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A quantitative version of the Kupka-Smale theorem

Published online by Cambridge University Press:  19 September 2008

Y. Yomdin
Y. Yomdin
Affiliation:
Ben Gurion University of the Negev, Beer-Sheva 84120, Israel
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Abstract

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Let Mm be a compact, m-dimensional smooth manifold. The n-periodic point x of a diffeomorphism f: M → M is called γ-hyperbolic, for γ≥O, if the eigenvalues λj, of dfn(x) satisfy . We prove that any Ck-diffeomorphism f: M → M, k≥3, for any ε>0 can be ε-approximated in Ck-norm by fe: M → M such that for any n each n-periodic point of fe is (a(ε))nα - hyperbolic. Here and ao>0 depends on f

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 3 , September 1985 , pp. 449 - 472
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Abraham, R. & Robbin, J.. Tranversal Mappings and Flows. W. A. Benjamin, Inc: New York-Amsterdam, (1967).Google Scholar
[2]Artin, M. & Masur, B.. On periodic points. Ann. Math. no. 81 (1965), 82–99.CrossRefGoogle Scholar
[3]Gromov., M.. On the entropy of holomorphic maps. Preprint.Google Scholar
[4]Kupka, I.. Contribution à la théorie des champs génériques, Contrib. Dig. Eqs. 2 (1963), 457–484;Google Scholar
Kupka, I.. Contribution à la théorie des champs génériques, Contrib. Dig. Eqs. 3 (1964), 411–420.Google Scholar
[5]Peixoto, M.. On an approximation theorem of Kupka and Smale. J. Diff. Eqs. 3, 2 (1967), 214–227.CrossRefGoogle Scholar
[6]Sard, A.. The measure of the critical values of differentiable maps. Bull Amer. Math. Soc. 48 (1942), 883–890.CrossRefGoogle Scholar
[7]Smale, S.. Stable manifolds for differential equations and diffeomorphisms. Ann. Sc. Norm. Sup. Pisa 18 (1963), 97–116.Google Scholar
[8]Thom, R.. Quelques propriétes globales des variétés différentiable. Comment. Math. Helv. 28 (1954), 17–86.CrossRefGoogle Scholar
[9]Vituskin, A. G.. O Mnogomernyh Variazijah. Gostehisdat: Moskva, 1955.Google Scholar
[10]Yomdin, Y.. The geometry of critical and near-critical values of differentiable mappings. Math. Ann. 264 (1983), 495–515.CrossRefGoogle Scholar
[11]Yomdin, Y.. Quantitative transversality. In preparation.Google Scholar