非混沌的树映射

设f是树T上的连续自映射,SAP(f),ω(f);Ω(f)分别是f的强几乎周期点集,ω-极限集,非游荡集.本文证明下面几条是等价的:(i)f是非混沌的;(ii)SAP(f)=ω(f);(iii)fΩ(f)是逐点等度连续的;(iv)f│ω(f)是逐点等度连续的;(v)f是一致非混沌的。     

非混沌的树映射

设f是树T上的连续自映射,SAP(f),ω(f),Ω(f)分别是f的强几乎周期点集,ω-极限集,非游荡集.本文证明下面几条是等价的(i)f是非混沌的;(ii)SAP(f)=ω(f);(iii)f|Ω(f)是逐点等度连续的;(iv)f|ω(f)是逐点等度连续的;(v)f是一致非混沌的.     

北大保送试题与代数基本定理

题目 已知f(x)是二次函数,且a,f(a),f(f(a)),f(f(f(a)))构成正项等比数列,求证:f(a)=a. 证明:设f(a)=qa(q＞0),则f(f(a))=q2a,即f(qa)=q2a;同理有f(q2a)=q3a.     

亚纯函数及其微分多项式的唯一性

设f是非常数亚纯函数,g是f的线性微分多项式．a和b是f的两个不同的小函数．本文证明如果f和g几乎CM分担a和b,则f≡g;此外,若f是非常数整函数,且f和f^(k)(k≥1)IM分担a和b,b-a≠Pe^λz,测f≡g．     

华沙圈上连续映射的distality与等度连续性

设W为华沙圈,f:W→W为连续映射.本文得到了f为distal的一个刻画并且讨论了f的distality与等度连续性的关系.证明了:(i)f是distal的当且仅当f为恒等映射.(ii)如果f为满射,则f是distal的当且仅当f为等度连续的.     

诱导超空间连续流的拓扑传递性

(X,f)为紧拓扑空间X上的连续流,(K(X),f)是由(X,f)诱导的紧超空间K(X)上的连续流.研究了(X,f)和(K(X),f)拓扑传递性之间的关系.证明了如果(X,f)是拓扑传递的,那么(K(X),f)也一定是拓扑传递的,并且举例证明了其逆命题不成立.进一步证明了(K(X),f)拓扑传递的当且仅当(X,f)是弱混合的.     

强预不变凸函数的充要条件

本文讨论了强预不变凸函数与预不变凸函数、严格预不变凸函数及半严格预不变凸函数之间的关系,得到它的三个充要条件：(i)在一定条件下,f是强预不变凸函数的充分必要条件是f是预不变凸函数且f满足中间点强预不变凸性；(ii)在一定条件下,f是强预不变凸函数的充分必要条件是f是严格预不变凸函数且f满足中间点强预不变凸性；(iii)在一定条件下,f是强预不变凸函数的充分必要条件是f是半严格预不变凸函数且f满足中间点强预不变凸性．     

周期函数基本结果概览北大核心

1基本定义为简单起见,我们不妨设所考虑的函数f(t)定义在整个数轴R=(-∞,+∞)上.定义1(周期函数)若实数T≠0使得f(t+T)=f(t)对一切t∈R恒成立,则称T是f的一个周期,并称f是周期为T的周期函数.注1设函数f的定义域是R的某个子集D,若对任意t∈D有t±T∈D且f(t+T)=f(t),则称f是D上的周期函数.     

Fengenbaum映射的搓揉序列与特征集

设f为Feigenbaum映射,亦即函数方程fp(λx)=λf(x)满足一定条件的单峰解．f的搓揉序列为0-1无限序列,f的特征集是临界点轨迹的闭包．本文研究f的性质进而证明．f的搓揉序列是某代换在符号空间中的不动点,f在特征集上的限制是某代换子移位的一个因子．     

你发现它们的周期了吗?

