Complexity of a kind of interval continuous self-map of finite type

An interval map is called finitely typal, if the restriction of the map to non-wandering set is topologically conjugate with a subshift of finite type. In this paper, we prove that there exists an interval continuous self-map of finite type such that the Hausdorff dimension is an arbitrary number in the interval (0, 1), discuss various chaotic properties of the map and the relations between chaotic set and the set of recurrent points.     

The Hausdorff dimension of chaotic sets for interval self-maps

We obtain two sufficient conditions for an interval self-map to have a chaotic set with positive Hausdorff dimension. Furthermore, we point out that for any interval Lipschitz maps with positive topological entropy there is a chaotic set with positive Hausdorff dimension.     

Almost periodicity,chain recurrence and chaos

In this paper we consider a continuous mapf: X→X, whereX is a compact metric space. The existence of chaotic sets off is discussed. For the special caseX=[0,1], we prove thatf has a positive topological entropy iff it has an uncountable chaotic set in which each point is almost periodic, and iff it has an uncountable chaotic set in which each point is chain recurrent. As an application, a uniform proof for some known results will be given.     

几乎周期性与SS混沌集

引进正则移位不变集的概念,证明了有正则移位不变集的紧致系统在几乎周期点集中存在SS混沌集,特别地,具有正拓扑熵的区间映射在几乎周期点集中存在SS混沌集.     

From a Floyd-Auslander minimal system to an odd triangular map

In 1989 A.N. Sharkovsky asked the question which of the properties characterizing continuous maps of the interval with zero topological entropy remain equivalent for triangular maps of the square. The problem is difficult and only partial results are known. However, in the case of triangular maps with nondecreasing fibres there are only few gaps in a classification (given by Z. Ko?an) of a set of 24 of these conditions. In the present paper we remove these gaps by giving an example of a triangular map in the square with the following properties:

(1)

all fibre maps are nondecreasing,

(2)

all recurrent points of the map are uniformly recurrent, and

(3)

the restriction of the map to the set of recurrent points has an uncountable scrambled set (and so is Li-Yorke chaotic).

The example is obtained by taking an appropriate Floyd-Auslander minimal system and then taking its appropriate continuous extension to a triangular map of the square.     

Chaos among self-maps of the Cantor space

Glasner and Weiss have shown that a generic homeomorphism of the Cantor space has zero topological entropy. Hochman has shown that a generic transitive homeomorphism of the Cantor space has the property that it is topologically conjugate to the universal odometer and hence far from being chaotic in any sense. We show that a generic self-map of the Cantor space has zero topological entropy. Moreover, we show that a generic self-map of the Cantor space has no periodic points and hence is not Devaney chaotic nor Devaney chaotic on any subsystem. However, we exhibit a dense subset of the space of all self-maps of the Cantor space each element of which has infinite topological entropy and is Devaney chaotic on some subsystem.     

Fixed points of continuous DCPOs

In this article we show that for a continuous DCPO D, the set of fixed points of every self-map is a continuous DCPO if and only if x<y implies x is way below y. We also prove that some classes of continuous functions have the property that if a self-map on a DCPO is in the class then the set of fixed points is a continuous DCPO. We also investigate when the set of fixed points is a retract.     

Uncountable equilateral sets in Banach spaces of the form <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)

The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin’s axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form C(K) has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form C(K) which has no uncountable equilateral set (or equivalently no uncountable (1+ε)-separated set in the unit sphere for any ε > 0) making another consistent combinatorial assumption. The compact K is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than C(K) which has no uncountable equilateral set. It follows from the results of S. Mercourakis and G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples of nonseparable Banach spaces which have no uncountable equilateral sets in any equivalent renorming but it follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in which it has an uncountable equilateral set.     

∑上的非弱几乎周期的回复点集和SS混沌集

该文对单边符号空间上的转移自映射进行了讨论,证明了在非弱几乎周期的回复点集上存在不可数的SS混沌集.     

On chaotic extensions of dynamical systems

In this paper, inspired by some results in linear dynamics, we will show that every dynamical system (X,f), where f is a continuous self-map on a separable metric space X, can be extended to a chaotic (in the sense of Devaney) dynamical system in an isometric way.     

关于两种混沌映射的有限乘积性质

首先在一般度量空间上给出有限积映射是Li-Yorke混沌的一个判据,并且用反倒展示:当有限积映射是Li-Yorke混沌时,未必一定存在因子映射是Li-Yorke混沌的.然后,利用上述判据,在[0,1]N上证明有限积映射有不可数scrsmbled集的一个充要条件.进而,推出关于有限积映射为Li-Yorke 混沌的一组等价...     

