# 28. The Variational Principle

In today's lecture we state and prove the famous *Variational Principle*, which tells us that for a topological dynamical system on a compact metric space:

- The topological entropy $\mathsf{h}_{\operatorname{top}}(f)$ is an upper bound for the measure-theoretic entropy $\mathsf{h}_{\mu}(f)$ for any $ \mu \in\mathcal{M}(f)$,
- The measure-theoretic entropy is maximised by looking for measures that assigns most weight to the regions of maximal complexity.

More precisely:

$$ \mathsf{h}_{\operatorname{top}}(f) = \sup_{ \mu \in \mathcal{M}(f)}\mathsf{h}_{\mu}(f).$$

This results connects the two halves of the course together, and thus serves as a nice culmination of all we have done this semester.

#### ☃️ 🎄 Enjoy your holidays! 🦄 🍷

Comments and questions?