This Problem Sheet is based on Lectures 5 and 6.

• Problem 1 gives a bunch more equivalent ways of formulating the weakly mixing property using two or three sets[1]. (Compare to Proposition 5.13.)
• Problem 2 is (yet) another cute characterisation of weakly mixing systems.
• Problem 3 introduces the notion of totally transitive systems, and asks you to prove that weakly mixing $\Rightarrow$ totally transitive, and that totally transitive + chaotic $\Rightarrow$ weakly mixing.
• Problem 4 is fun (and harder than it looks).

1. This also shows that after I messed up the proof of Proposition 5.13 in lecture and drew the picture backwards my claim that "it didn't ↩︎