Let Λ be an invariant set for a C r diffeomorphism f of a manifold M. We say that Λ is a hyperbolic set for f if there is a continuous splitting of the tangent bundle of M restricted to Λ, TM Λ, which is Tf invariant:
$$T{M_\Lambda } = {E^s} \oplus {E^u};\;Tf\left( {{E^s}} \right) = {E^s};\;Tf\left( {{E^u}} \right) = {E^u};$$
and for which there are constants c and λ, c>0 and 0<λ <1, such that
$$\begin{gathered} \left\| {{{\left. {T{f^n}} \right|}_{{E^s}}}} \right\| < c{\lambda ^n},\;n \geqslant 0, \hfill \\ \left\| {{{\left. {T{f^{ - n}}} \right|}_{{E^u}}}} \right\| < c{\lambda ^n},\;n \geqslant 0. \hfill \\ \end{gathered} $$


Periodic Orbit Periodic Point Global Stability Finite Sequence Vertical Band 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Michael Shub
    • 1
  1. 1.Thomas J. Watson Research CenterIBMYorktown HeightsUSA

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