Bowen估计熵的变分原理

设(X, f)是一个拓扑动力系统,其中X是紧致度量空间, f是X上的连续自映射.本文对(X, f)引入Bowen估计熵,给出了Bowen估计熵的Billingsley定理和变分原理.     

测度的上局部熵和集合的类Bowen熵

设f为度量空间(x,d)上的连续映射,该文主要讨论了x的任意子集关于f的类Bowen熵可以通过(x,d)上测度的上局部熵估计.     

熵极小动力系统的复杂性

设X是一个紧致度量空间,f:X→X是一个连续映射,(X,f)是熵极小的.该文首先证明了f是强遍历的;另外,如果还假设X中存在f的一个真的(拟)弱几乎周期点,则得到f具有正拓扑熵且对任意的n1,f~n是遍历敏感依赖的.因此,f在Li-Yorke和Takens-Ruelle意义下是混沌的.该文所得结论改进和推广了最近的一些结论.     

可数对一映射及相关问题

本文研究了■0-sn-度量空间与度量空间之间的关系.利用特殊映射,获得了在序列空间中下述命题等价:(1)空间X是■0-sn-度量空间;(2)存在从度量空间M到X可数对一、序列商、σ映射f;(3)存在从度量空间M到X可数对一、序列商、σ映射f使得对每一个x∈X,■f-1(x)是σ-紧.推广了参考文献[3,4]中的一些结果.     

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.     

Ponomarev系统中的2序列覆盖映射（英文）

本文讨论了Ponomarev系统中的一个逆问题.对于Ponomarev系统(f,M,X,P)(或(f,M,X,{P_n})),证明了f是2序列覆盖映射当且仅当P是X的sof网(或每一P_n是X的so覆盖).作为一个推论,本文得到了空间X是度量空间的2序列覆盖映射像(或2序列覆盖π映射像)当且仅当X有sof网(或so覆盖组成的点星网).     

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。     

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.     

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.     

拓扑群作用下度量空间中链回归点集的研究

仿造度量空间中链回归点的定义,给出了拓扑群作用下度量空间中G-链回归点的概念,并将度量空间中链回归点的一些结论,推广到拓扑群作用下度量空间中,得到如下结果:1)同胚伪等价映射f的G-链回点集等于它的逆映射f~(-1)的G-链回归点集;2)伪等价映射f的G-链回点集和G-链等价集对G强不变;3)同胚等价映射f的G-链回点集f对强不变.4)等价映射f限制在它的G-链回归点集上形成的G-链回归点集就是等价映射f在度量G-空间X上形成的G-链回归点集.这些结果丰富了拓扑群作用下度量空间中G-链回归点的理论.     

非自治动力系统的原像熵

本文对紧致度量空间上的连续自映射序列应用生成集和分离集引入了点原像熵、原像分枝熵以及原像关系熵等几类原像熵的定义并进行了研究.主要结果是:(1) 证明了这些熵都是等度拓扑共轭不变量.(2)讨论了这些原像熵之间及它们与拓扑熵之间的关系,得到了联系这些熵的不等式.(3)证明了对正向可扩的连续自映射序列而言, 两类点原像熵相等,原像分枝熵与原像关系熵也相等.(4)证明了对(a).由闭Riemann 流形上的一个扩张映射经充分小的C1-扰动生成的自映射序列,以及(b).有限图上等度连续的自映射序列,有零原像分枝熵.     

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．     

线段映射的局部变差增长与局部拓扑熵

本文考虑闭区间上变差有界的连续映射f:I→I的局部变差增长γ(x,f)与局部拓扑熵h(x,f)．将证明γ(x,f)≥h(x,f)对所有x∈I成立,并且局部变差增长映射γf(x)=γ(x,f)与局部拓扑熵映射sf(x)=h(x,f)都是上半连续的,得到一个变分原理:局部变差增长γ(x,f)与局部拓扑熵h(x,f)的上确界分别等于全局变差增长γ(f)=limn→∞1/nln Var(fn)与拓扑熵h(f)．当映射f:I→I拓扑传递时,与Brin 和Katok对局部(测度)熵的讨论类似,我们证明,至多除一个不动点外,局部变差增长γ(x,f)与局部拓扑熵h(x,f)在开区间I°内恒为常值．     

On the Rank of the Semigroup TE(X)

${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5.     

Entropy estimates for some C-endomorphisms

In this paper we compute the non-commutative topological entropy in the sense of Voiculescu for some endomorphisms of stationary inductive limits of circle algebras. These algebras are groupoid C*-algebras, and the endomorphisms restricted to the canonical diagonal are induced by some expansive maps, whose entropies provide a lower bound. For the upper bound, we use a result of Voiculescu, similar to the classical Kolmogorov-Sinai theorem. The same technique is used to compute the entropy of a non-commutative Markov shift.

     

Discussions around Voiculescu's free entropies

In this article, we set two analogous definitions of the free entropies χ and χ∗ introduced by Voiculescu (Invent. Math. 118 (1994) 411; 132 (1998) 189). We discuss their relations, improving the preceding results obtained in Cabanal-Duvillard and Guionnet (Ann. Probab. (2001), to appear), where a bound on the microstates entropy χ was established.     

Topological entropy for discontinuous semiflows

We study two variations of Bowen's definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces. We prove that these definitions reduce to Bowen's ones in the case of continuous semiflows. As a second result, we prove that our entropies give a lower bound for the τ-entropy defined by Alves, Carvalho and Vásquez (2015). Finally, we prove that for impulsive semiflows satisfying certain regularity condition, there exists a continuous semiflow defined on another compact metric space which is related to the first one by a semiconjugation, and whose topological entropy equals our extended notion of topological entropy by using separated sets for the original semiflow.     

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.     

Entropies and Flux-Splittings for the Isentropic Euler Equations

51. IntroductionThe Euler equations for an iselitropic compressible fluid readwhere p 2 0 denotes the density, v the velocity) and p(p) 2 0 the pressure. The equstiope(1.1) form a nonlinear hyperbolic system of conserVation laws. By definition, a msthematicalentropy n = n(p, v) and its corresponding elltropy flux-function q = q(p, v) satisfyfor any smooth solution (p,m) of (1.1). A weak entropy3 by definition, vanishes on thevacuum p = 0. Following Laxlll'lz], we are interested in measuxable…