Commun. Math. Phys. 213, 381 – 411 (2000) Communications in Mathematical Physics © Springer-Verlag 2000 The Generalized Milnor–Thurston Conjecture and Equal Topological Entropy Class in Symbolic Dynamics of Order Topological Space of Three Letters Shou-Li Peng 1, 2 , Xu-Sheng Zhang 2 1 CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, P. R. China 2 Center for Nonlinear Complex Systems, Department of Physics, College of Science,Yunnan University, Kunming,Yunnan 650091, P. R. China. E-mail: slpeng@ynu.edu.cn Received: 4 February 1998 / Accepted: 1 March 2000 Abstract: This paper presents an answer to an open problem in the dynamical systems of three letters: the generalized Milnor–Thurston conjecture on the existence of infinitely many plateaus of topological entropy in the two-dimensional parameter plane. The con- cept of equal topological entropy class is introduced by the dual star product which is a generalization of the Derrida–Gervois–Pomeau star product to the symbolic dynamics of three letters for the endomorphisms on the interval. The algebraic rules established by the dual star products for the doubly superstable kneading sequences are equivalent to the normal factorization of the Milnor–Thurston characteristic polynomials. Moreover, the classification theory of symbolic primitive and compound sequences based on the topological conjugacy in the meaning of equal entropy is completed in the topological space  3 of three letters. 1. Introduction About twenty years ago, Milnor and Thurston [1] proposed the kneading theory for the study of piecewise monotone maps on the one-dimensional interval, which has played a key role in establishing the symbolic dynamics of endomorphisms on the interval. At almost the same time Derrida, Gervois and Pomeau [2] presented their star product of endomorphisms on the ground of experimental symbolic description [3], which supplied a symbolic representation of two letters to real numbers. This star product provides an extremely important tool for studying the complete classification and the universalities in unimodal maps [4, 5]. These two works laid the foundation of the symbolic dynamics of one-dimensional endomorphisms, even if they were not thought to have any connection with each other in the initial stage of development of the symbolic dynamics of one- dimensional endomorphisms. Milnor and Thurston [1] conjectured that topological entropies increase monotoni- cally in the region of non-zero entropy and there exist infinitely many intervals of constant topological entropies in the one-dimensional parameter line of the quadratic maps. This