Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Functions not constant on fractal quasi-arcs of critical points


Author: Alec Norton
Journal: Proc. Amer. Math. Soc. 106 (1989), 397-405
MSC: Primary 28A75; Secondary 26B35, 58C25
DOI: https://doi.org/10.1090/S0002-9939-1989-0969524-8
MathSciNet review: 969524
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper provides geometric sufficient conditions for an arc to be a critical set for some function not constant along that arc--an example of which was first discovered by Whitney in 1935. In particular, any fractal subarc of a quasi-circle has this property. The maximum degree of differentiability of the function is closely connected to the arc's geometry.


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

  • [1] A. S. Besicovitch, On existence of subsets of finite measure of sets of infinite measure, Indag. Math. 14 (1952), 339-344. MR 0048540 (14:28e)
  • [2] A. S. Besicovitch and I. J. Schoenberg, On Jordan arcs and Lipschitz classes of functions defined on them, Acta Math. 106 (1961), 119-136. MR 0136693 (25:158)
  • [3] G. Choquet, L'isometrie des ensembles dans ses rapports avec la théorie du contact et la théorie de la measure, Mathematica, Bucharest, XX (1944), 29-64. MR 0012317 (7:9e)
  • [4] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Math., no. 85, Cambridge Univ. Press, 1985. MR 867284 (88d:28001)
  • [5] G. Glaeser, Étude de quelques algèebres Tayloriennes, J. Analyse Math. 6 (1958), 1-124. MR 0101294 (21:107)
  • [6] M. Laczkovich and G. Petruska, Whitney sets and sets of constancy: On a problem of Whitney, Real Anal. Exchange, Michigan State Univ. 10 (1985), 313-323. MR 790808 (86i:26014)
  • [7] A. Norton, A critical set with nonnull image has large Hausdorff dimension, Trans. Amer. Math. Soc. 296 (1986), 367-376. MR 837817 (87i:26011)
  • [8] -, The fractal geometry of critical sets with nonnull image and the diffrentiability of functions, Ph.D. Thesis, Univ. of California, Berkeley, 1987.
  • [9] C. A. Rogers, Hausdorff measure, Cambridge Univ. Press, 1970. MR 0281862 (43:7576)
  • [10] A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883-890. MR 0007523 (4:153c)
  • [11] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970. MR 0290095 (44:7280)
  • [12] H. Whitney, Analytic extensions of differentiable functions defined on closed sets, Trans. Amer. Math. Soc. 36(1934), 63-89. MR 1501735
  • [13] -, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), 514-517. MR 1545896

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28A75, 26B35, 58C25

Retrieve articles in all journals with MSC: 28A75, 26B35, 58C25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0969524-8
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society