Jörg Neunhäuserer
We show that for some simple classical chaotic dynamical systems the set of Li-Yorke pairs has full Hausdorff dimension on invariant sets.
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Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coupled expanding systems
本文研究了若干混沌动力系统的Li-Yorke对集的豪斯多夫维数，其中包括$A$耦合的膨胀系统.本文证明了PDMATHAMM耦合膨胀系统的Li-Yorke对在一定条件下在不变集上具有满豪斯多夫维数。本文将文献[8]关于拓扑共轭于全移位且具有自相似不变集的动力系统的Li-Yorke对的豪斯多夫维数的结果推广到共轭于某种子移位的动力系统的情形。给出了一类A-耦合扩张映射的"混沌不变集"的豪斯多夫维数。
On Li--Yorke chaotic transformation groups modulo an ideal
本文在拓扑变换半群中引入模理想的Li-Yorke混沌概念，刻画一类阿贝尔无限Li-Yorke混沌Fort变换群，并证明其可余分解为非Li-Yorke混沌变换群。
Completely Li--Yorke chaotic homeomorphisms with positive entropy
本文针对紧度量空间上完全Li--Yorke混沌同胚是否能有正熵的公开问题，证明对所有真点对非双侧渐近的正熵同胚，均可构造出相关的正熵完全Li--Yorke混沌同胚，给出肯定回答。
In this paper, we construct a homeomorphism on the unit closed disk to show that an invertible mapping on a compact metric space is Li-Yorke chaotic does not imply its inverse being Li-Yorke chaotic.
On a class between Devaney chaotic and Li-Yorke chaotic generalized shift dynamical systems
本文证明广义移位动力系统为稠密混沌当且仅当底映射无拟周期点，从而确立该类严格介于Devaney混沌与Li-Yorke混沌之间，并给出多种敏感性与混沌性质的等价刻画。
Li-Yorke chaos for invertible mappings on non-compact spaces
本文给出了两个例子，证明了一个可逆映射是Li-Yorke混沌的并不意味着它的逆是Li-Yorke混沌的，其中一个是无限维希尔伯特空间上的可逆有界线性算子，另一个是单位开圆盘上的同胚.此外，我们还利用最后一个例子证明了Li-Yorke混沌在拓扑共轭下是不保持的。
Recurrence determinism, one of the fundamental characteristics of recurrence quantification analysis, measures predictability of a trajectory of a dynamical system. It is tightly connected with the conditional probability that, given a recurrence, following states of the trajectory will be recurrences. In this paper we study recurrence determinism of interval dynamical systems. We show that recurrence determinism distinguishes three main types of $ω$-limit sets of zero entropy maps: finite, solenoidal without non-separable points, and solenoidal with non-separable points. As a corollary we obt…
Li-Yorke and Devaney chaotic uniform dynamical systems amongst weighted shifts
该研究刻画有限离散域上加权广义移位所诱导一致动力系统的 Li–Yorke 混沌与 Devaney 混沌性质，并分别给出充要条件。
Li-Yorke chaos for maps on G-Spaces
本文在G-空间上定义并研究G-Li-Yorke混沌，证明其可迭代性及与Li-Yorke混沌的关系，并给出G-传递且存在公共不动点条件下映射为G-Li-Yorke混沌的充分条件。
We prove that if a topological dynamical system is mean sensitive and contains a mean proximal pair consisting of a transitive point and a periodic point, then it is mean Li-Yorke chaotic (DC2 chaotic). On the other hand we show that a system is mean proximal if and only if it is uniquely ergodic and the unique measure is supported on one point.
Li-Yorke chaotic weighted composition operators on Hardy and Bergman spaces over the unit disk
研究加权复合算子在单位盘上的Hardy和Bergman空间中的李-约尔克混沌及均值李-约尔克混沌，证明算子混沌当且仅当非幂有界，均值混沌当且仅当非绝对Cesàro有界。
We study chaotic properties of uniformly convergent nonautonomous dynamical systems. We show that, contrary to the autonomous systems on the compact interval, positivity of topological sequence entropy and occurrence of Li-Yorke chaos are not equivalent, more precisely, neither of the two possible implications is true.
