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)


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
$$\begin{array}{*{20}{c}} {A\left( f \right) = \left\{ {x \in B:\,there\;exists\,a\,sequence} \right.\,\left( {x = {{x}_{1}},{{x}_{2}}, \ldots } \right) \in B,} \\ {\left. {\mathop{{\sup }}\limits_{j} \left\| {{{x}_{j}}} \right\| < \infty \;and\,f\left( {{{x}_{j}}} \right) = {{x}_{{j - 1}}},j = 2,3, \ldots } \right\}.} \\ \end{array}$$


Periodic Point Compact Manifold Unstable Manifold Neighborhood Versus Transversal Intersection 
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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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