# 19. Ergodicity

In this lecture we move onto the second “flavour” of dynamical systems that we will consider over the year: measure-preserving transformations on probability spaces.

The name “measure-theoretic dynamics” isn't very catchy, so this subject usually goes by the name *ergodic theory.*

We prove the famous *Poincaré Recurrence Theorem*, and then define what it means for a dynamical system on a probability space to be ergodic.

Ergodicity is the measure-theoretic analogue of (topological) transitivity—the rest of the lecture is devoted to explaining why.

Comments and questions?