On the image size of singular maps. I

Author:
S. M. Bates

Journal:
Proc. Amer. Math. Soc. **114** (1992), 699-705

MSC:
Primary 58C25; Secondary 26B05

MathSciNet review:
1074748

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Abstract | References | Similar Articles | Additional Information

Abstract: For fixed integers , we establish a sharp upper bound for the Hölder differentiability class of singular maps of subrank whose images have positive Lebesgue -measure.

**[Bates]***Towards a precise smoothness hypothesis in Sard's thoerem*, Proc. Amer. Math. Soc. (to appear).**[Falconer, 1985]***The geometry of fractal sets*, Cambridge Tracts in Math. 85, Cambridge Univ. Press, Cambridge and New York.**[Federer, 1969]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[Hirsch, 1976]**Morris W. Hirsch,*Differential topology*, Springer-Verlag, New York-Heidelberg, 1976. Graduate Texts in Mathematics, No. 33. MR**0448362****[Kaufman, 1979]**R. Kaufman,*A singular map of a cube onto a square*, J. Differential Geom.**14**(1979), no. 4, 593–594 (1981). MR**600614****[Norton, 1986]**Alec Norton,*A critical set with nonnull image has large Hausdorff dimension*, Trans. Amer. Math. Soc.**296**(1986), no. 1, 367–376. MR**837817**, 10.1090/S0002-9947-1986-0837817-2**[Norton, 1987]***The fractal geometry of critical sets with nonnull image and the differentiablity of functions*, Ph.d. Thesis, University of California, Berkeley.**[Sternberg, 1964]**Shlomo Sternberg,*Lectures on differential geometry*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0193578****[Whitney, 1935]**Hassler Whitney,*A function not constant on a connected set of critical points*, Duke Math. J.**1**(1935), no. 4, 514–517. MR**1545896**, 10.1215/S0012-7094-35-00138-7

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1992-1074748-8

Article copyright:
© Copyright 1992
American Mathematical Society