Abstract
We prove the quantitative Morse-Sard theorem for \(\mathcal{C}^k\) mappings with n variables, i.e. we bound the \(\epsilon\)-entropy of near-critical values. In particular, we give, for the entropy dimension of the rank-\(\nu\) set of critical values, a bound depending only on n, \(\nu\) and k. We then give examples showing that our statement is the best possible. We also give the \(\mathcal{C}^k\) version of the polynomial quantitative transversality of Chapter 8.
Keywords
- Maximal Radius
- Taylor Polynomial
- Entropy Dimension
- Transversality Theorem
- High Smoothness
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© 2004 Springer-Verlag
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Yomdin, Y., Comte, G. (2004). 9. Mappings of Finite Smoothness. In: Tame Geometry with Application in Smooth Analysis. Lecture Notes in Mathematics, vol 1834. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-40960-1_9
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DOI: https://doi.org/10.1007/978-3-540-40960-1_9
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20612-5
Online ISBN: 978-3-540-40960-1
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