Entropy and preimage sets
Entropy and preimage sets
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DOI：
10.1017/s0143385703000221
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发表时间：
2003-12
影响因子：
0.9
通讯作者：
D. Fiebig;U. Fiebig;Z. Nitecki
中科院分区：
文献类型：
--
作者：
D. Fiebig;U. Fiebig;Z. Nitecki
研究了拓扑熵与原像离散度之间的关系。符号动力学在我们的调查中起着至关重要的作用。对于正向扩张映射，我们证明了由Hurley定义的两个逐点前像不变量彼此一致且与拓扑熵一致，并且反映在单个点的前像数的增长率上，称为映射的前像增长点。我们将这一概念推广到系统的一个熵点，其中$-稳定集的前象的离散度度量拓扑熵。我们证明了对于满足规范性质的弱形式的映射，每个点都是一个熵点，并且每个渐近h-扩张同胚(特别是紧流形的每个光滑微分同胚)都有熵点。给出了Hurley不变量不同的映射和没有熵点的同胚映射的例子。
We study the relation between topological entropy and the dispersion of preimages. Symbolic dynamics plays a crucial role in our investigation. For forward expansive maps, we show that the two pointwise preimage entropy invariants defined by Hurley agree with each other and with topological entropy, and are reflected in the growth rate of the number of preimages of a single point, called a preimage growth point for the map. We extend this notion to that of an entropy point for a system, in which the dispersion of preimages of an $\varepsilon$-stable set measures topological entropy. We show that for maps satisfying a weak form of the specification property, every point is an entropy point and that every asymptotically h-expansive homeomorphism (in particular, every smooth diffeomorphism of a compact manifold) has entropy points. Examples are given of maps in which Hurley's invariants differ and of homeomorphisms with no entropy points.