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TOPOLOGICAL ENTROPY OF FREE SEMIGROUP ACTIONS FOR NONCOMPACT SETS

机译:非紧集的自由半群作用的拓扑熵

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In this paper we introduce the topological entropy and lower and upper capacity topological entropies of a free semigroup action, which extends the notion of the topological entropy of a free semigroup action defined by Bufetov [10], by using the Caratheodory-Pesin structure (C-P structure). We provide some properties of these notions and give three main results. The first is the relationship between the upper capacity topological entropy of a skew-product transformation and the upper capacity topological entropy of a free semigroup action with respect to arbitrary subset. The second are a lower and a upper estimations of the topological entropy of a free semigroup action by local entropies. The third is that for any free semigroup action with m generators of Lipschitz maps, topological entropy for any subset is upper bounded by the Hausdorff dimension of the subset multiplied by the maximum logarithm of the Lipschitz constants. The results of this paper generalize results of Bufetov [10], Ma et al. [26], and Misiurewicz [27].
机译:在本文中,我们介绍了一个自由半群作用的拓扑熵和上下容量拓扑熵,它们通过使用卡拉索多里-贝辛结构(CP)扩展了由布费托夫[10]定义的一个自由半群作用的拓扑熵的概念。结构体)。我们提供了这些概念的一些属性,并给出了三​​个主要结果。第一个是偏积乘积变换的上限容量拓扑熵与任意子集的自由半群作用的上限容量拓扑熵之间的关系。第二个是通过局部熵对自由半群作用的拓扑熵的上下估计。第三是对于具有m个Lipschitz映射生成器的任何自由半群作用,任何子集的拓扑熵都以子集的Hausdorff维数乘以Lipschitz常数的最大对数为上限。本文的结果推广了Bufetov [10],Ma等人的结果。 [26]和Misiurewicz [27]。

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