有些数学问题,表面上看与周期毫无关系,但实际上隐含着周期性,一旦揭示了周期,问题就迎刃而解。下面以函数和数列为例说明如下。 1．函数中的周期例1 设对任意整数x,都满足f(x)=f(x 1) f(x-1),且f(0)=19,f(4)=93,求f(59)的值。解∵ f(x)=f(x-1) f(x 1), ∴ f(x 1)=f(x) f(x 2), 两式相加并整理得f(x-1)=-f(x 2), ∴ f(x)=-f(x 3), ∴ f(x 6)=-f(x 3)=f(x), 从而f(x)是以6为周期的函数。∴ f(59)=f(6×9 5)=f(5) =f(4) f(6)=f(4) f(0)=112。例2 函数f(x)在R上是有定义的,且满足(1)f(x)是偶函数,且f(0)=2008；(2)g(x)=f(x-1)是奇函数。试求f(2004)的值。     

弱混合的一个等价条件

Let X be a compact metric space containing at least two points and let f ： X → X be continuous. In this paper, we introduce the notion of totally maximum sensitive （TMS, for short） and prove that f is weakly mixing if and only if it is TMS, and if X = [0, 1], then f is weakly mixing if and only if it is 2-maximum sensitive.     

On strictly weakly mixing <Emphasis Type="Italic">C</Emphasis>*-dynamical systems

We consider strictly ergodic and strictly weakly mixing C*-dynamical cystems. We establish that a system is strictly weakly mixing if and only if its tensor product is strictly ergodic and strictly weakly mixing. We also investigate some weighted uniform ergodic theorem with respect to S-Besicovitch sequences for strictly weakly mixing dynamical systems.     

Eventually independent sequences

The class of eventually independent sequences for a weakly mixing transformation is an isomorphism invariant that is useful for distinguishing zero entropy transformations. This invariant is used to distinguish certain weakly mixing transformations as well as certain partially mixing transformations. In memory of Shlomo Horowitz Research partially supported by National Science Foundation Grant MCS7606735A01.     

传递子集的性质及其在符号动力系统中的应用

讨论了传递子集的一些性质,并且应用这些性质研究符号动力系统的弱混合子集和传递子集之间的关系,给出了符号动力系统的传递子集是弱混合子集的一个充分条件.     

Mixing of all orders and pairwise independent joinings of systems with singular spectrum

We prove that every pairwise independent joining of weakly mixing systems with purely singular spectrum is independent; it follows that every mixing system with purely singular spectrum is mixing of all orders.     

Topologically weakly mixing homeomorphisms of the Klein bottle that are uniformly rigid

In this paper we prove that there is a large family of topologically weakly mixing homeomorphisms of the Klein bottle that are uniformly rigid. We do this by viewing the Klein bottle as the quotient of the two-torus by an appropriate group action and producing topologically weakly mixing homeomorphisms of the two-torus that are uniformly rigid and equivariant with respect to the action.     

Ergodic and Mixing Probability Measures on [SIN] Groups

We prove that the concepts of completely mixing, mixing, and weakly mixing probability measures on a locally compact group in the class [SIN] are mutually equivalent. A probability measure on such a group is completely mixing if and only if it is ergodic and aperiodic. For measures that are not necessarily aperiodic ergodicity is shown to be equivalent to a mixing-like property which we call almost mixing. Some results on when a probability measure on a [SIN] group is ergodic are also developed. Our methods are based on the theory of boundaries of random walks.     

Attracting current and equilibrium measure for attractors on ℙ<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

Let f be a holomorphic endomorphism of ℙ ^k having an attracting setA. We construct an attracting current and an equilibrium measure associated toA. The attracting current is weakly laminar and extremal in the cone of invariant currents. The equilibrium measure is mixing and has maximal entropy onA.     

Banach空间上的强混合算子

1999年,L．B．Gonzalez 证明了任意无限维可分 Banach 空间上存在拓扑传递的有界线性算子．这个结果肯定地回答了 S．Rolewicz 提出的问题．本文证明了由 L．B．Gonzalez 所给出的算子实际上是强混合的,同时,对加权移位算子的混合性利用权序列进行了刻划并指出任意无限维可分 Hilbert 空间上存在弱混合而非强混合的有界线性算子．     

A note on weakly Lindelöf frames

The notion of σ^?-properness of a subset of a frame is introduced. Using this notion, we give necessary and su?cient conditions for a frame to be weakly Lindelöf. We show that a frame is weakly Lindelöf if and only if its semiregularization is weakly Lindelöf. For a completely regular frame L, we introduce a condition equivalent to weak realcompactness based on maximal ideals of the cozero part of L. This enables us to show that every weakly realcompact almost P -frame is realcompact. A new characterization of weakly Lindelöf frames in terms of neighbourhood strongly divisible ideals of ?? is provided. The closed ideals of ?? equipped with the uniform topology are applied to describe weakly Lindelöf frames.