Topological entropy of continuous functions on topological spaces |
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| Authors: | Lei Liu Yangeng Wang Guo Wei |
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| Institution: | 1. Department of Mathematics, Northwest University, Xian, Shaanxi 710069, PR China;2. Department of Mathematics and Computer Science, University of North Carolina at Pembroke, Pembroke, NC 28372, USA;1. Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA;2. Department of Mathematics, South University of Science and Technology of China, 518055 Shenzhen, PR China;3. Department of Applied Mathematics, College of Science, China Agricultural University, Beijing, 100083, PR China;1. Instituto de Matemática, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile;2. Department of Mathematics, Southern University of Science and Technology, 1088 Xueyuan Rd., Xili, Nanshan District, Shenzhen, Guangdong, 518055, China;3. Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil;1. Department of Mathematics, South China University of Technology, Guangzhou 510641, PR China;2. Department of Applied Mathematics, Chinese Culture University, Yangmingshan, Taipei 11114, Taiwan;1. IMJ-PRG, CNRS (UMR 7586), Sorbonne Université, Paris Université, Campus Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 05, France;2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile;1. Department of Mathematics, Bradley University, Peoria, IL 61625, USA;2. Department of Mathematics, Christopher Newport University, Newport News, VA 23606, USA;1. Department of Mathematics, Howard University, Washington, DC 20059, USA;2. Department of Pure Mathematics, University of Leeds, Leeds LS2 9J2, UK;3. Université de Lyon, Université Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, F69622 Villeurbanne Cedex, France |
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| Abstract: | Adler, Konheim and McAndrew introduced the concept of topological entropy of a continuous mapping for compact dynamical systems. Bowen generalized the concept to non-compact metric spaces, but Walters indicated that Bowen’s entropy is metric-dependent. We propose a new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new entropy, and compare the new entropy with the existing ones. The defined entropy generates that of Adler, Konheim and McAndrew and is metric-independent for metrizable spaces. Yet, it holds various basic properties of Adler, Konheim and McAndrew’s entropy, e.g., the entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same entropy, the entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new entropy coincides with Adler, Konheim and McAndrew’s entropy for compact systems. |
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