Li-Yorke Chaotic Weighted Composition Operators
本文完整刻画了 $C_0(X)$ 与 $L^p(μ)$ 空间上加权复合算子的 Li-Yorke 混沌性，并由此给出 $c_0$、$\ell^p$ 及 Köthe 序列空间上加权移位算子 Li-Yorke 混沌的简化判别条件。
It is rigorously proved that the chaotic dynamics of the non-smooth system with relay function is persistent even if a chaotic perturbation is applied. We consider chaos in a modified Li-Yorke sense such that infinitely many almost periodic motions take place in its basis. It is demonstrated that the system under investigation possesses countable infinity of chaotic sets of solutions. Coupled Duffing oscillators are used to show the effectiveness of our technique, and simulations that support the theoretical results are represented. Moreover, a chaos control procedure based on the Ott-Grebogi-…
Compact non-uniformizable Li-Yorke chaotic dynamical systems via an example
本文通过一个例子，研究了在有限紧致拓扑空间中，单侧移位系统呈现李-约尔克混沌的条件与非渐近对、混乱对等概念的等价性。
The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor.
Auslander-Yorke Dichotomy and Its Generalizations for Non-Autonomous Dynamical Systems
本文研究了非自治动力系统中Auslander-Yorke分歧的扩展，探讨了敏感性和等连续性的拓扑类比及其在不同空间中的应用。
The equivalence relationship between Li-Yorke $δ$-chaos and distributional $δ$-chaos in a sequence
本文讨论了序列的Li-Yorke混沌与分布混沌的关系。指出了序列$Q$中所有分布$δ$ -扰码对的集合是一个$G_δ$集合，证明了Li-Yorke $δ$ -混沌等价于序列中的分布$δ$ -混沌，一致混沌集合是序列中的分布扰码集合，一类传递系统蕴涵着序列中的分布混沌.
Sufficient conditions under which a transitive system is chaotic
设（X，T）是一个拓扑传递的动力系统.证明了若存在（X，T）的子系统（Y，T）使得（X \times Y，T \times T）是传递的，则（X，T）是Li和Yorke意义下的强混沌.然后，我们表明，许多已知的充分条件在文献中，以及一些新的结果，这一声明的推论。事实上，我们在这些结果中推导出的混沌行为是一种更强的李约克混沌，我们称之为一致混沌。对于最小的系统，我们表明，除其他结果外，一致的混沌保持的扩展和最小的系统，这是不一致的混沌PI。
In this article, we show that a chaotic behavior can be found on a cube with arbitrary finite dimension. That is, the cube is a quasi-minimal set with Poincare chaos. Moreover, the dynamics is shown to be Devaney and Li-Yorke chaotic. It can be characterized as a domain-structured chaos for an associated map. Previously, this was known only for unit section and for Devaney and Li-Yorke chaos.
