Variational principle for topological entropy

Remarks

The set $\mathcal{M}(f)$ is nonempty by the Krylov–Bogolyubov theorem. The inequality $h_\mu(f) \le h_{\mathrm{top}}(f)$ follows by comparing a finite measurable partition with an open cover, and Misiurewicz gave a short proof of the reverse inequality. For the reverse inequality one builds, from large $(n, \varepsilon)$-separated sets, averaged measures whose weak-$\ast$ limit is invariant and has entropy close to $h_{\mathrm{top}}(f)$. The restriction to ergodic measures follows from the ergodic decomposition, since $h_\mu$ is the average of the entropies of the ergodic components. Example: for the full shift on $k$ symbols, $h_{\mathrm{top}} = \log k$, and the Bernoulli measure giving each symbol mass $1/k$ has entropy $\log k$, so the supremum is attained. The supremum is not always attained; a measure that attains it is called a measure of maximal entropy.

Going deeper

For a student

The theorem lets one compute topological entropy from a well-chosen invariant measure, and conversely bounds all measure-theoretic entropies by a topological quantity. It is the starting point of the thermodynamic formalism, where entropy is replaced by pressure and the maximizing measures become equilibrium states. Measures of maximal entropy exist when $\mu \mapsto h_\mu(f)$ is upper semicontinuous, for example for expansive homeomorphisms.