动力系统 (FnK(X),FnK(f))的动态特性

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2024-08-22 DOI:10.1016/j.topol.2024.109048

引用次数: 0

摘要

假设 (X,f) 是一个动力系统，其中 X 是一个非enerate 连续体，f 是一个映射。对于任意正整数 n，我们考虑具有 Vietoris 拓扑的超空间 Fn(X)。对于 n>1 和 K∈Fn(X)，Fn(X) 的子集 Fn(K,X) 被定义为 Fn(X) 中包含 K 的元素集合。我们考虑商超空间 FnK(X)=Fn(X)⧸Fn(K,X) ，它是通过将 Fn(K,X) 缩小到一个点集而从 Fn(X) 得到的。此外，我们还考虑了诱导映射 Fn(f):Fn(X)→Fn(X) 和 FnK(f):FnK(X)→FnK(X) 。本文引入动力系统 (FnK(X),FnK(f))，并研究条件 f∈M、Fn(f)∈M 和 FnK(f)∈M 之间的关系，其中 M 是以下几类映射之一：传递、混合、弱混合、完全传递、精确、阿金-奥斯兰德-纳加尔意义上的精确、阿金-奥斯兰德-纳加尔意义上的强传递、精确传递、完全精确、强精确传递、强积传递、轨道传递、德瓦尼混沌、不可还原、TT++、强传递和极强传递。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Dynamic properties of the dynamical system (FnK(X),FnK(f))

Let $(X, f)$ be a dynamical system, where X is a nondegenerate continuum and f is a map. For any positive integer n, we consider the hyperspace $F_{n} (X)$ with the Vietoris topology. For $n > 1$ and $K \in F_{n} (X)$ the subset $F_{n} (K, X)$ of $F_{n} (X)$ is defined as the collection of elements of $F_{n} (X)$ containing K. We consider the quotient hyperspace $F_{n}^{K} (X) = F_{n} (X) ⧸ F_{n} (K, X)$ , which is obtained from $F_{n} (X)$ by shrinking $F_{n} (K, X)$ to one point set. Furthermore, we consider the induced maps $F_{n} (f) : F_{n} (X) \to F_{n} (X)$ and $F_{n}^{K} (f) : F_{n}^{K} (X) \to F_{n}^{K} (X)$ . In this paper, we introduce the dynamical system $(F_{n}^{K} (X), F_{n}^{K} (f))$ and we study relationships between the conditions $f \in M$ , $F_{n} (f) \in M$ and $F_{n}^{K} (f) \in M$ , where $M$ is one of the following classes of maps: transitive, mixing, weakly mixing, totally transitive, exact, exact in the sense of Akin-Auslander-Nagar, strongly transitive in the sense of Akin-Auslander-Nagar, exact transitive, fully exact, strongly exact transitive, strongly product transitive, orbit-transitive, Devaney chaotic, irreducible, $T T_{+ +}$ , strongly transitive and very strongly transitive.

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.