环的自映射的最少周期点数

设Ｘ是一个紧致连通的ＡＮＲ＇ｓ，ｆ：Ｘ→Ｘ是连续映射。于是ｆ有Ｎｉｅｌｓｅｎ型数，ＮＦｎ（ｆ），它是数集｛＃Ｆｉｘ（ｇ＾ｎ）ｇ－ｆ｝的下界。本文在Ｘ是环时，给出ＮＦｎ（ｆ）是该数集的下确界，即存在连续映射ｇ＝ｆ，使得＃Ｆｉｘ（ｇ＾ｎ）＝ＮＦｎ（ｆ）的一个充分条件。     

相对映射的周期点

相对Nielsen周期点理论是讨论形如f:(X,A)→(X,A)映射的周期点个数估计问题,本文对已知的估计量给予统一的处理.利用这种方法,定义了两个新的Nielsen型数, NPn(f;X-A)和NΦn(f;X-A),它们分别是映射f在cl(X-A)中的n周期点和最小周期为n的周期点个数的下界.     

Least number of periodic points of self-maps of Lie groups

There are two algebraic lower bounds of the number of n-periodic points of a self-map f :M → M of a compact smooth manifold of dimension at least 3:N Fn(f) = min{#Fix(gn); g ～f; g is continuous} and N J Dn(f) = min{#Fix(gn); g ～ f; g is smooth}.In general,N J Dn(f) may be much greater than N Fn(f).If M is a torus,then the invariants are equal.We show that for a self-map of a nonabelian compact Lie group,with free fundamental group,the equality holds  all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.     

准幂零流形上映射同伦类的拓扑熵的下确界

设f:M→M是准幂零流形M上的连续自映射,N∞(f)是f的渐近Nielsen数.本文应用Nielsen不动点理论,给出log N∞(f)是f的同伦类中所有映射的拓扑熵的下确界的一个充要条件,该结果是关于幂零流形上类似结果的一个本质推广.     

关于映射同伦类的拓扑熵

设ｆ：Ｔ＾ｍ→Ｔ＾ｍ为ｍ维环面自映射，Ｎ＾∞（ｆ）是ｆ的渐近Ｎｉｅｌｓｅｎ数，本文应用Ｎｉｅｌｓｅｎ不动点理论，给出了ｌｏｇＮ＾∞（ｆ）是ｆ的同伦类的拓扑熵的最好下界的一个充要条件；并通过在齐性空间上引入等价度量，将此结论推广到了幂零流形自映射的情形。     

On the Least Number of Fixed Points of an Equivariant Map

The problem on the least number of fixed points of an equivariant map of a compact polyhedron on which a finite group acts is considered. For such a map, the least number of fixed points and the least number of fixed orbits are estimated in terms of invariants of the type of Nielsen numbers. The estimates obtained are sharp. The results are similar to those of P. Wong, but their assumptions are essentially weaker. Some notations are refined. The proofs are constructive.     

Coincidences of projections and linear n-valued maps of tori

We prove that the Nielsen fixed point number N(φ) of an n-valued map φ:X?X of a compact connected triangulated orientable q-manifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of φ, a subset of X×X, to the two factors. For certain q×q integer matrices A, there exist “linear” n-valued maps Φn,A,σ:Tq?Tq of q-tori that generalize the single-valued maps fA:Tq→Tq induced by the linear transformations TA:Rq→Rq defined by TA(v)=Av. By calculating the Nielsen coincidence number of the projections of its graph, we calculate N(Φn,A,σ) for a large class of linear n-valued maps.     

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.     

On some properties of orientation-preserving surjections on the circle

Some properties of orientation-preserving surjections with nonempty set of periodic points are studied. In particular, orientation-preserving homeomorphisms of the whole circle S ¹ are considered.      

一般三角帐篷映射混沌性与两种混沌互不蕴含性

将三角帐篷映射推广为一般的n-三角帐篷映射,并且借助于一般Bernoulli移位映射,Banks定理与Li-Yorke定理,首先证明：对于任意的正整数n,n-三角帐篷映射既是Devaney混沌的,也是Li-Yorke混沌的.然后,利用所得到的结果,通过实例展示：Devaney混沌与Li-Yorke混沌的互不蕴含性.     

一类Hamilton系统周期映射的单调性

本文证明了系统 x+α· x+β· xn=0 (α>0 ,β≠ 0 ,n 2 )的周期映射是单调的     

Coven和Hedlund一个定理的较简单的证明

Coven和Hedlund在[1]中证明：若区间映射的周期集为有限集,则它的每一个非游荡点都是周期点。本给出了这一定理的一个较简单的证明。此外,我们还用“同伦”的方法简化了该的一个关键引理的证明。     

无最小正周期的周期函数的构造及其性质

从群论的角度给出周期函数的等价定义.对任意给定的非循环群GR),构造出以G为周期集的无最小正周期的周期函数.讨论无最小正周期的周期函数的性质.     

A simple proof of the chaoticity of shift map under a new definition of chaos

Recently, Du has given a new strong definition of chaos by using the shift map. In this paper, we give a proof of the main theorem by constructing a dense uncountable invariant subset of the symbol space Σ₂ containing transitive points in a simpler way with the help of a different metric. We also provide two examples, which support this new definition.     

Realization of fixed point sets

Letf:XX be a selfmap of a compact connected polyhedron, andA a nonempty closed subset ofX. In this paper, we shall deal with the question whether or not there is a mapg:XX homotopic tof such that the fixed point set Fixg ofg equalsA. We introduce a necessary condition for the existence of such a mapg. It is shown that this condition is easy to check, and hence some sufficient conditions are obtained.Partially supported by the Natural Science Foundation of Liaoning University.     

树映射的等度连续性

讨论了树（即：不含圈的维紧致连通的分支流形）映射的等度连续性,得到了树映射是等度连续的两个充要条件。