Density of distributional chaos in non-autonomous systems

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2026-03-01 Epub Date: 2026-01-16 DOI:10.1016/j.topol.2026.109735

Francisco Balibrea , Lenka Rucká

引用次数: 0

Abstract

In this paper we are interested in two open problems concerning distributional chaos in non-autonomous discrete dynamical systems as stated in [4] and [18]. As a negative answer to the first problem, we show that positive topological entropy of a pointwise convergent non-autonomous system (as well as distributional chaos of this system) does not imply distributional chaos of its limit map. This disproves a conjecture in [18]. In the second open problem it is wondered if the distributional chaos is a generic property of pointwise convergent non-autonomous systems. We show that the answer is negative for convergent systems on the Cantor set. On the other hand we prove, that distributionally chaotic systems form a dense, but not open (nor closed) set in the space of non-autonomous convergent systems on the interval, independent of the metric we use.

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非自治系统中分布混沌的密度

在本文中，我们对[4]和[18]中所述的关于非自治离散动力系统中分布混沌的两个开放问题感兴趣。作为对第一个问题的否定回答，我们证明了点向收敛非自治系统的正拓扑熵（以及该系统的分布混沌）并不意味着其极限映射的分布混沌。这推翻了b[18]中的一个猜想。在第二个开放问题中，我们想知道分布混沌是否是点向收敛非自治系统的一般性质。我们证明了对于康托集上的收敛系统，答案是否定的。另一方面，我们证明了分布混沌系统在区间上的非自治收敛系统的空间中形成一个稠密但不开（也不闭）的集合，与我们使用的度量无关。

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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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.