圆周上有4周期轨的连续自映射的周期集

讨论了圆周上有4周期轨的连续自映射的周期集.首先按相对共轭以及相对同伦的关系对圆周上所有有4周期轨的连续自映射分类,再利用映射覆盖图来讨论每一类映射的周期集.最后按同伦最小周期集对圆周上所有有4周期轨的连续自映射进行了分类.将此结果与线段上的Sharkovskii定理对比时可以发现,儿乎所有圆周上有4周期轨的连续自映射的周期集都是全体自然数集.     

无周期点的圆周自映射

<正> 圆周自映射按有无周期点可以分成两大类,对有周期点的一类,我们在[1—3]中已作了讨论,本文讨论无周期点的一类. 文[4]讨论了这类自映射,证明它们都是唯一遍历的(uniquely ergodic).该文虽然不涉及非游荡集的概念,但在实际上却隐含了关于非游荡集结构的某些重要结论.该文完     

周期点集不空的圆周自映射

<正> 本文是[1]的续篇,继续讨论圆周自映射所产生的动力系统性质.圆周自映射按有无周期点可以分成两大类,也可以按拓扑度分成四种情形,即|deg|≥2,deg=0,deg=-1和 deg=1.其中前三种情形的映射都有不动点,属于周期点集不空的一类.第四种情形的映射较为复杂,它们可以有不动点,无不动点但有周期点,也可以无周期点.在文[1]中我们对|deg|≥2的映射讨论了周期集合,周期点集,非游荡集和拓扑熵之间的种种联     

周期点集为非空闭集的圆周自映射

<正> 对于周期点集为闭集的线段自映射周期点的周期以及周期点集和回复点集的关系问题引起了许多人的兴趣,并得到了满意的结果.相应地,对于圆周自映射我们已在文献[4]和[5]中证明了:定理 A 设 f 为圆周到自身的连续映射,如果 degf=0,则 f 的周期点集的闭包等     

三维幂零流形上映射的 Nielsen 型数和同伦最小周期集

对于三维幂零流形上的所有映射, 给出了完整计算 Nielsen 型数 $NP_n(f)$ 和 $N\Phi_n(f)$ 的显式公式. 最一般的情形已被 Heath 和 Keppelmann 讨论过, 我们研究剩余的部分. 而在三维幂零流形映射的同伦最小周期集的研究中, 给出了三维幂零流形上所有映射的最小周期集的完整描述, 并包含了对Jezierski和Marzantowicz 结果的改正.     

周期轨间蕴含关系的判定算法（Ⅱ）

近年来,Sarkovskii定理及其有关研究引起很大兴趣.按Sarkovskii定理,若闭区间上连续自映射f有3-周期点,则对任意正整数n有n周期点.但f不可能有所有类型的n-周期轨.例如:则f仅有两种类型的3-周期轨中的一类.这表明Sarkovskii定理远远没有给出周期轨之间的关系的全部信息.本文(Ⅰ)中将给出周期轨的型的概念,并证明可以建立机械方法来判断一种周期轨是否蕴含另一类型的周期轨.本文(Ⅱ)中将给出这个判断方法的计算机程序,并列出一些计算结果.     

完备稠序线性序拓扑空间上的奇周期轨序关系

研究完备稠序线性序拓扑空间上连续自映射的周期轨,指出当连续自映射有（2n＋1）-周期轨而没有（2n-1）-周期轨时,该（2n＋1）-周期轨上各点的序关系.利用这个关系将Sharkovskii定理从实直线推广到完备稠序线性序拓扑空间上。     

周期轨间蕴含关系的判定算法（Ⅰ）

近年来,Sarkovskii定理及其有关研究引起很大兴趣.按Sarkovskii定理,若闭区向上连续自映射f有3-周期点,则对任意正整数n有n周期点.但f不可能有所有类型的n-周期轨.例如:则f仅有两种类型的3-周期轨中的一类.这表明Sarkovskii定理远远没有给出周期轨之间的关系的全部信息.本文(Ⅰ)中将给出周期轨的型的概念,并证明可从建立机械方法来判断一种周期轨是否蕴含另一类型的周期轨。本文(Ⅱ)中将给出这个判断方法的计算机程序,并列出一些计算结果.     

