Positively expansive homeomorphisms of compact spaces
David Richeson, Jim Wiseman
给出了一个新的初等证明：紧度量空间的同胚是正扩张的当且仅当空间是有限的。
We give a new and elementary proof showing that a homeomorphism of a compact metric space is positively expansive if and only if the space is finite.
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N-expansive homeomorphisms on surfaces
我们利用在[乐]中发展的技术来研究曲面上的N-扩张同胚。证明了当f是定义在无游荡点的紧致无边界曲面M上的2-扩张同胚时，f是可扩张的.这个条件的徘徊集不能放松：我们提出了一个例子的2-扩张同胚的双与游荡点，这是不扩张的。
Partially Expansive Homeomorphisms
在这篇文章中，我们考虑几种形式的膨胀。引入了两个与拓扑维数有关的新定义。研究了三维流形的cw-扩张曲面同胚和扩张同胚下的局部稳定集的拓扑。还考虑了$C^r$ -稳定膨胀率问题。
Expansive and fixed point free homeomorphisms of the plane
本文的目的是在一定的扩张条件下刻画平面的不动点自由同胚的集合。
Anomalous cw-expansive surface homeomorphisms
证明了亏格2曲面存在一个cw-扩张同胚，其不动点的局部独立集不局部连通。
Hyper-expansive Homeomorphisms
紧度量空间上的同胚称为超扩张的，如果每对不同的紧集都被Hausdorff度量上的同胚所分离。我们描述了这样的动力学，作为那些有限数量的轨道和非徘徊集的排斥和吸引的周期轨道的工会。给出了紧度量空间允许超扩张同胚的一个刻划。
In this article we characterize monotone extensions of cw-expansive homeomorphisms of compact metric spaces. We study the topology of its quotient space in the case of a compact surface. These results are applied to prove that there are cw-expansive homeomorphisms of compact surfaces with infinitely many fixed points and empty wandering set. The examples are quotients of topological perturbations of pseudo-Anosov diffeomorphisms. We also show that there is a cw-expansive homeomorphism with the shadowing property of the 2-sphere.
In this article we construct an expansive homeomorphism of a compact three-dimensional manifold with a fixed point whose local stable set is not locally connected. This homeomorphism is obtained as a topological perturbation of a quasi-Anosov diffeomorphism that is not Anosov.
There are no stable points for continuum-wise expansive homeomorphisms
我们得到了一些关于连续扩张同胚的结果，如局部稳定集和不稳定集不存在稳定点和存在非平凡连通分支。这些事实在扩张同胚的分类定理中具有重要意义。然而，它们现在是通过新的和独立的技术来实现的，这些技术可以保持更一般的设置。
Positively n-expansive homeomorphisms and the L-shadowing property
本文研究了具有L-阴影性质的n-扩张同胚的动力学，证明了正n-扩张同胚若具有L-阴影性质则只能定义在有限度量空间上。
N-expansive homeomorphisms with the shadowing property
本文研究紧度量空间上具有阴影性质的n-扩张同胚，构造了非(n-1)-扩张且有无穷多链回归类的例子，并利用局部稳定集证明极限阴影性质，为拓扑混合n-扩张同胚的存在性问题提供方向。
Continuum-wise expansive homeomorphisms on Peano continua
在Peano连续统上，连续扩张同胚的所有局部稳定和不稳定分量都是非平凡的。特别是，存在对初始条件的敏感依赖。这推广了\cite{h,l}中关于Peano连续体上扩张同胚缺少李雅普诺夫稳定点（弱汇）和存在非平凡稳定和不稳定分量的结果.我们还利用这一事实推广了\cite{ktt}中的一个结果：允许连续扩张同胚的Peano曲线$X$在任何地方都是边可数的。然而，这是没有解决的问题，这样的动力学是否可能在局部平面Peano曲线。还提出了其他一些问题。
Downarowicz and Maass (2008) have shown that every Cantor minimal homeomorphism with finite topological rank $K > 1$ is expansive. Bezuglyi, Kwiatkowski, and Medynets (2009) extended the result to non-minimal aperiodic cases. In this paper, we show that all finite-rank zero-dimensional systems are expansive or have infinite odometer systems; this is an extension of the two aforementioned results. Nevertheless, the methods follow similar approaches.
