Verlag, Berlin, 1971.
    Lecture Notes in Mathematics, Vol. 229. MR 56:12260
  • 15. W. P. Ziemer.
    Extremal length and $p$-capacity.
    Mich. Math. J., 16:43-51, 1969. MR 40:346

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Additional Information

Jeremy T. Tyson
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651
Email: jttyson@math.lsa.umich.edu, tyson@math.sunysb.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05433-2
Keywords: Hausdorff dimension, quasiconformal maps, generalized modulus
Received by editor(s): October 15, 1998
Received by editor(s) in revised form: January 15, 1999
Published electronically: May 18, 2000
Additional Notes: The results of this paper form part of the author’s doctoral dissertation at the University of Michigan. The author was supported by a National Science Foundation Graduate Research Fellowship and a Sloan Doctoral Dissertation Fellowship.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society

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