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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
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.

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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