This Problem Sheet is based on Lectures 40-45.

  • Problem 1 gives alternative characterisations of the global stable manifolds $W^s(x,f)$ and $W^s(A,f)$.
  • Problem 2 shows that if one has an invariant splitting that restricts to a hyperbolic splitting on the non-wandering set, then in fact the splitting is hyperbolic everywhere.
  • Problem 3 asks you to improve the Shadowing Theorem under the additional hypothesis that the hyperbolic set is isolated.
  • Problem 4 relates Anosov and mixing dynamical systems.
  • Problem 5 asks you to show that if the non-wandering set $ \mathsf{nw}(f)$ is hyperbolic, then the closure of the periodic points can be identified with the non-wandering set of the restriction $f|_{\mathsf{nw}(f)}$.

Comments and questions?