Li-Yorke Chaos for Composition Operators on $L^p$-Spaces
该研究在线性动力学已有Li-Yorke混沌刻画的基础上，针对$L^p$空间上的复合算子给出更简洁实用的Li-Yorke混沌刻画，推导出加权移位算子混沌的简洁判据，并以实例验证结果的紧性。
Li-Yorke chaos for dendrite maps with zero topological entropy and $ω$-limit sets
设$X$是一个端点集$E(X)$闭的树状映射，$f:~X \to X$是一个拓扑熵为零的连续映射.令$P(f)$是$f$的周期点的集合。证明了若$L$是$f$的无限$ω$ -极限集，则有$L\cap P(f)\subset E(X)^{\prime}$，其中$E(X)^{\prime}$是$E(X)$的所有聚集点的集合.此外，如果$E(X)$是可数的，而$L$是不可数的，则$L\cap P(f)=\emptyset$。我们还表明，如果$E(X)^{\prime}$是有限的，那么$f$的任何不可数的$ω$ -极限集有一个分解，因此，如果$f$有一个李约克对$(x,y)$与$ω\_f(x)$或$ω\_f(y)$是不可数的，那么$f$是李约克混沌。
Omega-limit sets and invariant chaos in dimension one
Ω极限集在一维动力学中起着重要的作用。在过去的50年里，至少出现了三种基本集合的定义。作者经常使用不同定义的结果。在这里，我们填补了缺少这些定义的等价性证明的空白。利用基集上的结果，推广了文[P.~ Oprocha，不变的混乱集和分布混沌，Dyn。24（2009），no. 1，31- 43.]有限图的连续映射。Li-Yorke混沌弱于正拓扑熵。当我们在Li-Yorke置乱集上加上不变性条件时，等价性就产生了。本文证明了连续图映射的正拓扑熵、马蹄形、不变Li-Yorke混沌集、一致不变分布混沌集和分布混沌对是等价的。
Inheriting of chaos in nonautonomous dynamical systems
考虑非自治离散动力系统$\{ f_n\}_{n\ge 1}$，其中每个$f_n$都是满射连续映射$[0,1]\to [0,1]$，使得$f_n$一致收敛到一个映射$f$ .我们表明，除其他外，如果$f$是混沌的意义上的李和约克然后非自治系统$ \{ f_n\}_{n\ge 1}$是李约克混沌，以及，同样是真实的分布混沌。如果$f$的拓扑熵为零，则非自治系统继承其无穷$ω$ -极限集.
For graph maps, one scrambled pair implies Li-Yorke chaos
对于动力系统$(X,f)$，$X$是具有度量$d$的紧致度量空间，并且$f$是连续映射$X\to X$，如果$S$中的每对$(x,y)$都被置乱，则集合$S\subseteq X$被置乱，即，$\liminf_{n\to+\infty}d(f^n(x),f^n(y))=0$和$\limsup_{n\to+\infty}d(f^n(x),f^n(y))>0$。如果系统$(X,f)$具有不可数的置乱集，则系统是Li-Yorke混沌的.众所周知，对于区间和圆映射，扰码对的存在意味着Li-Yorke混沌，实际上是Cantor扰码集的存在。我们证明了同样的结果适用于图映射。我们进一步证明了在紧可数度量空间上，一个加扰对蕴涵着一个无限加扰集的存在。
If a topological dynamical system $(X,T)$ has positive topological entropy, then it is multivariant mean Li-Yorke chaotic along a sequence $\{a_k\}_{k=1}^\infty$ of positive integers which is "good" for pointwise ergodic convergence with a mild condition; more specifically, there exists a Cantor subset $K$ of $X$ such that for every $n\ge2$ and pairwise distinct points $x_1,x_2,\dotsc,x_n$ in $K$ we have \[\liminf_{N\to\infty}\frac{1}{N}\sum_{k=1}^N\max_{1\leq i<j\leq n} d(T^{a_k}x_i,T^{a_k}x_j)=0\] and \[\limsup_{N\to\infty}\frac{1}{N}\sum_{k=1}^N\min_{1\leq i<j\leq n} d(T^{a_k}x_i,T^{a_k}x_j…
We provide new similarities between regular continued fractions and Lüroth series in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete analogue for the Lüroth transformation of results by W. Liu, B. Li and W. Liu, S. Wang on the distal, asymptotic and Li-Yorke pairs for the Gauss map.
We construct a multiply Xiong chaotic set with full Hausdorff dimension everywhere that is contained in some multiply proximal cell for the full shift over finite symbols and the Gauss system respectively.
