Global Regularities Of Fractal Dimensions And Topological Entropy In One-Dimensional Symbolic Dynamics

In this paper, the global regularities of fractal dimensions and topological entropy in one-dimensional maps are studied by using the method of symbolic dynamics. For m-modal maps with m turning points C1,C2,…,Cm, by defining equivalent scalingfactors α_e as the geometric average of all scaling factors α_Ci (i=1,2,…,m),|α_e|= |α_C1α_C2…α_Cm|1/m, global superuniversal relationship between equivalent scaling factors α_e and fractal dimensions d of Feigenbaum-type attractors on all the infinitely many critical (accumulation) points of transitions to chaos is obtained: d(W)log_|W||α_e(W)|=β^e, where|W| is the basic period of the m-tuply superstablesequences W, and β^e is not only independent of the concrete sequences W, but also independent of the concrete maps. This is a general superuniversality for arbitrary multimodal maps, which is verified numerically for the cases of m ≤3 and can also be extended to Lorenz maps with a discontinuous point.By choosing bimodal maps as an entrance for investigating multimodal ones, a global regularity of topological entropy, i.e., the multifractal devil's carpet of topological entropy, is found. The devil's carpet possesses a perfect subregion similarity, through which the generalized Milnor-Thurston conjecture is visually realized. The research conducted in this paper solves the symbolic structure of the devil's carpet and the metrization of the plateau of an equal topological entropy class (ETEC), and presents the symbolic description and visual realization of equal entropy fractals. The physical purpose of the devil's carpet lies in the fact that it can answer how many generalized Feigenbaum's universal scaling constants exist (infinitelymany) and where they distribute (within the ETEC plateaus). An elementary exploration of the fractal characteristic of the topological entropy "flow" on the devil's carpet is also carried out. Based on these, a classification of all admissible sequences in the complete topological space Σ₃ of three letters is suggested. Theseresults reveal the complexity of global dynamical behavior in the whole parameter plane of bimodal systems.Moreover, an effective numerical method of the word-lifting technique for calculating parameters of symbolic sequences is proposed, which can overcome the difficulty of lacking explicit expressions of the inverse functions of the maps with orders higher than or equal to 5. This lays a practical operational foundation for the study of metric universalities and global regularities in arbitrary multimodal maps.The results presented in this paper, i.e., the global regularities of fractal dimensions and topological entropy in one-dimensional maps, will promote the study of global regularities and universalities in higher dimensional systems, and will be of positive significance to the further study of regularities and universalities of nonlinear phenomena in celestial systems.