Skip to main content
SpringerLink
Log in

β-spread of Sets in Metric Spaces and Critical Values of Smooth Functions

  • Chapter
Book cover Characteristic Functions, Scattering Functions and Transfer Functions

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 197))

  • 610 Accesses

Abstract

This paper provides new geometric restrictions on the set of critical values of differentiable functions. The classical and widely used condition on critical values of differentiable mappings is given by the Morse-Sard theorem ([20, 38, 39]): if the mapping is C k-smooth, with k sufficiently big, then the set of its critical values has the Lebesgue measure (or, more precisely, the Hausdorff measure of an appropriate dimension) zero.

In a work of the author since 1981 ([43]–[46], [50], and others) which was strongly inspired by questions, remarks, and constructive criticism of Moshe Livsic, it was shown that in fact the critical values of any differentiable mapping satisfy geometric restrictions much stronger than just the property to be of measure zero. These restrictions are given in terms of the metric entropy and they turn out to be pretty close to a complete characterization of the possible sets of critical values. Still a gap between the necessary and sufficient conditions remained.

In the present paper we close (partially) this gap, introducing into consideration of critical values a certain geometric invariant (we call it β-spread) which was previously studied in quite different relations. We show that in many important cases β-spread provides a complete characterization of critical values of differentiable functions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V.I. Bakhtin, Properties of the entropy and the Hausdorff dimensions, Vestnik Mosk. Gos. Univ., Ser. 1, no. 5, 1982, 25–29.

    Google Scholar 

  2. S. Bates, On the image size of singular maps. I. Proc. Amer.Math. Soc. 114 (1992), no. 3, 699–705.

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Bates, On the image size of singular maps. II. Duke Math. J. 68 (1992), no. 3, 463–476.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Bates, Toward a precise smoothness hypothesis in Sard’s theorem. Proc. Amer. Math. Soc. 117 (1993), no. 1, 279–283.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Bates, C. Moreira, De nouvelles perspectives sur le théorème de Morse-Sard. (French) [Some new perspectives on the Morse-Sard theorem] C.R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 1, 13–17.

    MATH  MathSciNet  Google Scholar 

  6. S. Bates, A. Norton, On sets of critical values in the real line. Duke Math. J. 83 (1996), no. 2, 399–413.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Beck, J. Spencer, Unit distances, J. Combin. Theory Ser. A 37 (1984), no. 3, 231-238.

    Google Scholar 

  8. J. Beck, Uniformity and irregularity, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 1400–1407, Amer. Math. Soc., Providence, RI, 1987.

    Google Scholar 

  9. J. Beck, Irregularities of distribution. I, Acta Math. 159 (1987), no. 1-2, 1–49.

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Beck, Irregularities of distribution. II, Proc. London Math. Soc. (3) 56 (1988), no. 1, 1–50.

    Article  MATH  MathSciNet  Google Scholar 

  11. J. Beck, W. Chen, Irregularities of point distribution relative to half-planes I, Mathematika 40 (1993), no. 1, 102–126.

    MATH  MathSciNet  Google Scholar 

  12. J. Beck, W. Chen, Irregularities of point distribution relative to convex polygons II, Mathematika 40 (1993), no. 1, 127–136.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Beck, W. Chen, Irregularities of point distribution relative to convex polygons III, J. London Math. Soc. (2) 56 (1997), no. 2, 222–230.

    Article  MATH  MathSciNet  Google Scholar 

  14. A.S. Besicovitch, SA. J. Taylor, on the complementary intervals of a linear closed set of zero Lebesgue measure, J. London Math. Soc. 29 (1954), 449–459.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Bouligand, Ensembles impropres et nombre dimensionnel, Bull. Sci. Math. (2) 52 (1928), 320–344 and 361–376.

    Google Scholar 

  16. L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand Math. Studies, 13, 1967.

    Google Scholar 

  17. K. Falconer, Techniques in fractal geometry, John Wiley and Sons, Ltd., Chichester, 1997. xviii+256 pp.

    MATH  Google Scholar 

  18. K. Falconer, Fractal geometry. Mathematical foundations and applications, Second edition. John Wiley and Sons, Inc., Hoboken, NJ, 2003. xxviii+337 pp.

    MATH  Google Scholar 

  19. K. Falconer, M. Jarvenpaa, Packing dimensions of sections of sets, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 1, 89–104.

    Article  MATH  MathSciNet  Google Scholar 

  20. H. Federer, Geometric Measure Theory, Die Grundlehren der Math.Wissenschafften, Vol. 153, Springer-Verlag, 1969.

    Google Scholar 

  21. L.T. Fejes-Toth, Über einen geometrischen Satz, Mathematische Zeitschrift 46 (1940), 83–85.

    Article  MathSciNet  Google Scholar 

  22. J. Goodman, R. Pollack, B. Sturmfels, The intrinsic spread of a configuration in R d. J. Amer. Math. Soc. 3 (1990), no. 3, 639–651.

    Article  MATH  MathSciNet  Google Scholar 

  23. J. Harrison, A. Norton, Geometric integration on fractal curves in the plane. Indiana Univ. Math. J. 40 (1991), no. 2, 567–594.

    Article  MATH  MathSciNet  Google Scholar 

  24. J. Harrison, A. Norton, The Gauss-Green theorem for fractal boundaries. Duke Math. J. 67 (1992), no. 3, 575–588.

    Article  MATH  MathSciNet  Google Scholar 

  25. H. Minkowski, Theorie der konvexen Körper, insbesondere Begründung ihres Oberfl ächenbegriffs, in: Gesammelte Abhandlungen von Hermann Minkowski (part II, Chapter XXV), Chelsea, New York, 1967, 131–229.