Fixed points of Scott continuous self-maps

In this paper we study the properties of the set consisting of all fixed points of a Scott continuous self-map between domains. All the results give an answer to an open problem posed by Lawson and Mislove in 1990.     

Chaotic sets of continuous and discontinuous maps

This paper discusses Li-Yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps. Scrambled sets and maximal scrambled sets are introduced to characterize Li-Yorke chaotic sets. The orbit invariant for a scrambled set is discussed. Some properties about maximality, equivalence and uniqueness of maximal scrambled sets are also discussed. It is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps, such as the Gauss map, interval exchange transformations, and a class of planar piecewise isometries, are studied. Finally, some open problems on scrambled sets are listed and remarked.     

Syndetically proximal pairs

For continuous self-maps of compact metric spaces, we study the syndetically proximal relation, and in particular we identify certain sufficient conditions for the syndetically proximal cell of each point to be small. We show that any interval map f with positive topological entropy has a syndetically scrambled Cantor set, and an uncountable syndetically scrambled set invariant under some power of f. In the process of proving this, we improve a classical result about interval maps and establish that if f is an interval map with positive topological entropy and m?2, then there is n∈N such that the one-sided full shift on m symbols is topologically conjugate to a subsystem of fn² (the classical result gives only semi-conjugacy).     

不含混沌真子系统的Li-Yorke混沌

研究了一类Li-Yorke混沌系统,该系统没有真子系统是Li-Yorke混沌的,我们称之为混沌极小系统.本文证明混沌极小系统是拓扑传递的,而且该系统每个非空开集都包含一个不可数混乱集.混沌极小系统不一定是极小的,本文构造了一个这样的反例.特别地,我们考察了线段连续自映射,指出该类系统都不是混沌极小的,线段上混沌极小子系统的存在性和该系统有正熵是等价的.     

区间上三段单调扩张自映射的周期轨道

设ｎ３是奇数，ｍ０是整数，Ｓｎ及ｒｎ分别是方程ｘｎ－２ｘｎ－１－１＝０及ｘｎ＋２－３ｘｎ－ｘ２－１＝０的唯一正根，记ｔｎ０＝ｒｎ，ｔｎｉ＝ｓｎ（ｉ１）．又设ｆ及ｇ分别是闭区间Ｉ上的Ｎ型（即增－减－增型）及反Ｎ型（即减－增－减型）扩张自映射．本文证明了，若ｆ（或ｇ）的扩张常数或则ｆ（或ｇ）有２ｍ·ｎ周期点．此外，本文还指出，当或时，在［０，１］上存在着具有扩张常数λ却无２ｍ·ｎ周期轨道的Ｎ型（或反Ｎ型）扩张自映射．     

On the centre and the set of ω-limit points of continuous maps on dendrites

It is well known that for dynamical systems generated by continuous maps of a graph, the centre of the dynamical system is a subset of the set of ω-limit points.In this paper we provide an example of a continuous self-map f₁ of a dendrite such that ω(f₁) is a proper subset of C(f₁).The second example is a continuous self-map f₂ of a dendrite having a strictly increasing sequence of ω-limit sets which is not contained in any maximal one. Again, this is impossible for continuous maps on graphs.     

树映射的ω-极限集与湍流

设Ｔ是个树，ｆ：Ｔ→Ｔ是个连续自映射，　ｎ是Ｔ的端点数，ｘ∈Ｔ．本文证明：（１）如果存在ｚ∈ω（ｘ，ｆ）∩Ｆ（ｆ），使ｚ是ω（ｘ，ｆ）的一个单侧聚点，那么一定存在ｒ∈Ｎｎ－１，使ｆｒ含有湍流；（２）如果ω（ｘ，ｆ）是个含有不动点的无穷集，那么必存在ｒ∈Ｎｎ，使ｆｒ含有湍流．     

关于迭序列的子列极限点

本文讨论了度量空间上一个连续自映射击一点处的迭代序列的子列极限点集的结构，所得的结果统一和推广了Ｄｉａｚ和Ｍｅｔｃａｌｆ，Ｍａｉｔｉ和Ｂａｂｕ和Ｐａｒｋ的若干结果．     

Distributional chaos in a sequence

In this paper we prove a sufficient condition for the continuous map of a compact metric space for being distributively chaotic in a sequence. As an application, it is proved that a continuous map of an interval is chaotic in the Li–Yorke sense if and only if it is distributively chaotic in a sequence.