Topological permutation entropy

Permutation entropy quantifies the diversity of possible ordering of the successively observed values a random or deterministic system can take, just as Shannon entropy quantifies the diversity of the values themselves. When the observable or state variable has a natural order relation, making permutation entropy possible to compute, then the asymptotic rate of growth in permutation entropy with word length forms an alternative means of describing the intrinsic entropy rate of a source. Herein, extending a previous result on metric entropy rate, we show that the topological permutation entropy rate for expansive maps equals the conventional topological entropy rate familiar from symbolic dynamics. This result is not limited to one-dimensional maps.