    Google Scholar 

  26. L.D. Ivanov, Variazii mnozestv i funkzii, Moscow, Nauka 1975.

    Google Scholar 

  27. A.N. Kolmogorov, Asymptotic characteristics of some completely bounded metric spaces, Dokl. Akad. Nauk USSR 108, 1956, 585–589.

    Google Scholar 

  28. A.N. Kolmogorov, V.M. Tihomirov, ∈-entropy and ∈-capacity of sets in functional spaces, Uspehi Mat. Nauk 2 (86), 14, 1959, 3–86. English transl., Amer. Math. Soc. Transl. (2) 17, 1961, 277–364.

    MathSciNet  Google Scholar 

  29. G. Kozma, Z. Lotker, G. Stupp, The minimal spanning tree and the upper box dimension, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1183–1187.

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Lapidus, M. van Frankenhuysen, Fractal Geometry and Number Theory, Birkh äuser, 2000.

    Google Scholar 

  31. G.G. Lorentz, Metric entropy and approximation, Bull. Amer. Math. Soc. 72, 1966, 903–937.

    Article  MATH  MathSciNet  Google Scholar 

  32. S.C. Milne, Peano curves and smoothness of functions, Adv. in Math. 35, no.2, 1980, 129–157.

    Article  MATH  MathSciNet  Google Scholar 

  33. A. Norton, A critical set with nonnull image has large Hausdorff dimension, Trans. Amer. Math. Soc. 296 (1986), no. 1, 367–376.

    Article  MATH  MathSciNet  Google Scholar 

  34. A. Norton, A C 1×∞ function with an interval of critical values Indiana Univ. Math. J. 40 (1991), no. 4, 1483–1488.

    Article  MATH  MathSciNet  Google Scholar 

  35. A. Norton, The Zygmund Morse-Sard theorem, J. Geom. Anal. 4 (1994), no. 3, 403–424.

    MATH  MathSciNet  Google Scholar 

  36. A. Norton, C. Pugh, Critical sets in the plane, Michigan Math. J. 38 (1991), no. 3, 441–459.

    Article  MATH  MathSciNet  Google Scholar 

  37. R. Rumely, Capacity Theory on Algebraic Curves, Lecture Notes in Mathematics, 1378, Springer-Verlag, 1989. i+437 pp.

    MathSciNet  Google Scholar 

  38. A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48, 1942, 883–890.

    Article  MATH  MathSciNet  Google Scholar 

  39. A. Sard, Hausdorff measure of critical images on Banach manifolds, Amer. J. of Math., 87, 1965, 158–174.

    Article  MathSciNet  Google Scholar 

  40. C. Tricot, Formule intégrale pour la dimension fractale dans le plan. (French) [Integral formulation for the fractal dimension in the plane] C.R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 169–172.

    MATH  MathSciNet  Google Scholar 

  41. C. Tricot, Curves and fractal dimension. With a foreword by Michel Mendes France. Translated from the 1993 French original. Springer-Verlag, New York, 1995. xiv+323 pp.

    Google Scholar 

  42. A.G. Vitushkin, On representation of functions by means of superpositions and related topics, L’Enseignement Math. XXIII, 1977, 255–320.

    MathSciNet  Google Scholar 

  43. Y. Yomdin, The geometry of critical and near-critical values of differentiable mappings. Math. Ann. 264, (1983), n. 4. 495–515.

    Article  MATH  MathSciNet  Google Scholar 

  44. Y. Yomdin, β-spread of sets in metric spaces and critical values of smooth functions, Preprint, MPI Bonn, (1983).

    Google Scholar 

  45. Y. Yomdin, Global bounds for the Betti numbers of regular fibers of differentiable mappings. Topology 24, (1985), n. 2, 145–152.

    Article  MATH  MathSciNet  Google Scholar 

  46. Y. Yomdin, Sard’s theorem and its improved versions in numerical analysis. Proc. A.M.S. Conf., Colorado, 1988. Lectures in Applied Math. 29, (1990), 701–706.

    MathSciNet  Google Scholar 

  47. Y. Yomdin, β-spread of sets in metric spaces and capacity, in preparation.

    Google Scholar 

  48. Y. Yomdin, Discrete subsets of semialgebraic sets, in preparation.

    Google Scholar 

  49. Y. Yomdin, Remez-type inequalities for polynomials bounded on discrete sets, submitted.

    Google Scholar 

  50. Y. Yomdin, G. Comte, Tame Geometry with Applications in Smooth Analysis, Lecture Notes in Mathematics, 1834, Springer, Berlin, Heidelberg, New York, 2004.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Weizmann Institute, Rehovot, 76100, Israel

    Yosef Yomdin

Authors
  1. Yosef Yomdin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva, 84105, Israel

    Daniel Alpay  & Victor Vinnikov  & 

Additional information

To the memory of Moshe Livsic

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Yomdin, Y. (2009). β-spread of Sets in Metric Spaces and Critical Values of Smooth Functions. In: Alpay, D., Vinnikov, V. (eds) Characteristic Functions, Scattering Functions and Transfer Functions. Operator Theory: Advances and Applications, vol 197. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0183-2_13

Download citation

Publish with us

Policies and ethics

Search

Navigation