This Problem Sheet is based on Lectures 19 and 20.

• Problem 1 asks you to prove that the doubling map $e_2 \colon S^1 \to S^1$ is ergodic with respect to Lebesgue measure.
• Problem 2 asks you to prove that the circle rotation $\rho_{ \theta} \colon S^1 \to S^1$ is ergodic with respect to Lebesgue measure if and only if $\theta$ is irrational.
• Problem 3 gives another criterion for ergodicity.
• Problems 4 and 5 introduce the notion of the first return time and the Poincaré return map.