Israel Journal of Mathematics

, Volume 57, Issue 3, pp 285–300 | Cite as

Volume growth and entropy

  • Y. Yomdin


An inequality is proved, bounding the growth rates of the volumes of iterates of smooth submanifolds in terms of the topological entropy. ForC x-smooth mappings this inequality implies the entropy conjecture, and, together with the opposite inequality, obtained by S. Newhouse, proves the coincidence of the growth rate of volumes and the topological entropy, as well as the upper semicontinuity of the entropy.


Volume Growth Topological Entropy Algebraic Function Opposite Inequality Smooth Submanifolds 
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  1. 1.
    R. Bowen,Entropy for group automorphisms and homogeneous spaces, Trans. Am. Math. Soc.153 (1971), 401–414.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. Coste,Ensembles semi-algébriques, Lecture Notes in Math.959, Springer-Verlag, Berlin, 1982, pp. 109–138.Google Scholar
  3. 3.
    E. I. Dinaburg,On the relations among various entropy characteristics of dynamical systems, Math. USSR-Isv.5 (1971), 337–378.CrossRefGoogle Scholar
  4. 4.
    D. Fried,Entropy and twisted cohomology, Topology, to appear.Google Scholar
  5. 5.
    D. Fried and M. Shub,Entropy, linearity and chain-recurrence, Publ. Math. IHES50 (1979), 203–214.MATHMathSciNetGoogle Scholar
  6. 6.
    L. D. Ivanov,Variations of Sets and Functions, Nauka, 1975 (in Russian).Google Scholar
  7. 7.
    O. D. Kellogg,On bounded polynomials in several variables, Math. Z.27 (1928), 55–64.CrossRefMathSciNetGoogle Scholar
  8. 8.
    S. Newhouse,Entropy and volume, preprint.Google Scholar
  9. 9.
    M. Shub,Dynamical systems, filtrations and entropy, Bull. Am. Math. Soc.80 (1974), 27–41.MATHMathSciNetGoogle Scholar
  10. 10.
    A. G. Vitushkin,On Multidimensional Variations, Gostehisdat, 1955 (in Russian).Google Scholar
  11. 11.
    Y. Yomdin,Global bounds for the Betti numbers of regular fibers of differentiable mappings, Topology24 (1985), 145–152.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Y. Yomdin,C k -resolution of semialgebraic mappings. Addendum to “Volume growth and entropy”, Isr. J. Math.57 (1987), 301–317 (this issue).MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew Univeristy 1987

Authors and Affiliations

  • Y. Yomdin
    • 1
  1. 1.Department of MathematicsBen Gurion Univesity of the NegevBeer ShevaIsrael

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