This Problem Sheet is based on Lectures 9 and 10.

• Problem 1 shows that for (weakly) expansive systems, the growth rate of the number of periodic points provides a lower bound for the topological entropy.
• Problem 2 asks you to verify directly (i.e. without appealing to $\mathsf{h}_{\operatorname{top}}^* = \mathsf{h}_{\operatorname{top}}$) that $\mathsf{h}_{\operatorname{top}}^*$ is a dynamical invariant.
• Problem 3 asks you to prove that if $f$ is a reversible system on $S^1$ such that $\mathsf{per}(f) \ne \emptyset$ then $f$ is not weakly expansive. Remark: In fact, the assumption that $\mathsf{per}(f) \ne \emptyset$ is not needed—there are no weakly expansive reversible systems on $S^1$ at all. At the moment we lack the machinery to prove this when $\mathsf{per}(f) = \emptyset$, but by the end of the course we will be able to do so.
• Problem 4 concerns systems whose topological entropy grows linearly when iterated.
• Problem 5 shows how powerful an invariant topological entropy is.