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Application of Gray codes to the study of the theory of symbolic dynamics of unimodal maps
Institution:1. Grupo de Neurocomputación Biológica, Dpto. de Ingeniería Informática, Escuela Politécnica Superior, Universidad Autónoma de Madrid, 28049 Madrid, Spain;2. Instituto de Seguridad de la Información, Consejo Superior de Investigaciones Científicas, Serrano 144, 28006 Madrid, Spain;1. Department of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, Jiangsu 210023, PR China;2. School of Automation, Southeast University, Nanjing, 210096, PR China;3. Institute of Information and Control Engineering Technology, Nanjing Normal University, Nanjing, Jiangsu 210042, PR China;4. College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing, Jiangsu 210042, PR China;1. Department of Mathematics, Eslamabad-E-Gharb Branch, Islamic Azad University, Eslamabad-E-Gharb, Iran;2. School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran;1. College of Science, Shandong University of Science and Technology, Qingdao 266590, PR China;2. Shandong Key Laboratory of Robotics and Intelligent Technology, Qingdao 266590, PR China;1. MOE Key Lab for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China;2. College of Information Engineering, Tarim University, Alar, Xinjiang 843300, China;3. School of Science, Xi’an Polytechnic University, Xi’an, Shaanxi 710048, China
Abstract:In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works.
Keywords:Unimodal maps  Kneading sequences  Symbolic sequences  Gray Ordering Number  GON  Mandelbrot map
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