Distributional chaos in random dynamical systems

In this paper, we introduce the notion of distributional chaos and the measure of chaos for random dynamical systems generated by two interval maps. We give some sufficient conditions for a zero measure of chaos and examples of chaotic systems. We demonstrate that the chaoticity of the functions that generate a system does not, in general, affect the chaoticity of the system, i.e. a chaotic system can arise from two nonchaotic functions and vice versa. Finally, we show that distributional chaos for random dynamical system is, in some sense, unstable.     

关于周期吸附系统的分布混沌的注记

设X是一个紧致度量空间,f X→X是一个连续映射.若存在f的一个m-周期点p和另一个m'-周期点q(p≠q),使得对任意非空开集V(C)X,都有{p,q}(C)∞ Un=0fn(V),则称动力系统(X,f)是一个(m,m')型周期吸附系统.证明了:1)若(X,f)是一个(m,m')型周期吸附系统且X是自密的,则对任一给定的正整数k,存在一个fk的的分布混沌集S,使得S与X的任一非空开集之交均含有一个Cantor集;2)若(X,f)是一个(m,m')型周期吸附系统且拓扑共轭于(X ',f'),则(X ',f')也是一个(m,m')型周期吸附系统.改进和推广了已有结果.     

Distributionally scrambled set and minimal set

We investigate the relation between distributional chaos and minimal sets, and discuss how to obtain various distributionally scrambled sets by using least and simplest minimal sets. We show: i) an uncountable extremal distributionally scrambled set can appear in a system with just one simple minimal set: a periodic orbit with period 2; ii) an uncountable dense invariant distributionally scrambled set can occur in a system with just two minimal sets: a fixed point and an infinite minimal set; iii) infinitely many minimal sets are necessary to generate a uniform invariant distributionally scrambled set, and an uncountable dense extremal invariant distributionally scrambled set can be constructed by using just countably infinitely many periodic orbits.     

Chaos in a topologically transitive system

The chaotic phenomena have been studied in a topologically transitive system and it has been shown that the erratic time dependence of orbits in such a topologically transitive system is more complicated than what described by the well-known technology "Li-Yorke chaos". The concept "sensitive dependency on initial conditions" has been generalized, and the chaotic phenomena has been discussed for transitive systems with the generalized sensitive dependency property.     

A Devaney Chaotic System Which Is Neither Distributively nor Topologically Chaotic

Weiss proved that Devaney chaos does not imply topological chaos and Oprocha pointed out that Devaney chaos does not imply distributional chaos. In this paper, by constructing a simple example which is Devaney chaotic but neither distributively nor topologically chaotic, we give a unified proof for the results of Weiss and Oprocha.     

疯狂动力系统的一些性质

针对纤维映射为旋转的疯狂动力系统,比较系统的研究了它们的回复性、混合性、混沌性等基本的动力学性质.     

Distributional chaos and factors

We show the existence of a dynamical system without any distributionally scrambled pair which is semiconjugated to a distributionally chaotic factor.     

周期吸附系统的分布混沌

由一个紧致度量空间X以及连续映射f:X→X所组成的偶对(X,f)称之为一个动力系统.若存在f的不动点p以及另一周期点q,使得对于任一非空开集U(?)X,都有∪_(n=0)~∞f~n(U)含有p和q,则称(X,f)是一个周期吸附系统,其中f~i表示f的i次迭代.本文指出:若(X,f)是一个周期吸附系统并且X是自密的,则存在一个f的分布混沌集D,使得D与每一非空开集之交都包含着一个Cantor集.     

CHAOS AND TOPOLOGICAL MIXING OF TREE MAPS

In the present paper we show that a tree map is totally transitive iff it is topologically mixing. Using this result, we prove that the tree maps having a chaotic (or scrambled) subset with full Lebesgue measure is dense in the space consisting of all topologically mixing (transitive, respectively) maps.     

ON F-SENSITIVE PAIRS

In the present paper, we define sensitive pairs via Furstenberg families and discuss the relation of three definitions: sensitivity, F -sensitivity and F -sensitive pairs, see Theorem 1. For transitive systems, we give some sufficient conditions to ensure the existence of F -sensitive pairs. In particular, each non-minimal E system (M system, P system) has positive lower density ( Fs , Fr resp.)-sensitive pairs almost everywhere. Moreover, each non-minimal M system is Fts -sensitive. Finally, by some examples we show that: (1) F -sensitivity can not imply the existence of F -sensitive pairs. That means there exists an F -sensitive system, which has no F -sensitive pairs. (2) There is no immediate relation between the existence of sensitive pairs and Li-Yorke chaos, i.e., there exists a system (X, f ) without Li-Yorke scrambled pairs, which has κ B -sensitive pairs almost everywhere. (3) If the system (G, f ) is sensitive, where G is a finite graph, then it has κ B -sensitive pairs almost everywhere.     

完备度量空间中的混沌判定

本文研究完备度量空间上的离散动力系统的混沌标准,证明了如果完备度量空间X上的连续映射f具有正则非退化返回排斥子或连接不动点的正则非退化异宿环,则存在f的不变闭子集A,使得f限制在此不变闭子集上的子系统与两个符号的符号动力系统拓扑共轭,从而获得具有这类结构的连续映射f具有Devaney混沌、分布混沌、正拓扑熵及ω-混沌,...     

Henstock-Kurzweil可积函数空间的紧性特征

本文研究Henstock-Kurzweil可积(HK可积)函数空间中的一个经典问题.文章通过研究分布Henstock-Kurzweil积分(DHK积分)的性质,给出了该问题的否定答案.进一步,利用收敛性获得了函数HK可积的一个充分必要条件.最后,在上述结论的基础上刻画了HK可积函数空间的紧性.所得结果丰富和推广了HK可...     

从广义函数观点看布纹闪影问题

当两片尼龙布或两扇铁纱窗被重迭在一起时,我们看到了明暗交替的条纹.这就是布纹闪影 (moirc) 现象.本文用非标准分析和广义函数理论,给出了产生布纹闪影现象的数学上的根据.     

ON F-SENSITIVE PAIRS

In the present paper, we define sensitive pairs via Furstenberg families and discuss the relation of three definitions: sensitivity, F -sensitivity and F -sensitive pairs, see Theorem 1. For transitive systems, we give some sufficient conditions to ensure the existence of F -sensitive pairs. In particular, each non-minimal E system (M system, P system) has positive lower density ( Fs , Fr resp.)-sensitive pairs almost everywhere. Moreover, each non-minimal M system is Fts -sensitive. Finally, by some exampl...