Today's result really belongs in Dynamical Systems II, since it concerns differentiable dynamical systems. However it makes more sense to cover it now, given that (a) the “manifold” we are working on is the circle $S^1$[1], and (b) it is a natural extension of the Poincaré Classification Theorem from the last lecture.

The Denjoy Theorem can be loosely stated as: If $f \colon S^1 \to S^1$ is a $C^2$ diffeomorphism with irrational rotation number then $f$ is automatically transitive.

Combining this with the Poincaré Classification Theorem tell us that any $C^2$ diffeomorphism of $S^1$ with irrational rotation number is conjugate to the corresponding irrational rotation.

This concludes our study of topological dynamics.

##### Bonus Review of Measure Theory

The second half of today's notes contain a summary (without any proofs) of all the necessary background results in measure theory that we will need for the rest of the course.

1. One-dimensional manifolds are easy to understand! ↩︎