Approximation of chaotic operators
众所周知，算子理论中的“超循环”概念与动力系统中的“传递”概念是相同的。目前，超循环算子类的研究已相当深入。遵循超循环算子研究的思路，本文研究了一类具有混沌性质的算子。首先讨论了所有Li-Yorke混沌算子或分布混沌算子的集合的闭包。我们给出了它们的谱描述，并证明了这两个闭包是一致的。此外，所有Li-Yorke混沌算子的集合和所有分布混沌算子的集合都具有非空内部，并且它们也是一致的。本文还讨论了所有超循环算子集的闭包与所有分布混沌算子集的闭包之间的包含关系。最后，我们得到了上面考虑的集合的连通性。
In this paper, we study the mean Li-Yorke chaotic phenomenon along any infinite positive integer sequence for infinite-dimensional random dynamical systems. To be precise, we prove that if an injective continuous infinite-dimensional random dynamical system $(X,ϕ)$ over an invertible ergodic Polish system $(Ω,\mathcal{F},\mathbb{P},θ)$ admits a $ϕ$-invariant random compact subset $K$ with $h_{top}(K,ϕ)>0$, then given a positive integer sequence $\mathbf{a}=\{a_i\}_{i\in\mathbb{N}}$ with $\lim_{i\to+\infty}a_i=+\infty$, for $\mathbb{P}$-a.s. $ω\inΩ$ there exists an uncountable subset $S(ω)\subs…
For a continuous self-map of a star graph to be Li-Yorke chaotic and to have full periodicity, we prove some new sufficient conditions on the orbit of the center.
Chaotic behavior of uniformly convergent nonautonomous systems with randomly perturbed trajectories
研究具有随机扰动轨迹的非自治离散动力系统。我们假设这样一个系统是由一致收敛到一个映射$f$的连续映射序列生成的。给出了由极限函数$f$生成的（标准）自治离散动力系统的回复点对于具有随机扰动轨迹的非自治系统也是回归点的条件.我们还提供了一个必要条件的非自治离散动力系统是非混沌的意义下，李和约克关于小的随机扰动。
In the present paper the classical ideas of Hausdorff and Lebesgue are combined and the Hausdorff--Lebesgue measure is introduced. This makes it possible to obtain new results in chaotic dynamics.
Let $σ_{K}: \sum_{K}\rightarrow \sum_{K}$ be a shift map. For an interval $[p,q]\subset[0,1]$, let $D_{σ_{K}}([p,q])$ denote the set of pairs for which the density spectrum of the $ε$-approach time set equals $[p,q]$ when $ε$ is small and $E_{σ_{K}}([p,q])$ the set of pairs for which the density spectrum of the $ε$-approach time set converges to $[p,q]$ when $ε\rightarrow 0^+$. Then $\dim_{H} D_{σ_{K}}([p,q])=\dim_{H} E_{σ_{K}}([p,q])=2-q$. Moreover, $\mathscr{H}^{2-q}(E_{σ_{K}}([p,q]))=1$ when $q=0$ and $\mathscr{H}^{2-q}(E_{σ_{K}}([p,q]))=+\infty$ when $q>0$. Meanwhile, $\mathscr{H}^{2-q}(D_…
Chaotic discretization theorems for forced linear and nonlinear coupled oscillators
本文研究了受外力作用的耦合振子系统，证明其在Li-Yorke意义下的混沌性，并通过数值模拟和分岔分析验证了系统的复杂动力学行为。
Morphogenesis of Chaos
形态发生，正如勒内·托姆在广义上所理解的那样，被认为是各种类型的混沌。即倍周期级联、Devaney混沌和Li-Yorke混沌。此外，在讨论的形式中，我们考虑了非线性的遗传，双涡卷蔡氏吸引子和准周期运动作为混沌吸引子的可能骨架。为了使我们的介绍更清楚，我们不得不说，我们可以考虑其他各种伴随的概念，混沌吸引子的结构，它的分形维数，分岔图的形式，李雅普诺夫指数的频谱，等等。我们比较了我们的主要概念与图灵的形态发生和约翰冯诺依曼自动机，考虑到这可能不仅是形式上的，而且将为图灵形态发生中的混沌发展和自我复制机器提供思路。为了提供严格的研究的主题，我们引入了新的定义，如混沌函数集，混沌的生成器和复制器，并精确描述成分Devaney和Li-Yorke混沌连续动力学。适当的模拟，说明形态发生的现象。
We prove that there exists a scrambled set for the Gauss map with full Hausdorff dimension. Meanwhile, we also investigate the topological properties of the sets of points with dense or non-dense orbits.
