Topological entropy of fixed-point free flows
Topological entropy of fixed-point free flows
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DOI：
10.1090/s0002-9947-1990-1010414-5
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发表时间：
1990-02
影响因子：
1.3
通讯作者：
Romeo F. Thomas
中科院分区：
文献类型：
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作者：
Romeo F. Thomas
引入拓扑熵作为拓扑共轭的一个不变量，同时也是测度论的一个类比。紧度量空间上单参数流的拓扑熵由Bowen定义。关于拓扑熵的一般说法得到了证明，但拓扑熵的计算并不容易，并证明了它在共轭下是不变的。尽管如此，我想尝试提出一个新的方向，并研究拓扑熵的定义，它涉及处理由于允许轨道重新参数化而产生的技术困难。利用该定义还证明了一些著名的结果。这些结果使我们能够证明一些用Bowen定义难以证明的结果。我们还证明了该定义等价于Bowen对紧致度量空间上没有不动点的任何流的定义。最后，证明了扩张流的拓扑熵可以在局部截面上全局定义。
Topological entropy was introduced as an invariant of topological conjugacy and also as an analogue of measure theoretic entropy. Topological entropy for one parameter flows on a compact metric spaces is defined by Bowen. General statements are proved about this entropy, but it is not easy to calculate the topological entropy, and to show it is invariant under conjugacy. For all this I would like to try to pose a new direction and study a definition for the topological entropy that involves handling the technical difficulties that arise from allowing reparametrizations of orbits. Some well-known results are proved as well using this definition. These results enable us to prove some results which seem difficult to prove using Bowen's definition. Also we show here that this definition is equivalent to Bowen's definition for any flow without fixed points on a compact metric space. Finally, it is shown that the topological entropy of an expansive flow can be defined globally on a local cross sections.