Anomalous cw-expansive homeomorphisms on compact surfaces of higher genus
本文构造了高亏格紧致表面上的异常cw-扩张同胚，解决了Artigue提出的问题，并给出了非扩张同胚的例子。
Countably and entropy expansive homeomorphisms with the shadowing property
本文研究具有伪轨跟踪性质的可数膨胀与熵膨胀同胚，证明传递可数膨胀同胚在传递点集上膨胀，并证明在曲面情形下可数膨胀和熵cw-膨胀均等价于拓扑共轭于阿诺索夫微分同胚。
Let $X$ be a compact metric space and let $f:X\rightarrow X$ be a homeomorphism on $X$. We show that if $f$ is both pointwise recurrent and expansive, then the dynamical system $(X, f)$ is topologically conjugate to a subshift of some symbolic system. Moreover, if $f$ is pointwise positively recurrent, then the subshift is semisimple; a counterexample is given to show the necessity of positive recurrence to ensure the semisimilicity.
We show that on a totally disconnected compact metric space every separating homeomorphisms is expansive except at periodic points. We conclude that minimal separating homeomorphisms are expansive and that every separating homeomorphism has asymptotic points. We show that the only spaces admitting separating (or finite expansive) and recurrent homeomorphisms are finite sets. We apply our results to give a characterization of expansivity in terms of the expansivity of the cyclic group of powers of the homeomorphism.
Every compact metric space that supports a positively expansive homeomorphism is finite
我们给出了标题的一个简单证明。
We show that if there exists a topologically expansive homeomorphism on a uniform space, then the space is always a regular space. Through examples we show that in general composition of topologically expansive homeomorphisms need not be topological expansive and also that conjugate of topologically expansive homeomorphism need not be topological expansive. Further, we obtain a characterization of orbit expansivity in terms of topological expansivity and conclude that if there exists a topologically expansive homeomorphism on a compact uniform space then the space must be metrizable. We also s…
Generic surface homeomorphisms are almost continuum-wise expansive
本文证明了无边紧致曲面上cw-膨胀同胚在 $C^0$ 拓扑下的稠密性，并进一步证明一般同胚可被半共轭且满足连通性、不分隔性与直径约束的cw-膨胀同胚逼近。
Entropy, pseudo-orbit tracing property and positively expansive measures
研究了紧度量空间的同胚，其对非游荡集的限制具有伪轨跟踪性质。证明了若存在正扩张测度，则拓扑熵为正.这个结果的一些简短的应用。
We prove that a homeomorphism of a compact metric space has an expansive measure \cite{ms} if and only if it has many ones with invariant support. We also study homeomorphisms for which the expansive measures are dense in the space of Borel probability measures. It is proved that these homeomorphisms exhibit a dense set of Borel probability measures which are expansive with full support. Therefore, their sets of heteroclinic points has no interior and the spaces supporting them have no isolated points.
Dimensions and entropies for an expansive homeomorphism
针对膨胀同胚，受Young曲面微分同胚公式启发构造具双Lyapunov指数的双曲度量，研究多种熵与维数关系，证明双曲度量膨胀拓扑系统的Eckmann-Ruelle猜想并建立熵的变分原理。
Expansive homeomorphisms of the plane
本文讨论了平面的可扩同胚的分类问题。给出了同胚与线性双曲自同构共轭的充要条件。这些技术涉及平面的拓扑和度量方面。使用的李雅普诺夫度量函数定义了相同的拓扑作为一个诱导的通常的度量，但在一般情况下，是不等同于它是这样的技术的一个例子。关于李雅普诺夫函数在无穷远处的行为的一个假设的发现使我们能够推广一些在紧致上下文中有效的结果。附加的局部性质允许我们得到另一个分类定理。
Various questions around finitely positively expansive dynamical systems
本文研究有限正膨胀动力系统，证明在可逆条件下加上阴影性质可推出底空间有限，而极小性不能替代该条件，并证明此类系统在区间上不存在且蕴含零拓扑熵。
Characterizing expansivity through $C^*$-algebras
本文以交换$C^*$-代数$C(X)$中的膨胀可观测量刻画紧度量空间上的膨胀同胚，证明其代数结构与共轭不变性，并导出周期点可数及区间、圆周无膨胀同胚等结论。
Measure-expansive systems
该研究提出可测度量空间上的测度膨胀动力系统，推广紧度量空间膨胀系统的相关结果，刻画其同胚特征，证明区间无此类同胚、圆周上为Denjoy同胚，给出概率证明及连续映射的类似结果。
We study expansive dynamical systems in the setting of distributive lattices and their automorphisms, the usual notion of expansiveness for a homeomorphism of a compact metric space being the particular case when the lattice is the topology of the phase space ordered by inclusion and the automorphism the one induced by the homeomorphism, mapping open sets to open sets. We prove in this context generalizations of Mañé's Theorem and Utz's Theorem about the finite dimensionality of the phase space of an expansive system, and the finiteness of a space supporting a positively expansive homeomorphis…
We introduce minimally expansive and GH-stable points for homeomorphisms on metric spaces and $μ$-uniformly expansive, $μ$-shadowable and strong $μ$-topologically stable points for Borel measures (with respect to a homeomorphism on a metric space). We prove that: (i) minimally expansive shadowable point of a homeomorphism on a compact metric space is topologically stable and GH-stable. (ii) $μ$-uniformly expansive $μ$-shadowable point for a Borel measure $μ$ (with respect to a homeomorphism on a compact metric space) is strong $μ$-topologically stable.
