论文标题
无限产品空间的任意子集的拓扑熵
Topological Entropy for Arbitrary Subsets of Infinite Product Spaces
论文作者
论文摘要
在本说明中，引入了紧凑型拓扑空间中所有序列空间的任意子集的广义拓扑熵的概念。结果表明，对于紧凑型空间上的连续地图，地图的所有轨道集的广义拓扑熵与地图的经典拓扑熵一致。考虑了这种新的熵概念的一些基本特性；其中包括：熵关于脱节联合，笛卡尔产物，组件限制和扩张，偏移映射以及与越野拓扑的某些连续性特性有关的行为。例如，显示出有限收缩家族给出的分形的任何自相似结构都会引起分形子集的内在拓扑熵的概念。引入了与任何增加的兼容半学分序列相关的鲍恩熵概念，并引入了拓扑空间上的任何兼容半学分序列，并考虑了其某些基本特性。作为$ 1 \ leq p \ leq \ infty $的特殊情况，介绍了任何公制空间的序列的bowen $ p $ entropy。结果表明，针对紧凑的度量空间的广义拓扑熵和鲍恩$ \ infty $ - entropy的概念一致。
In this note a notion of generalized topological entropy for arbitrary subsets of the space of all sequences in a compact topological space is introduced. It is shown that for a continuous map on a compact space the generalized topological entropy of the set of all orbits of the map coincides with the classical topological entropy of the map. Some basic properties of this new notion of entropy are considered; among them are: the behavior of the entropy with respect to disjoint union, cartesian product, component restriction and dilation, shift mapping, and some continuity properties with respect to Vietoris topology. As an example, it is shown that any self-similar structure of a fractal given by a finite family of contractions gives rise to a notion of intrinsic topological entropy for subsets of the fractal. A generalized notion of Bowen's entropy associated to any increasing sequence of compatible semimetrics on a topological space is introduced and some of its basic properties are considered. As a special case for $1\leq p\leq\infty$ the Bowen $p$-entropy of sets of sequences of any metric space is introduced. It is shown that the notions of generalized topological entropy and Bowen $\infty$-entropy for compact metric spaces coincide.
