This Problem Sheet is based on Lectures 31, 32 and 33.

  • Problem 1 shows that the local stable and unstable manifolds of a hyperbolic fixed point satisfy$$W^s_{\operatorname{loc},r}(u,f) \cap W^u_{\operatorname{loc},r}(u,f) = \{ u \}.$$
  • Problem 2 investigates periodic points close to a hyperbolic fixed points.
  • Problem 3 will no doubt seem completely random...[1]
  • Problems 4 and 5 are about hyperbolic toral automorphisms (cf. Lecture 8).

  1. Spoiler: We will use this result in the proof of the Stable Manifold Theorem next lecture. ↩︎


Comments and questions?