In this paper we investigate the relation between measure expansiveness and hyperbolicity. We prove that non atomic invariant ergodic measures with all of its Lyapunov exponents positive is positively measure-expansive. We also prove that local diffeomorphisms robustly positively measure-expansive is expanding. Finally, we prove that if a $C^1$ volume preserving diffeomorphism that. can not be accumulated by positively measure expansive diffeomorphis have a dominated sppliting.
Local product structure for expansive homeomorphisms
设$f\colon M\to M$是具有稠密拓扑双曲周期点的扩张同胚，$M$是紧致流形.然后在$M$的开稠密子集中存在局部乘积结构。此外，如果某个拓扑双曲周期点有余维1，则此局部乘积结构是一致的。特别是，我们得出结论，同胚是共轭的线性Anosov同胚的环面。
For an $α$-expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the $α$-shadowing property for one-jump pseudo orbits (known as the local product structure property). The proof relies on a reformulation of the property of expansiveness in terms of the pseudo orbits of the system.
Let (M,d) be a compact metric space and f:M --> M an expansive homeomorphism. We define Lyapunov exponents L(f,m)_{max} and l(f,mu)_{min} for an f-invariant measure m. When L(f,m)_{max} > 0 and l(f,mu)_{min} < 0 can be interpreted as a weak form of hyperbolicity for f. We prove that if M is a Peano space then there is g>0 such that L(f,m)_{max} > g and l(f,m)_{min}< - g. We also show that the hypothesis that M is a Peano space is necessary to obtain the maximal Lyapunov exponent positive and the minimal Lyapunov exponent negative. Moreover we define Lyapunov exponents for K, a compact f-invari…
We consider the problem of the observability of positively expansive maps by the time series associated to continuous real functions. For this purpose we prove a general result on the generic observability of a locally injective map of a compact metric space of finite topological dimension, extending earlier work by Gutman [6]. We apply this result to partially solve the problem of finding the minimal number of functions needed to observe a positively expansive map. We prove that two functions are necessary and sufficient for positively expansive maps on tori.
Lyapunov functions via Whitney's size functions
在本文中，我们提出了一种技术，用于构建基于惠特尼的大小函数的李雅普诺夫函数。给出了渐近稳定平衡点、孤立集、扩张同胚和连续扩张同胚的应用。
A Note on Measure-Expansive Diffeomorphisms
本文证明了同胚是可数扩张的当且仅当它是测度扩张的。这一结果用于证明可扩、测度可扩和连续可扩$C^1$ -同态的集合的$C^1$ -内部是一致的。
Minimal expansive systems and spiral points
已知如果紧致度量空间X中存在极小扩张同胚，则X是全不连通的.本文给出了这个结果的一个简短证明，并分析了它在可扩流上的推广。
In this paper we consider expansive homeomorphisms of compact spaces with a hyperbolic metric presenting a self-similar behavior on stable and unstable sets. Several application are given related to Hausdorff dimension, entropy, intrinsically ergodic measures and the transitivity of expansive homeomorphisms with canonical coordinates.
Typical expansive measures for infinite-dimensional maps
本文否定回答了Morales和Sirvent关于膨胀性维度约束是否扩展到测度论概念的问题，证明Hilbert立方体上的移位映射关于乘积Lebesgue测度是膨胀的，并建立了一般充分条件，揭示了无限维相空间中丰富的测度论结构。
We characterize and describe the extensions of expansive and Anosov homeomorphisms on compact spaces. As an application we obtain a stability result for extensions of Anosov systems, and show a construction that embeds any expansive system inside an expansive system having the shadowing property for the pseudo orbits of the original space.
