一簇Lorenz映射的混沌行为与统计稳定性

该文研究一簇Ｌｏｒｅｎｚ映射犛犪：［０,１］→［０,１］（０＜犪＜１）犛犪（狓）＝狓＋犪　狓∈ ［０,１－犪）｛（狓＋犪－１）／犪　狓∈ ［１－犪,１］．从拓扑的角度考虑了犛犪的混沌行为,证明了：犛犪有稠密轨道;犛犪的周期的集合犘犘（犛犪）＝｛１,犿＋１,犿＋２,…｝,其中犿为使犪犿＜１－犪成立的最小正整数;犛犪的拓扑熵犺（犛犪）＞０;几乎所有（关于Ｌｅｂｅｓｇｕｅ测度）的点狓的Ｌｙａｐｕｎｏｖ指数λ（犛犪,狓）＝λ犪＞０．从统计的角度讨论了犛犪的稳定性．我们用下界函数方法证明了犛犪是统计稳定的,并且狌犵犪（犃）＝∫犃犵犪（狓）ｄ狓（犃∈犅）为犛犪的唯一绝对连续（关于Ｌｅｂｅｓｇｕｅ测度）不变概率测度．同时,不变密度犵犪在参数扰动和随机作用的随机扰动下是稳定的．     

Hénon映射的周期过程

本文研究了Ｈｅｎｏｎ映射Ｈ（ｘ，ｙ）＝（ａ＋ｂｙ－ｘ２，ｘ）的周期过程，给出了Ｈ有某些周期的充分必要条件或充分条件，得到其周期集的一段序结构．     

覆盖映射的双曲Birkhoff中心

设ｆ是不带边的紧致流形Ｍ上的覆盖映射，本文证明了，若ｆ的Ｂｉｒｋｈｏｆｆ中心Ｃ（ｆ）是双曲的且无环，则ｆ满足公理Ａ且Ω－单一化稳定；若Ｃ（ｆ）是双曲的且ｆ是Ω－单一化稳定的，则Ｃ（ｆ）是无环的。     

连续树映射非游荡集的拓扑结构

本文研究树（即不含有圈的一维紧致连通的分支流形）上连续自映射的非游荡集的拓扑结构．证明了孤立的周期点都是孤立的非游荡点；具有无限轨道的非游荡点集的聚点都是周期点集的二阶聚点，以及ω－极限集的导集等于周期点集的导集和非游荡集的二阶导集等于周期点集的二阶导集．     

非线性多值算子扰动的映射定理

本文讨论非线性多值算子的非紧扰动的映射定理,并给出非线性泛函方程ｚ∈Ｔ（ｘ）＋Ｆ（ｘ）可解性的最新结果,其中Ｔ是多值算子且（Ｔ＋１/ｎＩ）^－１是１－集压缩,而Ｆ是１－集压缩或γ－凝聚．所得的结果改善了［５,８,１２］中的主要结果     

标准椭圆与双曲线的两点式方程及其应用

一问题的提出众所周知,不重合的两点确定一条直线,设这两点的坐标为Ａ（ｘ１,ｙ１）,Ｂ（ｘ２,ｙ２）,其中ｘ１≠ｘ２且ｙ１≠ｙ２,则Ａ,Ｂ所在直线的方程可表示为ｙ－ｙ１ｙ２－ｙ１＝ｘ－ｘ１ｘ２－ｘ１．我们称之为直线方程的两点式．知道了直线上两个不同点的...     

