Is there a connection between topological entropy and stationary distributions?

Question

In a book I read the following:

"The topological entropy is the supremum over all stationary distributions of the entropy of the corresponding stationary sequence."

I did not find this definition anywhere else, do you know this definition?

Answer 1

This is not a definition of topological entropy, but rather a version of a result called the "variational principle". I can't remember who it is due to. This result states that topological entropy of a map $T$ coincides with its largest Kolmogorov-Sinai entropy, maximised over all $T$-invariant distributions. Note that the K-S entropy of $T$ is basically the same as the Shannon entropy rate of the sequence generated by iterating $T$ from a random initial state with a stationary (i.e. $T$-invariant) distribution.