Transitive cylinder flows whose set of discrete points is of full Hausdorff dimension
对于每个无理$α\in[0,1)$，我们构造一个连续函数$f\: [0,1)\to \R$，使得相应的柱面变换$[0,1)\times\R \ni (x,t) \mapsto (x+α, t+ f(x)) \in [0,1)\times\R$是传递的，并且其轨道是离散的点集的豪斯多夫维数是2。这种圆柱变换显示出一定的Devaney型的混沌行为。
Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing
研究了由双曲曲面映射T（如Anosov曲面同态或baker映射）驱动的斜积动力系统中的不变图在乘法强迫下的分支，当强迫被参数r=e^{-t}标度时.对于一系列的参数两个不变的图形（平凡的和非平凡的）共存，我们使用热力学形式主义的参数依赖的Hausdorff和包装尺寸的一组点，这两个图重合。作为推论，我们刻画了全局吸引子A_t的维数与参数的依赖关系：Hausdorff维数和packing维数有一个公共值dim（A_t），并且存在一个由T^{-1}的SRB测度确定的临界参数t_c，使得当t < t_c时，dim（A_t）=3，当t_c < t < t_{max}时，t --> dim（A_t）是严格递减的.
Some dynamical properties for linear operators
在我们最近的另一篇文章中，我们介绍了线性算子的一个新的动力学性质，称为范数单峰性，这意味着分布混沌。本文将进一步讨论范数单峰性。证明了范数单峰是相似不变量，并给出了范数单峰算子的谱。作为应用，在每个套代数中存在分布混沌算子。此外，还考虑了关于范数单峰和Li-Yorke混沌的正规算子和紧算子。特别地，单位算子的一个小的紧扰动可能是分布混沌的。
Dynamics on Hyperspace of Pointwise Periodic Homeomorphisms
研究点周期性同胚诱导映射的超空间动力学，证明几乎周期点与均匀复发点集相等，且无限ω-极限集内存在唯一最小集，该集为添加机。同时，超空间系统要么等连续要么具有李-约尔凯混沌和ω混沌。
Nowadays there are a number of surveys and theoretical works devoted to the Lyapunov exponents and Lyapunov dimension, however most of them are devoted to infinite dimensional systems or rely on special ergodic properties of the system. At the same time the provided illustrative examples are often finite dimensional systems and the rigorous proof of their ergodic properties can be a difficult task. Also the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. The survey is dev…
We show that the existence of a dense set of periodic points for a topologically transitive non-minimal continuous group action on a Hausdorff uniform space with an infinite acting group does not necessarily imply a sensitive dependence to the initial conditions in such a system. This leads to define the chaos in the sense of Devaney for a continuous group action on a Hausdorff uniform spaces with an infinite acting group in the original way, i.e. a non-minimal topologically transitive and sensitive system with a dense set of periodic points is a chaotic system in the sense of Devaney.
