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.



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