# Problem Sheet M

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$.

Comments and questions?