关于赋范空间上连续自映射的回归点

在赋范空间中讨论回归点的性质,主要得到了结果：（1）如果,是序列紧赋范空间X上的连续双射,x是f的任一回归点,则对于任意整数N〉0都存在f的回归点x0∈X使得f^n（x0）=x;（2）序列紧赋范空间上连续自映射的回归点集是f的强不变子集;（3）如果f是局部连通赋范空间X上的连续自映射,则f的每一个回归点或是类周期点或是类周期点的聚点．作为推论,在实直线段上得到了类似的结论．     

A note on the continuous self-maps of the ladder system space

We give a partial characterization of the continuous self-maps of the ladder system space  ${K_{\mathcal{S}}}$ . Our results show that ${K_{\mathcal{S}}}$ is highly nonrigid. We also discuss reasonable notions of “few operators” for spaces C(K) with scattered K and we show that $C({K_{\mathcal{S}}})$ does not have few operators for such notions.     

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.     

On the Lefschetz periodic point free continuous self-maps on connected compact manifolds

We characterize the Lefschetz periodic point free self-continuous maps on the following connected compact manifolds: CPn the n-dimensional complex projective space, HPn the n-dimensional quaternion projective space, Sn the n-dimensional sphere and Sp×Sq the product space of the p-dimensional with the q-dimensional spheres.     

Two theorems on poisson measures on a topological group

Our first theorem is concerned with the convergence of nets of Poisson measures on a topological group. As a corollary we obtain a characterization of Poisson measures. The second theorem gives a characterization of elementary Poisson measures.     

The dualizing spectrum of a topological group

To a topological group G, we assign a naive G-spectrum , called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that detects Poincaré duality in the sense that BG is a Poincaré duality space if and only if is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behaves multiplicatively on certain topological group extensions. In proving these results we introduce a new tool: a norm map which is defined for any G and for any naive G-spectrum E. Applications of the dualizing spectrum come in two flavors: (i) applications in the theory of Poincaré duality spaces, and (ii) applications in the theory of group cohomology. On the Poincaré duality space side, we derive a homotopy theoretic solution to a problem posed by Wall which says that in a fibration sequence of fini the total space satisfies Poincaré duality if and only if the base and fiber do. The dualizing spectrum can also be used to give an entirely homotopy theoretic construction of the Spivak fibration of a finitely dominated Poincaré duality space. We also include a new proof of Browder's theorem that every finite H-space satisfies Poincaré duality. In connection with group cohomology, we show how to define a variant of Farrell-Tate cohomology for any topological or discrete group G, with coefficients in any naive equivariant cohomology theory E. When E is connective, and when G admits a subgroup H of finite index such that BH is finitely dominated, we show that this cohomology coincides with the ordinary cohomology of G with coefficients in E in degrees greater than the cohomological dimension of H. In an appendix, we identify the homotopy type of for certain kinds of groups. The class includes all compact Lie groups, torsion free arithmetic groups and Bieri-Eckmann duality groups. Received July 14, 1999 / Revised May 17, 2000 / Published online February 5, 2001     

A new Lindelöf topological group

We show that the subsemigroup of the product of ω₁-many circles generated by the L-space constructed by J. Moore is again an L-space. This leads to a new example of a Lindelöf topological group. The question whether all finite powers of this group are Lindelöf remains open.     

The upgrade of topological group based on a new hyper-topology

Wang et al [1] have put forward the problem of upgrade of topology and obtained a series of systematic theories and applicable results. Li et al [2] have put forwarded the problem of upgrade of group and also obtained a series of results. Consequently, the problem of upgrade of mathematic structure has drawn more attention of researchers. However, for the work to upgrade topology and group at the same time, little progress has been made although there are someone studying on it. Based on a kind of intuitive convergent way, the paper has proposed a new hyper-topology and upgraded two mathematic structures of topological group to power set and fuzzy power set respectively. It has proved that multiplication and contradiction operations continues in the new hyper-topology after upgrading and a series of results are obtained creating hyper-topological group and fuzzy hyper-topological group. Accordingly, hyper-topological group and fuzzy hyper-topological group can be created, obtaining a breakthrough to upgrade topological group to its power set and fuzzy power set.     

A remark on weakly convex continuous mappings in topological linear spaces

Let C be a compact convex subset of a Hausdorff topological linear space and T:C→C a continuous mapping. We characterize those mappings T for which T(C) is convexly totally bounded.     

Topological entropy of continuous functions on topological spaces

Adler, Konheim and McAndrew introduced the concept of topological entropy of a continuous mapping for compact dynamical systems. Bowen generalized the concept to non-compact metric spaces, but Walters indicated that Bowen’s entropy is metric-dependent. We propose a new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new entropy, and compare the new entropy with the existing ones. The defined entropy generates that of Adler, Konheim and McAndrew and is metric-independent for metrizable spaces. Yet, it holds various basic properties of Adler, Konheim and McAndrew’s entropy, e.g., the entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same entropy, the entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new entropy coincides with Adler, Konheim and McAndrew’s entropy for compact systems.