RENYI ENTROPY OF MAPS: APPLICATIONS TO FUZZY SETS. PATTERN RECOGNITION, AND CHAOTIC DYNAMICS

Abstract

One of the main and possibly only interest in Renyi entropy is to provide a model to measure uncertainty when the observer involves subjectivity via prior knowledge or prior misknowledge. Here we extend this concept and derive a Renyi entropy and a “cross-entropy of orders” for differentiable maps. This suggests a new approach to the entropy of fuzzy sets identified with the entropy of their membership functions considered as maps; and it provides an extension of Liapunov exponent, say the Liapunov exponent of order s, which is useful for analyzing chaotic dynamics as observed by an observer involving subjectivity. The theory applies to discrete maps by using the so-called concept of complete entropy introduced in earlier work.