In this paper we study the polynomial entropy of homeomorphism on compact metric space. We construct a homeomorphism on a compact metric space with vanishing polynomial entropy that it is not equicontinuous. Also we give examples with arbitrarily small polynomial entropy. Finally, we show that expansive homeomorphisms and positively expansive maps of compact metric spaces with infinitely many points have polynomial entropy greater or equal than 1.
First-time sensitive homeomorphisms
本文提出同胚的首次敏感性概念，证明其涵盖连续统式膨胀同胚、Hilbert方体移位映射及部分部分双曲微分同胚，并在附加技术假设下推出正拓扑熵。
Rotation Vectors for Homeomorphisms of Non-Positively Curved Manifolds
旋转向量，定义为同胚的环面是同位素的身份，推广到这样的同胚的任何完整的黎曼流形与非正截面曲率。这些广义旋转向量存在于几乎每一个这样的动力系统的轨道上的任何不变测度与紧支持。然后，这个概念扩展到流，作为一个应用程序，它示出了如何非零旋转向量可以用来构建一个可测的半共轭之间的一个给定的流和测地流的流形。
Lipschitz perturbations of expansive systems
本文将光滑动力系统的一些已知结果推广到紧度量空间的Lipschitz同胚范畴。我们考虑的动力学性质的鲁棒扩张性和结构稳定性允许Lipschitz扰动的双曲度量。我们还研究了光滑流形上Lipschitz拓扑与$C^1$拓扑之间的关系。
S-Expansiveness and Zip Shift Maps in Symbolic Dynamics
本文提出局部同胚的S-扩张性概念，证明拉链移位映射具有该性质及跟踪性质，并表明任何S-扩张局部同胚是拉链移位映射的因子。
In this paper, we define bi-asymptotically $c$-expansive maps on metric spaces and study its relationship with other variants of expansivity such as bi-asymptotically expansive maps and $N$-expansive maps. We also provide an example to establish that expansive homeomorphisms need not be bi-asymptotically expansive. Finally we prove a spectral decomposition theorem for bi-asymptotically $c$-expansive continuous surjective maps with the shadowing property on compact metric spaces.
On Stronger Forms of Expansivity
本研究提出正扩张映射的更强形式$p \mathscr{F}-$扩张映射的概念，构造相关实例、推导成立条件与性质，并研究半群下的$\mathscr{F}-$扩张同胚及逆极限空间移位映射的关联。
Expansive homeomorphisms on complexity quasi-metric spaces
本文研究复杂度准度量空间上的扩展同胚理论，证明缩放变换在该空间上扩张当且仅当α≠1，并将轨道分离与经典时间层级定理联系起来。
Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved it does. We provide examples showing that Chung and Li's result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive act…
Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces
我们介绍了同胚的扩张性，跟踪和链递归的拓扑定义。它们推广了度量空间的通常定义。我们证明了各种定理拓扑Anosov同胚（地图是扩张的，并具有跟踪属性）的非紧和非度量空间，推广定理，这样的同胚紧度量空间。主要结果是将Smale的谱分解定理推广到第一可数局部紧仿紧T2空间上的拓扑Anosov同胚.
