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Topological entropy of Devaney chaotic maps

机译：Devaney混沌映射的拓扑熵

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摘要

The infimum respectively minimum of the topological entropies in different spaces are studied for maps which are transitive or chaotic in the sense of Devaney (i.e., transitive with dense periodic points). After a short survey of results explicitly or implicitly known in the literature for zero and one-dimensional spaces the paper deals with chaotic maps in some higher-dimensional spaces. The key role is played by the result saying that a chaotic map f in a compact metric space X without isolated points can always be extended to a triangular (skew product) map F in X x [0,1] in such a way that F is also chaotic and has the same topological entropy as f. Moreover, the sets X x {0} and X x {1} are F-invariant which enables to use the factorization and obtain in such a way dynamical systems in the cone and in the suspension over X or in the space X x S. This has several consequences. Among others, the best lower bounds for the topological entropy of chaotic maps on disks, tori and spheres of any dimensions are proved to be zero.
机译：对于在Devaney意义上具有传递性或混沌性的地图（即具有密集周期点的传递性），研究了不同空间中拓扑熵的最小或最小值。在对文献中显式或隐式已知的零和一维空间的结果进行简短调查之后，本文处理了一些高维空间中的混沌映射。结果的关键作用在于，紧凑度量空间X中没有孤立点的混沌映射f总是可以扩展为X x [0,1]中的三角形（歪积）映射F，使得F也是混沌的，并且具有与f相同的拓扑熵。此外，集合X x {0}和X x {1}是F不变的，这使得可以使用分解并以这种方式获得圆锥体以及X上或空间X x S中的悬浮体中的动力学系统。这有几个后果。其中，磁盘，托里和任何尺寸的球面上的混沌图的拓扑熵的最佳下界被证明为零。

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