This Problem Sheet is based on Lectures 25 and 26.

• Problem 1 shows that for a uniquely ergodic topological dynamical system on a compact metric space, minimality is equivalent to asking that the measure is positive on open sets.
• Problem 2 asks you to prove that reversible topological dynamical systems on $S^1$ with irrational rotation number are uniquely ergodic.
• Problem 3 fills in the parts of Proposition 26.10 that I skipped.
• Problem 4 asks you to check that $\tilde{d}_p$ is a metric.
• Problem 5 proves a converse to Proposition 26.17, and shows that the two metrics $\tilde{d}_p$ and $d_{ \operatorname{R}}$ are strongly equivalent on $\mathscr{P}_p$.