Topological entropy of continuous functions on topological spaces
Topological entropy of continuous functions on topological spaces
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DOI：
10.1016/j.chaos.2007.04.008
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发表时间：
2009-01
影响因子：
7.8
通讯作者：
Lei Liu;Yangeng Wang;Guo Wei
中科院分区：
文献类型：
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作者：
Lei Liu;Yangeng Wang;Guo Wei
Adler、Konheim 和 McAndrew 引入了紧凑动力系统连续映射的拓扑熵概念。鲍文将这个概念推广到非紧度量空间，但沃尔特斯指出鲍文的熵是度量相关的。我们提出了拓扑熵的新定义，用于任意拓扑空间上的连续映射（紧致性、可度量性，甚至不一定需要分离公理），研究新熵的基本属性，并将新熵与现有熵进行比较。定义的熵生成 Adler、Konheim 和 McAndrew 的熵，并且对于可度量空间来说是度量无关的。然而，它具有阿德勒、康海姆和麦克安德鲁熵的各种基本性质，例如，子系统的熵受原始系统熵的限制，拓扑共轭系统具有相同的熵，诱导超空间系统的熵大于或等于原始系统的熵，特别是这个新熵与阿德勒，康海姆和麦克安德鲁紧凑系统的熵一致。
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.