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$\Omega $-inverse limit stability theorem


Author: Hiroshi Ikeda
Journal: Trans. Amer. Math. Soc. 348 (1996), 2183-2200
MSC (1991): Primary 58F10; Secondary 58F15
DOI: https://doi.org/10.1090/S0002-9947-96-01629-7
MathSciNet review: 1355074
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Abstract: We prove that if an endomorphism $f$ satisfies weak Axiom A and the no-cycles condition then $f$ is $\Omega $-inverse limit stable. This result is a generalization of Smale's $\Omega $-stability theorem from diffeomorphisms to endomorphisms.


References [Enhancements On Off] (What's this?)

  • 1. R.Bowen, On Axiom A Diffeomorphisms, Regional Conference Ser. Math. 35, A.M.S., Providence, Rhode Island, 1978. MR 58:2888
  • 2. M.Hirsch, C.Pugh, M.Shub, Invariant Manifolds, Lecture Notes in Math. 583, Springer-Verlag, New York (1977). MR 45:1188
  • 3. M.Hirsch, J.Palis, C.Pugh, M.Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9(1970), 121-134. MR 41:7232
  • 4. H.Ikeda, On infinitesimal stability of endomorphisms, The Study of Dynamical Systems, vol. 7, World Scientific, Singapore, 1989, pp. 59-84. MR 92e:58114
  • 5. H.Ikeda, Infinitesimally stable endomorphisms, Trans. Amer. Math. Soc. 344 (1994), 823-833. MR 95c:58116
  • 6. R.Mañé, Axiom A for endomorphisms, Lecture Notes in Math. 597, Springer-Verlag, New York (1977), 379-388. MR 57:14059
  • 7. R.Mañé and C.Pugh, Stability of endomorphisms, Lecture Notes in Math. 468, Springer-Verlag, New York (1975) 175-184. MR 58:31264
  • 8. F.Przytycki, Anosov endomorphisms, Studia Math. 58(1976), 249-285. MR 56:3893
  • 9. F.Przytycki, On $\Omega $-stability and structural stability of endomorphisms satisfying Axiom A, Studia Math. 60(1977), 61-77. MR 56:3891
  • 10. J.Quandt, Stability of Anosov maps, Proc. Amer. Math. Soc. 104(1988), 303-309. MR 89m:58118
  • 11. J.Quandt, On inverse limit stability for maps, J. Differential Equations 79 (1989), 316-339. MR 91a:58142
  • 12. M.Shub, Global Stability of Dynamical Systems, Springer-Verlag, New York, 1987. MR 87m:58086
  • 13. S.Smale, The $\Omega $-stability theorem, in Global Analysis, Proc. Symp. Pure Math. 14, pp. 289-297, A.M.S., Providence, Rhode Island, 1970. MR 42:6852

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Additional Information

Hiroshi Ikeda
Affiliation: Department of Mathematics, School of Education, Waseda University, Shinjuku, Tokyo, 169-50, Japan

DOI: https://doi.org/10.1090/S0002-9947-96-01629-7
Keywords: Inverse limit stability, weak Axiom A, prehyperbolic sets
Received by editor(s): November 12, 1993
Received by editor(s) in revised form: July 18, 1994
Dedicated: Dedicated to the memory of my father
Article copyright: © Copyright 1996 American Mathematical Society

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