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Ω-Stability is not dense in Axiom A

Published online by Cambridge University Press:  19 September 2008

S. E. Patterson
S. E. Patterson
Affiliation:
Northwestern University, Evanston, Illinois, USA
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Abstract

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An example of a diffeomorphism f on a three dimensional manifold M3 is constructed so that f satisfies Axiom A, has a cycle and f has a neighborhood N in Diffr (M3) so that each g in N is not Ω-stable. Existing techniques allow one to extend this example to any compact manifold of dimension greater than two.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 4 , December 1988 , pp. 621 - 632
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

[1]Dankner, A.. Axiom A dynamical systems, cycles, and stability. Topology 19 (1980), 163–177.CrossRefGoogle Scholar
[2]Newhouse, S. & Palis, J.. Hyperbolic Non-Wandering Sets on Two-Dimensional Manifolds, Dynamical Systems, ed. Peixoto, M., Academic Press: New York, 1973, 293–301.Google Scholar
[3]Newhouse, S.. The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. Publ. Math. IHES 50 (1979), 101–151.CrossRefGoogle Scholar
[4]Newhouse, S.. Lectures on Dynamical Systems, Progress in Math. 8, Birkhauser: 1980, Boston, 1–114.Google Scholar
[5]Palis, J.. A Note on Ω-Stability, Global Analysis eds Chern, S. S. & Smale, S., Amer. Math. Soc.: Providence, RI, 1970, 221–222.Google Scholar
[6]Patterson, S.. Thesis, University of North Carolina (1982).Google Scholar
[7]Peixoto, M.. Structural stability on two-dimensional manifolds. Topology 1 (1962).CrossRefGoogle Scholar
[8]Pugh, C., Walker, R. & Wilson, W.. On Morse-Smale Approximation - a Counter Example, J. Differential Equations 23 (1977), 173–182.CrossRefGoogle Scholar
[9]Smale, S.. The Ω-Stability Theorem, Proc. Symp. in Pure Math. 14 (1970), 289–298.CrossRefGoogle Scholar