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

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


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.

Mathematics Subject Classification (2000)

58K05 57R45 31B15 


Critical values Sard theorem metric entropy spanning trees 


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Yosef Yomdin
    • 1
  1. 1.Department of MathematicsThe Weizmann InstituteRehovotIsrael

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