平均熵

设Ｔ为紧度量空间Ｘ上的连续自映射,ｍ为Ｘ上的Ｂｏｒｅｌ概率测度,通过把测度（拓扑）摘局部化,引入了Ｔ关于ｍ的平均测度（拓扑）熵的概念,它们分别为相应ｍ－测度（拓扑）混沌吸引子熵的加权平均,从而Ｔ关于ｍ的平均测度（拓扑）熵大于零当且仅当Ｔ有ｍ－测度（拓扑）混沌吸引子．证明了线段Ｉ上关于Ｌｅｂｅｓｇｕｅ测度平均拓扑熵大于Ｃ与等于零的连续自映射都在Ｃ０（Ｉ,Ｉ）中稠密．     

广义Newton法求解非线性互补问题

１引言考虑非线性互补问题ＮＣＰ（ｆ）：的求解，即我们要寻求某ｘ∈Ｒｎ，使其满足（１．１）．其中映射ｆ：Ｒｎ→Ｒｎ为具有连续Ｆ－导数的非线性映射．众所周知，问题（１．ｌ）可以等价地转化为Ｂ－可微方程组：求解，其中：容易证明，由（１．３）定义的映射Ｇ处处Ｂ－可微，且其在点ｘ∈Ｒｎ处的Ｂ－导数ＢＧ（ｘ）为而对于问题（１．２）（１．３），我们希望直接用经典的广义Ｎｅｗｔｏｎ法进行求解．但是，由于由（１．３）定义映射Ｇ在（１．１）的解ｘ∈Ｒｎ处，没有可逆的强Ｆ－导数存在，因此，关于算法（１．５）（１．６）…     

复迭代映射z_(n 1)= c的 广义Mandelbrot集研究

研究了复迭代映射ｚ（ｎ＋１）＝／ｚｎｍ＋ｃ的广义Ｍａｎｄｅｌｂｒｏｔ集,指出其关于实轴是对称的,并且具有ｍ＋１次的旋转对称性,得出周期轨道的稳定性条件及一周期轨道的稳定区域的边界方程．利用逃逸时间算法和周期点查找的算法构造Ｍａｎｄｅｌｂｒｏｔ集,可以更清楚地了解Ｍａｎｄｅｌｂｒｏｔ集的结构．     

关于临界图的一个定理的推广

叶宏博证明了当Δ≥５时没有度序列是２ｒΔ２ｒ的Δ－临界图．Ｋａｙａｔｈｒｉ推广了上述结果,证明了当Δ≥５时,没有同时满足下列两个条件的Δ－临界图：（ａ）Ｇ有一个２度点ｘ;设ｙ,ｚ是ｘ的两个邻接点;（ｂ）有一主项点ｙ１∈ＮＧ（ｙ）（ｙ１≠ｙ）与－２度点邻接．我们对上述结果进一步推广,证明了条件（ｂ）不是必要的;只要ｙ１与一个度数小于Δ－１的点邻接即可（可以不是２度点）．     

Lauwerier吸引子结构和SBR测度

Abstract. In this paper,the Lauwerier map     

Invariant measure and Lyapunov exponents for birational maps of P2

In this paper we construct and study a natural invariant measure for a birational self-map of the complex projective plane. Our main hypothesis - that the birational map be "separating" - is a condition on the indeterminacy set of the map. We prove that the measure is mixing and that it has distinct Lyapunov exponents. Under a further hypothesis on the indeterminacy set we show that the measure is hyperbolic in the sense of Pesin theory. In this case, we also prove that saddle periodic points are dense in the support of the measure.     

Kupka-Smale theorem for polynomial automorphisms of ?2 and persistence of heteroclinic intersections

A map is Kupka-Smale if all periodic points are hyperbolic and the stable and unstable manifolds of any two saddle points are transverse. Here we prove that Kupka-Smale maps form a residual set of full Lebesgue measure in the space of polynomial automorphisms of ² of fixed dynamical degree d2. We also prove that a heteroclinic point of two saddle periodic orbits may be continued over (almost) the entire parameter space for this set of maps. This is one of the first persistence theorems proved in holomorphic dynamics in several variables.     

Hausdorff dimensions of the Julia sets of reluctantly recurrent rational maps

In this paper, we consider a rational map f of degree at least two acting on Riemman sphere that is expanding away from critical points. Assuming that all critical points of f in the Julia set J(f) are reluctantly recurrent, we prove that the Hausdorff dimension of the Julia set J(f) is equal to the hyperbolic dimension, and the Lebesgue measure of Julia set is zero when the Julia set J(f) ≠ .     

