(R)型准可解流形上映射同伦类的拓扑熵的下确界

设$f$是(R)型准可解流形$M$上的连续自映射, $N^\infty(f)$是$f$的渐近Nielsen数,本文将利用Nielsen不动点理论,给出$\log\,N^\infty(f)$是$f$的映射同伦类的拓扑熵的下确界的充分条件,这些条件将推广准幂零流形上的类似结果.     

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。     

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 $NPhi_n(f)$ 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.     

幂零流形上自映射的点态原像熵的可加性

类似于拓扑熵,点态原像熵作为动力系统的不变量,也度量了紧度量空间上系统的复杂性.但至今不知其性质与拓扑熵是否完全一致,例如映射笛卡尔积的点态原像熵的可加性等.本文将把环面自映射笛卡尔积的点态原像熵的可加性,推广到紧幂零流形自映射的情形.     

序有界模糊n-方体数集的上确界与下确界

本文中,我们讨论了在定义模糊n-方体数值映射的某种类型的积分时要用到的序有界模糊n-方体数集的上确界与下确界的问题。我们证明了序有界模糊n-方体数集的上、下确界的存在性,并给出了它们的表示。     

k维环面上坐标自映射下拓扑熵的一个下界

给出了ｋ维环面上坐标自映射下拓扑熵的一个下界,最后,还指出了ｋ维环面上渐近Ｒｅｉｄｅｍｅｉｓｔｅｒ数严格大于渐近Ｎｉｅｌｓｅｎ数的情形,并说明了文（３）（或文（４）中引理１为该文的一个特例。     

迹非零的布尔矩阵的幂敛指数的上确界

设是恰含ｄ个正对角元的ｎ阶布尔矩阵的集合，１≤ｄ≤ｎ．本文在柳柏濂、邵嘉裕１９９１年工作的基础上进一步证明了同时证明：这个界是最好可能的。从而，完全解决了的最大幂敛指数问题。     

关于拓扑熵下界的两点注记

设ｆ：Ｘ→Ｘ是紧连通多面体自映射，应用Ｎｉｅｌｓｅｎ不动点理论，我们给出了ｆ的拓扑熵ｈ（ｆ）的一个更好下界。另外，若ｆ：Ｔ^m→T^m是ｍ－环面自映射，我们还得到了ｌｏｇＮ^∞（ｆ）是{h(g)|g≈f:T^m→T^m}的下确界的一个充要条件，这里Ｎ（ｆ）是ｆ的渐近Ｎｉｅｌｓｅｎ数，从而局部解答了姜伯驹教授提出的一个问题。     

闭轨线上模映射的不动点类与拓扑熵下界估计

本文在自治系统dx/dt=f(x),f∈C(DRn,Rn)的闭轨线Γ上定义了模映射,并利用闭轨线Γ与单位圆周S1的同胚关系,给出了模映射的Reidemeister数、Nielsen数,以及模映射的拓扑熵下界估计.     

确界原理的一个修正证明

分析了华东师大版《数学分析》（第四版）教材中关于确界原理证明的不足之处,给出更为细致的修正证明.     

A note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

     

拓扑熵的一个下界估计

设Ｘ为局部闭路可缩的紧致空间，ｆ为Ｘ的自映射，ｈ（ｆ）为ｆ的拓扑熵，Ｒ∞（ｆ）为ｆ的渐近Ｒｅｉｄｅｍｅｉｓｔｅｒ数，则有ｈ（ｆ）≥ｌｏｇＲ∞（ｆ）．     

Probability-one homotopy maps for mixed complementarity problems

Probability-one homotopy algorithms have strong convergence characteristics under mild assumptions. Such algorithms for mixed complementarity problems (MCPs) have potentially wide impact because MCPs are pervasive in science and engineering. A probability-one homotopy algorithm for MCPs was developed earlier by Billups and Watson based on the default homotopy mapping. This algorithm had guaranteed global convergence under some mild conditions, and was able to solve most of the MCPs from the MCPLIB test library. This paper extends that work by presenting some other homotopy mappings, enabling the solution of all the remaining problems from MCPLIB. The homotopy maps employed are the Newton homotopy and homotopy parameter embeddings.     

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.     

On the measure of scrambled sets of tree maps with zero topological entropy

In this paper, we show that if f is a tree map of zero topological entropy and μ is an f-invariant Borel measure then any scrambled set S has zero outer μ-measure (hence μ-measurable). In particular, if S is measurable, it has zero μ-measure.     

Expanding maps on infra-nilmanifolds of homogeneous type

In this paper we investigate expanding maps on infra-nilmanifolds. Such manifolds are obtained as a quotient , where is a connected and simply connected nilpotent Lie group and is a torsion-free uniform discrete subgroup of , with a compact subgroup of . We show that if the Lie algebra of is homogeneous (i.e., graded and generated by elements of degree 1), then the corresponding infra-nilmanifolds admit an expanding map. This is a generalization of the result of H. Lee and K. B. Lee, who treated the 2-step nilpotent case.