$\Omega$-inverse limit stability theorem
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- by Hiroshi Ikeda PDF
- Trans. Amer. Math. Soc. 348 (1996), 2183-2200 Request permission
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
- Rufus Bowen, On Axiom A diffeomorphisms, Regional Conference Series in Mathematics, No. 35, American Mathematical Society, Providence, R.I., 1978. MR 0482842
- M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc. 76 (1970), 1015–1019. MR 292101, DOI 10.1090/S0002-9904-1970-12537-X
- M. Hirsch, J. Palis, C. Pugh, and M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1969/70), 121–134. MR 262627, DOI 10.1007/BF01404552
- Hiroshi Ikeda, On infinitesimal stability of endomorphisms, The study of dynamical systems (Kyoto, 1989) World Sci. Adv. Ser. Dynam. Systems, vol. 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 59–84. MR 1117286
- Hiroshi Ikeda, Infinitesimally stable endomorphisms, Trans. Amer. Math. Soc. 344 (1994), no. 2, 823–833. MR 1250821, DOI 10.1090/S0002-9947-1994-1250821-2
- Ricardo Mañé, Axiom A for endomorphisms, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Lecture Notes in Math., Vol. 597, Springer,#Berlin, 1977, pp. 379–388. MR 0474419
- Ricardo Mañé and Charles Pugh, Stability of endomorphisms, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, pp. 175–184. MR 0650659
- Feliks Przytycki, Anosov endomorphisms, Studia Math. 58 (1976), no. 3, 249–285. MR 445555, DOI 10.4064/sm-58-3-249-285
- Feliks Przytycki, On $U$-stability and structural stability of endomorphisms satisfying Axiom A, Studia Math. 60 (1977), no. 1, 61–77. MR 445553, DOI 10.4064/sm-60-1-61-77
- Juergen Quandt, Stability of Anosov maps, Proc. Amer. Math. Soc. 104 (1988), no. 1, 303–309. MR 958088, DOI 10.1090/S0002-9939-1988-0958088-X
- Juergen Quandt, On inverse limit stability for maps, J. Differential Equations 79 (1989), no. 2, 316–339. MR 1000693, DOI 10.1016/0022-0396(89)90106-X
- Michael Shub, Global stability of dynamical systems, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR 869255, DOI 10.1007/978-1-4757-1947-5
- S. Smale, The $\Omega$-stability theorem, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 289–297. MR 0271971
Additional Information
- Hiroshi Ikeda
- Affiliation: Department of Mathematics, School of Education, Waseda University, Shinjuku, Tokyo, 169-50, Japan
- Received by editor(s): November 12, 1993
- Received by editor(s) in revised form: July 18, 1994
- © Copyright 1996 American Mathematical Society
- 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
Dedicated: Dedicated to the memory of my father