The Topological R-entropy And The Measure-theoretic R-entropy Of A Continuous Map

In this paper, we introduce the concepts of the topological r-entropy and the measure-theoretic r-entropy of a continuous map on a compact metric space. After considering some properties of topological r-entropy, we prove the results as follows: 1. Measure-theoretic entropy is the limit of measure-theoretic r-entropy and topological entropy is the limit of topological r-entropy (r→0); 2. Topological r-entropy is more than or equal to the supremum of 6r-entropy in the sense of Feldman's definition, where the measure varies among all the ergodic Borel probability measures.