Some Notes on Topological and Measure-Theoretic Entropy

Abstract

In this note, we introduce four measure-theoretic quantities for Borel probability measures to characterize upper and lower Katok measure-theoretic entropies and establish two new variational principles for Bowen topological entropy and packing topological entropy. Besides, we prove that the packing topological entropy of the set of generic points for any invariant ergodic Borel probability measure is equal to the measure-theoretic entropy.