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Transactions of the American Mathematical Society

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Rate of approach to minima and sinks--the Morse-Smale case


Author: Helena S. Wisniewski
Journal: Trans. Amer. Math. Soc. 284 (1984), 567-581
MSC: Primary 58F09
DOI: https://doi.org/10.1090/S0002-9947-1984-0743733-5
MathSciNet review: 743733
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Abstract: The dynamical systems herein are Morse-Smale diffeomorphisms and flows on $ {C^\infty }$ compact manifolds. We show the asymptotic rate of approach of orbits to the sinks of the systems to be bounded by an expression of the form $ K\;\exp ( - DN)$, where $ D$ may be any number smaller than $ C = {\min_p}\{ 1/m\;\log \;\operatorname{Jac}\;{D_P}\,{f^m}\vert{W^u}(P)\} $. Here the minimum is taken over all nonsink $ P$ in the nonwandering set of $ f$, and $ m$ is the period of $ P$. We extend our theorems to the entire manifold, so that there is no restriction on the location of the initial points of trajectories.


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

  • [1] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math, vol. 470, Springer-Verlag, Berlin and New York, 1975. MR 0442989 (56:1364)
  • [2] R. Bowen and D. Ruelle, The ergodic theory of Axiom $ {\text{A}}$ flows, Invent. Math. 29 (1975), 181-202. MR 0380889 (52:1786)
  • [3] R. Bowen, Periodic orbits of hyperbolic flows, Amer. J. Math. 94 (1972), 1-37. MR 0298700 (45:7749)
  • [4] -, A horseshoe with positive measure, Invent. Math. 29 (1975), 203-204. MR 0380890 (52:1787)
  • [5] D. Fried and Michael Shub, Entropy, linearity, and chain recurrence, Extrait des Publications Mathematiques, No. 50, 1978, pp. 203-214. MR 556587 (81a:58033)
  • [6] J. Palis, On Morse-Smale dynamical systems, Topology, Vol. 8, Pergamon Press, New York and Oxford, 1969, 385-405. MR 0246316 (39:7620)
  • [7] J. Palis and S. Smale, Structural stability theorems, Proc. Sympos. Pure Math, Vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 223-232. MR 0267603 (42:2505)
  • [8] C. Pugh and M. Shub, The $ \Omega $-stability theorem for flows, Invent. Math. 11 (1970), 150-158. MR 0287579 (44:4782)
  • [9] M. Shub, Stability and genericity for diffeomorphisms, Dynamical Systems (M. Peixoto, ed.), Academic Press, New York, 1973, pp. 493-514. MR 0331431 (48:9764)
  • [10] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 71 (1967), 747-814. MR 0228014 (37:3598)
  • [11] H. Wisniewski, Rate of approach to minima and sinks--the $ {C^2}$ Axiom $ {\text{A}}$ no cycles case, Geometric Dynamics (Proc. Internat. Sympos. in Dynamical Systems, Rio de Janiero, Brazil, 1981), Lecture Notes in Math, vol. 1007, Springer-Verlag, Berlin and New York. MR 730300 (85b:58102)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0743733-5
Keywords: Dynamical systems, diffeomorphism, Morse-Smale systems, Axiom $ {\text{A}}$ systems, no-cycles, transversality, filtration, hyperbolic invariant set, basic set attractor, flow, unstable and stable manifold
Article copyright: © Copyright 1984 American Mathematical Society

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