紧度量空间上轨道的分类和极限集有向同伦不变性

本文利用正半轨道的ω极限集对紧度量空间的正半轨道进行分类,并讨论不动点和周期点的存在性.最后,引入轨道的正半同伦和负半同伦的概念,证明ω极限集和ω极限集在正半同伦和负半同伦的条件下是不变的,从而导出不动点和周期点在正半同伦和负半同伦的条件下保持不变.     

deg≥2的圆周自映射

<正> 记 R=(-∞,+∞),I=[0,1]和 S~1{e~(2πxi)|x∈I}.S~1是复平面上中心在原点的单位圆周.S~1上全体连续自映射的集合记为 C~0(S~1,S~1).设 f∈C~0(S~1,S~1),f 的周期集合,不动点集,周期点集,非游荡集和拓扑熵分别记为 p(f),F(f),P(f),Ω(f)和ent(f).此外,用 deg(f)记 f 的拓扑度或层数(一种定义见§2).关于圆周自映射所产生的动力系统性质已有很多人进行了讨论.据作者所知,所有这种讨论还仅限于在某种条件下寻求拓扑熵下限的最好估计以及 Sharkovskii 和 Li Yorke     

A note on minimal sets of periods for cofrontier maps

Using an earlier fixed point result of the author and Nielsen-type arguments patterned on related results for torus self-maps by Alsedá et al. we describe minimal sets of periods for self-maps of cofrontiers that are inverse limits of circles. Special attention is given to the pseudocircle, ever present in dynamical systems. Our result extends a property of circle maps, proved by Efremova, and independently by Block et al. in the late 1970s. We explain why our natural generalization is rather unexpected and shows potential of Nielsen theory for new applications.     

Homotopical theory of periodic points of periodic homeomorphisms on closed surfaces

In this work we show that the Wecken theorem for periodic points holds for periodic homeomorphisms on closed surfaces, which therefore completes the periodic point theory in such a special case. Using it we derive the set of homotopy minimal periods for such homeomorphisms. Moreover we show that the results hold for homotopically periodic self-maps of closed surfaces. This let us to re-formulate our results as a statement on properties of elements of finite order in the group of outer automorphisms of the fundamental group of a surface with non-positive Euler characteristic.     

Nielsen theory on 3-manifolds covered by S 2 × ℝ

Let f: M → M be a self-map of a closed manifold M of dimension dim M ≥ 3. The Nielsen number N(f) of f is equal to the minimal number of fixed points of f among all self-maps f in the homotopy class of f. In this paper, we determine N(f) for all self-maps f when M is a closed 3-manifold with S²×R geometry. The calculation of N(f) relies on the induced homomorphisms of f on the fundamental group and on the second homotopy group of M.     

Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH ₂(M; ?) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.     

Minimal periods of holomorphic maps on complex tori

We study the set of minimal periods of holomorphic self-maps of one- and two-dimensional complex tori. In particular, we characterize when the set of minimal periods of such maps is finite. In fact, we have an algorithm for doing this characterization for holomorphic self-maps of an arbitrary dimensional complex torus.     

Homotopy minimal period self-maps on flat manifolds

This paper studies the homotopical minimal period set for self-maps on flat manifolds. We introduce a new invariant: the density of the homotopical minimal period set in the natural numbers. Some conditions on self-maps and flat manifolds are obtained so that the corresponding density is positive.     

Rational obstruction theory and rational homotopy sets

We develop an obstruction theory for homotopy of homomorphisms between minimal differential graded algebras. We assume that has an obstruction decomposition given by and that f and g are homotopic on . An obstruction is then obtained as a vector space homomorphism . We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes . This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] whenA and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps. Received February 22, 1999; in final form July 7, 1999 / Published online September 14, 2000     

Limit sets for maps of the circle

A trick which reduces some questions concerning the dynamics of self-maps of the circle to similar questions about self-maps of the interval is suggested and applied to answer two questions of Block and Coppel.     

圆周上逆极限可扩的连续自映射

本文研究了圆周上一类自映射ｆ的正向可扩性与其道极限的可扩性间的联系，得出圆周上的连续满射ｆ的逆极限可扩等价于ｆ拓扑共轭于扩张映射．