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Stability of Morse-Smale Maps

  • Jack K. Hale
  • Luis T. Magalhães
  • Waldyr M. Oliva
Part of the Applied Mathematical Sciences book series (AMS, volume 47)

Abstract

We will deal in this section with smooth maps f: B → E, B being a Banach manifold imbedded in a Banach space E. The maps f belong to Cr(B, E), the Banach space of all E-valued Cr-maps defined on B which are bounded together with their derivatives up to the order r ≥ l. Let Cr(B, B) be the subspace of Cr(B, E) of all maps leaving B invariant, that is, f(B) ⊂ B. Denote by A(f) the set
$$\begin{array}{*{20}{c}} {A\left( f \right) = \left\{ {x \in B:\,there\;exists\,a\,sequence} \right.\,\left( {x = {{x}_{1}},{{x}_{2}}, \ldots } \right) \in B,} \\ {\left. {\mathop{{\sup }}\limits_{j} \left\| {{{x}_{j}}} \right\| < \infty \;and\,f\left( {{{x}_{j}}} \right) = {{x}_{{j - 1}}},j = 2,3, \ldots } \right\}.} \\ \end{array}$$

Keywords

Periodic Point Compact Manifold Unstable Manifold Neighborhood Versus Transversal Intersection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Jack K. Hale
    • 1
  • Luis T. Magalhães
    • 2
  • Waldyr M. Oliva
    • 3
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Universidade Tecnica de LisbõaLisbonPortugal
  3. 3.Departmento de Matemática Aplicada, Instituto de Matemática e EstatisticaUniversidade de São PauloSão PauloBrasil

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