This Problem Sheet is based on Lectures 27 and 28.

  • Problems 1 and 2 show that measure-theoretic entropy behaves nicely with respect to iterations and products.
  • Problem 3 asks you to prove that a reversible measure-preserving dynamical system that admits a generator necessarily has zero measure-theoretic entropy.
  • Problem 4 asks you to show that circle rotations have zero measure-theoretic entropy with respect to Lebesgue measure.
  • Problem 5 asks you to prove that the topological entropy of any dynamical system $f$ is equal to the topological entropy of its restriction $f|_{ \mathsf{nw}(f)}$ to its non wandering set $ \mathsf{nw}(f)$. This is a cute application of the Variational Principle.

Comments and questions?