Construction of maps with generating partitions for entropy evaluation.

Abstract

In this paper we study the topological entropy and the metric entropy of one-dimensional maps. Let T(x,s) be a map with a parameter s. Introduced in the paper is a class of maps defined over a special set of parameter values. These maps are characterized by possession of generating partitions and will be referred to as generating maps. They have some very attractive properties. According to their topological properties and with the aid of kneading sequences, the generating maps may be classified in a systematic way and their topological entropy values can be determined readily by using a theorem on kneading determinant by Milnor and Thurston. The generating partitions also allow us to compute the metric entropy of the maps with relative ease. After some general discussions this method of entropy evaluation is applied to the tent map, the logistic map, the spike map, and a discontinuous map. For the logistic map, the topological entropy values for a set of 306 generating maps and the metric entropy values for a set of 244 generating maps have been computed. The data show that the topological entropy never decreases with an increasing s, and that the computed metric entropy of each map agrees with the Liapunov number of the map. From the results given in this paper one sees that the generating maps can play a significant role in delineating the ergodic properties of maps.