Six Modes of Triply Super Stable Kneading Sequences in 1D Trimodal Maps

Abstract

Triply super stable kneading sequences (TSSKS) are very important kernel concept in the study of symbolic dynamics of 1D trimodal maps.  For a given period n, there are six types of TSSKS in which two of them decide the six cyclic star products, others supplemented the ‘joints’ in the symbolic space. For the former, start products provide the method to research metric universalities in the period-n-tupling process, the devil’s staircase of topological entropy and self-similar bifurcation structure in classical dynamical systems, the later will be calculated and obtained the corresponding parameters which can occupy  so called the admissibe  region. In this paper, firstly, for a period n takes 3-12, we produce all the permutations of six types , , , ,  and , by the famous admissibilty conditions, the admissible sets  are obtained respectively, here m stands for the mode of the TSSKS and takes integer 0-5; second, for ,  detemines a system of nonlinear equations uniquely by passing three critical points , m ensures six different modes of equations, for an proper initial value, the newton-iteration method is applied to get the three parameters of . For m takes 2-5, these parameters of TSSKS in  are calculated firstly in the paper, it would describe the parameter space and boundaries and enhence the knowledge of symbolic dynamics of 1D trimodal maps.