The equality of Kolmogorov–Sinai entropy and metric permutation entropy generalized

In a 2005 paper, the author and collaborators proposed an approach to permutation entropy based on symbolic dynamics. This approach allowed us to prove the equality of metric permutation entropy to the conventional metric entropy for symbolic dynamics and, as a consequence, also for nn-dimensional interval maps, under the assumption of ergodicity. In this paper we generalize our approach and extend that equality both to general (i.e., not necessarily ergodic) symbolic dynamics and to just measurable maps on (not necessarily ordered) finite-measure spaces—arguably the most general setting possible.

► Definition of permutation entropy (h∗)(h∗) for maps via symbolic dynamics.
► Kolmogorov–Sinai entropy=h∗ for ergodic maps.
► The same for non-ergodic maps using ergodic decomposition of measures.