Statistical properties of topological Collet-Eckmann maps

We study geometric and statistical properties of complex rational maps satisfying a non-uniform hyperbolicity condition called “Topological Collet-Eckmann”. This condition is weaker than the “Collet-Eckmann” condition. We show that every such map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic, and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and that this invariant measure is exponentially mixing (it has exponential decay of correlations) and satisfies the Central Limit Theorem.We also show that for a complex rational map the existence of such invariant measure characterizes the Topological Collet-Eckmann condition: a rational map satisfies the Topological Collet-Eckmann condition if, and only if, it possesses an exponentially mixing invariant measure that is absolutely continuous with respect to some conformal measure, and whose topological support contains at least 2 points.     

真拟弱几乎周期点和拟正则点

T为紧致度量空间X上的连续映射,M（X）为X上所有Borel概率测度.设x∈X,记Mx（T）为概率测度序列{1n∑n 1i=0δTi（x）}在M（X）中的极限点的集合,其中δx表示支撑集是{x}的点测度.记W（T）和QW（T）分别为T的弱几乎周期点和拟弱几乎周期点集.本文证明,如果（X,T）非平凡且满足specifcation性质,则存在x,y∈QW（T）／W（T）（称为真拟弱几乎周期点）,分别满足μ∈Mx（T）,x∈Supp（μ）和ν∈My（T）,y∈/Supp（ν）,回答了周作领等提出的公开问题.Mx（T）在弱拓扑中是紧致连通集,所以,要么是单点集,要么是不可数集.如果x∈QW（T）／W（T）,则Mx（T）是不可数集.一个自然的问题是,怎么刻画M x（T）是单点集的点x（这时x称为拟正则点）.本文给出M x（T）是单点集的充要条件.     

Reverse bifurcation and fractal of the compound logistic map

The nature of the fixed points of the compound logistic map is researched and the boundary equation of the first bifurcation of the map in the parameter space is given out. Using the quantitative criterion and rule of chaotic system, the paper reveal the general features of the compound logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the map may emerge out of double-periodic bifurcation and (2) the chaotic crisis phenomena and the reverse bifurcation are found. At the same time, we analyze the orbit of critical point of the compound logistic map and put forward the definition of Mandelbrot–Julia set of compound logistic map. We generalize the Welstead and Cromer’s periodic scanning technology and using this technology construct a series of Mandelbrot–Julia sets of compound logistic map. We investigate the symmetry of Mandelbrot–Julia set and study the topological inflexibility of distributing of period region in the Mandelbrot set, and finds that Mandelbrot set contain abundant information of structure of Julia sets by founding the whole portray of Julia sets based on Mandelbrot set qualitatively.     

Measure of noninvertibility of minimal maps

Minimal maps in compact metric spaces are known to be almost one-to-one. Thus, the set of points with more than one preimage is of first category. In the present paper we study the measure of this set with respect to the invariant measures of the considered minimal map. Among others, we give an example of a minimal self-mapping of a continuum such that the set of points with more than one preimage has positive measure for every invariant measure.     

Unstable directions and fractal dimension for skew products with overlaps in fibers

A unique feature of smooth hyperbolic non-invertible maps is that of having different unstable directions corresponding to different prehistories of the same point. In this paper we construct a new class of examples of non-invertible hyperbolic skew products with thick fibers for which we prove that there exist uncountably many points in the locally maximal invariant set ?? (actually a Cantor set in each fiber), having different unstable directions corresponding to different prehistories; also we estimate the angle between such unstable directions. We discuss then the Hausdorff dimension of the fibers of ?? for these maps by employing the thickness of Cantor sets, the inverse pressure, and also by use of continuous bounds for the preimage counting function. We prove that in certain examples, there are uncountably many points in ?? with two preimages belonging to ??, as well as uncountably many points having only one preimage in ??. In the end we give examples which, also from the point of view of Hausdorff dimension, are far from being homeomorphisms on ??, as well as far from being constant-to-1 maps on ??.