Abstract
A diffeomorphism of a smooth (compact) manifold may exhibit a globally (uniformly) hyperbolic behaviour, like Morse-Smale, Anosov, or Axiom A diffeomorphisms. But, even when the global behaviour is not hyperbolic, it occurs very frequently that the set of points whose orbits are constrained to stay in some appropriate open subset of the manifold is compact and has a hyperbolic structure. Such hyperbolic compact invariant sets then provide a good starting point for a global understanding of the dynamics.
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