This Problem Sheet is based on Lectures 1 and 2.

  • Problems 1 and 2 ask you to verify claims I made in class regarding the tent map, the logistic map, and the circle rotation.
  • Problem 3 is an example of a dynamical system with exactly one compact invariant set.
  • Problem 4 shows that transitive dynamical systems on metric spaces with isolated points are not very interesting.
  • Problem 5 asks to you to prove the “stronger” version of transitivity I mentioned in class today (which is in fact not stronger at all).
  • Problem 6 concerns transitive maps that are also contractions.
  • Problem 7 gives another equivalent way of formulating transitivity on metric spaces without isolated points.
  • Problem 8 is about transitive flows. The analogue of this result is not true for discrete dynamical systems.

Comments and questions?