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Whitney's example by way of Assouad's embedding

Author: Piotr Hajlasz
Journal: Proc. Amer. Math. Soc. 131 (2003), 3463-3467
MSC (2000): Primary 26B05; Secondary 26B35, 28A80
Published electronically: February 6, 2003
MathSciNet review: 1991757
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Abstract: In this note we show how to use the Assouad embedding theorem (about almost bi-Lipschitz embeddings) to construct examples of $C^m$ functions which are not constant on a critical set homeomorphic to the $n$-dimensional cube. This extends the famous example of Whitney. Our examples are shown to be sharp.

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  • 1. P. Assouad, Espaces Métriques, Plongements, Facteurs, Thèse de Doctorat. Université de Paris XI, 91405 Orsay, France, 1977. MR 58:30989
  • 2. P. Assouad, Étude d'une dimension métrique liée à la possibilité de plongement dans $\mathbb{R} ^n$, C. R. Acad. Sci. Paris Sér. A. 288 (1979), 731-734. MR 80f:54030
  • 3. P. Assouad, Plongements Lipschitziens dans $\mathbb{R} ^n$, Bulletin Société Mathématique de France, 111 (1983), 429-448. MR 86f:54050
  • 4. M. Bonk and J. Heinonen, In preparation.
  • 5. M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), 266-306. MR 2001g:53077
  • 6. G. David and T. Toro, Reifenberg flat metric spaces, snowballs, and embeddings, Math. Ann. 315 (1999), 641-710. MR 2001c:49067
  • 7. A. Ya. Dubovickii, On the structure of level sets of differentiable mappings of an $n$-dimensional cube into a $k$-dimensional cube (Russian), Izv. Akad. Nauk SSSR. Ser. Mat. 21 (1957), 371-418. MR 20:942
  • 8. K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986. MR 88d:28001
  • 9. J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext. Springer-Verlag, New York, 2001. MR 2002c:30028
  • 10. T. W. Körner, A dense arcwise connected set of critical points--molehills out of mountains, J. London Math. Soc. 38 (1988), 442-452. MR 90f:26011
  • 11. P. Koskela, The degree of regularity of a quasiconformal mapping, Proc. Amer. Math. Soc. 122 (1994), 769-772. MR 95a:30020
  • 12. B. Malgrange, Ideals of differentiable functions, Oxford Univ. Press, London, 1966. MR 35:3446
  • 13. A. Norton, A critical set with nonnull image has large Hausdorff dimension, Trans. Amer. Math. Soc. 296 (1986), 367-376. MR 87i:26011
  • 14. A. Norton, Functions not constant on fractal quasi-arcs of critical points, Proc. Amer. Math. Soc. 106 (1989), 397-405. MR 89m:28013
  • 15. A. Norton and C. Pugh, Critical sets in the plane, Michigan Math. J. 38 (1991), 441-459. MR 92f:57032
  • 16. S. Semmes, On the nonexistence of bi-Lipschitz parametrizations and geometric problems about $A_\infty$ weights, Rev. Mat. Iberoamericana 12 (1996), 227-410. MR 97e:30040
  • 17. M. Sion, On the existence of functions having given partial derivatives on a curve, Trans. Amer. Math. Soc. 77 (1954), 179-201. MR 16:344a
  • 18. P. Tukia, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 149-160. MR 83b:30019
  • 19. H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89.
  • 20. H. Whitney, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), 514-517.
  • 21. Y. Yomdin, Surjective mappings whose differential is nowhere surjective, Proc. Amer. Math. Soc. 111 (1991), 267-270. MR 91g:58025

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Additional Information

Piotr Hajlasz
Affiliation: Institute of Mathematics, Warsaw University, ul. Banacha 2, 02–097 Warszawa, Poland

Keywords: Critical set, Whitney's example, Whitney's extension theorem, Van Koch snowflake, Assouad's embedding
Received by editor(s): October 16, 2001
Received by editor(s) in revised form: May 29, 2002
Published electronically: February 6, 2003
Additional Notes: This work was supported by the KBN grant no. 2 PO3A 028 22.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2003 American Mathematical Society