Isolated sets, catenary Lyapunov functions and expansive systems
这是一份关于模型的孤立集和建设的特殊双曲李雅普诺夫函数。我们证明，经过适当的手术后，每一个孤立的集合是吸引子和排斥子的交集。我们给出了吸引子和排斥子的线性模型。有了这些工具，我们构建双曲型李雅普诺夫函数和度量的孤立集，其值沿轨道的悬链线曲线。应用扩展流和同胚，获得，除其他事项外，一个双曲度量局部截面的任意扩张流的紧凑度量空间。
Expansive Minimal Flows
本文将 Mañé 关于扩张同胚的经典结果推广到流的情形，完整刻画了紧度量空间上无不动点的极小扩张流，证明其必为一维集合上极小子移位的悬挂流，并由此推出高维正则扩张流必含无穷多个极小子集。
Finite Sets with Fake Observable Cardinality
设$X$是一个紧度量空间，$|A|$表示集合$A$的基数。我们证明了如果$f\colon X\to X$是同胚且$|X|=\infty$，则对所有的$δ>0$存在$A\subset X$使得$|A|=4$，对所有的$k\in Z$存在$x,y\in f^k(A)$，$x\neq y$，使得$dist(x,y)<δ$ .如果一个观察者只能区分两个点，如果它们的距离大于$δ$，那么他肯定会说$A$最多有3个点，即使他知道$A$的每一个点，$f$也是一个同胚。我们表明，对于超扩张同胚的相同$δ$ -观察者不会失败的$A$的基数，如果我们开始与$|A|=3$，而不是$4$。这个问题的推广被认为是通过我们所谓的$(m,n)$ -可扩展性。
On the work of Jorge Lewowicz on expansive systems
我们将尝试给一个概述的一个里程碑式的结果豪尔赫Lewowicz：他的分类膨胀同胚的表面。我们的目标将是提出的主要思想，希望给证据的深刻和美丽的贡献，他所作的动力系统。我们将避免被技术，并试图集中在Lewowicz介绍的工具，以获得这些分类结果，如李雅普诺夫函数和持久性的概念，动力系统。主要的贡献，我们将试图集中在他的概念框架和方法，以数学反映了前面提到的工具，并从根本上微妙的相互作用之间的拓扑结构和动态膨胀同胚的表面，他发现，以建立他的结果。
We introduce the property of pre-expansivity for cellular automata (CA): it is the property of being expansive on asymptotic pairs of configurations (i.e. configurations that differ in only finitely many positions). Pre-expansivity therefore lies between expansivity and pre-injectivity, two important notions of CA theory. We show that there exist one-dimensional positively pre-expansive CAs which are not positively expansive and they can be chosen reversible (while positive expansivity is impossible for reversible CAs). We show however that no bi-dimensional CA which is linear over an Abelian …
Density of spectral gap property for positively expansive dynamics and smooth potentials, with applications to the phase transition problem
本文针对高维正膨胀局部微分同胚，证明一大类正则势下关联转移算子具有谱隙性质，且一类间歇斜积映射的相变结果与圆周情形类似，推进了非一致膨胀动力学的相变与谱隙研究。
Let $f\colon X\to X$ be a continuous function on a compact metric space. We show that shadowing is equivalent to backwards shadowing and two-sided shadowing when the map $f$ is onto. Using this we go on to show that, for expansive surjective maps the properties shadowing, two-sided shadowing, s-limit shadowing and two-sided s-limit shadowing are equivalent. We show that $f$ is positively expansive and has shadowing if and only if it has unique shadowing (i.e.\ each pseudo-orbit is shadowed by a unique point), extending a result implicit in Walter's proof that positively expansive maps with sha…
On the Invariance of Expansive Measures for Flows
本文研究了无固定点连续流在紧致度量空间上的扩张测度，通过动态球的新特征化方法，证明了正熵的不变测度具有正扩张性，并展示了扩张测度的稳定类测度为零。
Expanding and expansive time-dependent dynamics
本文研究由映射序列给出的不定常动力系统。对于由紧致黎曼流形M上的扩张C^2-映射所构造的系统，我们给出了度量和拓扑熵的计算公式。如果我们只假设映射是C^1，但以同样的方式作用在M的基本群上，我们可以证明自治系统的等共轭性的存在，这意味着熵的完全变分原理。最后，我们引入了强一致膨胀性的概念，推广了正膨胀性的经典概念，我们证明了一些著名的结果的不定常类似物。特别地，我们推广了Reddy的结果，该结果表明，一个正扩张系统在一个等价的度量中局部扩展距离。
Expansive actions with specification of sofic groups, strong topological Markov property, and surjunctivity
本文证明了对于 sofic 群作用的膨胀动力系统，在非负 sofic 拓扑熵且满足弱 specification 与强拓扑马尔可夫性质的条件下，系统必为满射的。
Let $(X,d)$ be a metric space and $f: X \rightarrow X$ be a homeomorphism. We say that a dynamical system $(X,f)$ is \emph{expansive}, with constant of expansivity $c \in \mathbb{R{^+}}$, if for all $x,y \in X$ , $x \neq y$, exists $n \in \mathbb{Z}$, such that $d(f^n(x), f^n(y)) >c$. In this paper we will use the theory of Nonstandard Analysis to study a subfamily of these dynamics, which verify that for all $x,y \in X$, if $x\neq y$ then the set $\lbrace n \in \mathbb{Z} : d(f^n(x), f^n(y) > c \rbrace$ is infinite.
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