On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets
本文对Filippov系统中Shilnikov滑动连接的首回归映射进行局部分析，揭示了其共形迭代函数系统结构，证明了其局部不变集的闭包为Cantor集，并精确估计了其Hausdorff维数严格介于0和1之间且Lebesgue测度为0。
As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension three, we introduce a $C^2$-open set of diffeomorphisms of $\mathbb R^3$ having two horseshoes with different dimensions of instability. We prove that: the unstable set of one horseshoe and the stable set of the other are of Hausdorff dimension nearly $2$ whose cross sections are Cantor sets; the intersection of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly $1$. As a corollary we detect $C^2$-robust heterodimensional cycles. Our proof employs the theory of normally hy…
Let $X \subset \mathbb{R}^N$ be a Borel set, $μ$ a Borel probability measure on $X$ and $T:X \to X$ a Lipschitz and injective map. Fix $k \in \mathbb{N}$ greater than the (Hausdorff) dimension of $X$ and assume that the set of $p$-periodic points has dimension smaller than $p$ for $p=1, \ldots, k-1$. We prove that for a typical polynomial perturbation $\tilde{h}$ of a given Lipschitz map $h : X \to \mathbb{R}$, the $k$-delay coordinate map $x \mapsto (\tilde{h}(x), \tilde{h}(Tx), \ldots, \tilde{h}(T^{k-1}x))$ is injective on a set of full measure $μ$. This is a probabilistic version of the Tak…
We define a family B(t) of compact subsets of the unit interval which generalizes the sets of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as one moves the parameter t, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behavior as the usual family of quadratic polynomials. The set E of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension 1. We also show that the Hausdorff dimension of B(t) varies continuously with the parameter, and the d…
On proximality with Banach density one
设$(X,T)$是一个拓扑动力系统.一对点$(x,y)\in X^2$称为Banach邻近的，如果对于任何$ε>0$，集合$\{n\in\mathbb{Z}:\ d(T^nx,T^ny)<ε\}$具有Banach密度1。我们研究了Banach邻近关系的结构。一个有用的工具是拓扑动力系统的支撑的概念。我们证明了一个动力系统是强邻近的当且仅当$X^2$中的每一对都是Banach邻近的。$X$的子集$S$是Banach置乱的，如果$S$中的每两个不同的点形成一个Banach邻近对，但不是渐近的.我们构造了一个动力系统，使整个空间是一个Banach混沌集。即使全移位的Banach邻近关系是第一范畴，它也有一个稠密的Mycielski不变的Banach乱集.证明了区间映射是Li-Yorke混沌的当且仅当它有一个Cantor Banach混沌集。
Predicting dynamical systems with too few time-delay measurements: error estimates
该研究针对时滞测量不足的动力系统重构预测问题，证明概率框架下不变测度的Hausdorff维数是最小时滞嵌入维数的精确阈值，还修正并确立了经典预测误差猜想的下界，证明了条件测度维数的一般性定理。
Non-smooth saddle-node bifurcations II: dimensions of strange attractors
研究了拟周期强迫区间映射非光滑鞍结分岔中奇异非混沌吸引子的几何和拓扑性质。通过解释的吸引子作为限制对象的迭代的连续曲线和控制后者的几何形状，我们确定其Hausdorff和盒计数维数，并表明，这些采取不同的值。此外，同样的方法可以描述的吸引子的拓扑结构，并证明其极小。
Dimensions of attractors in pinched skew products
我们研究的尺寸奇怪的非混沌吸引子和相关的物理措施，在所谓的捏斜产品，介绍了Grebogi和他的同事在1984年。我们的主要结果是，豪斯多夫维数，逐点维数和信息维数都等于1，虽然盒计数维数已知是2。关于逐点维数的论断是从物理量可求正的较强结果中推导出来的。我们的研究结果证实了丁，Grebogi和Ott从1989年的猜想。
Prediction of dynamical systems from time-delayed measurements with self-intersections
本文针对紧致集上具有任意Borel概率测度的所有Lipschitz系统证明了关于混沌系统时延测量预测的猜想，给出了预测不佳点集测度衰减率上界，并推广至局部Lipschitz或Hölder系统。
Dimension results for inhomogeneous Moran set constructions
我们计算的Hausdorff，上盒和包装尺寸的某些非齐次Moran集的建设。这些构造超越了迭代函数系统的经典理论，因为在每一步都应用了不同的非线性压缩变换。此外，我们还允许收缩是弱共形的，并考虑收缩率的下确界为零的情况。此外，允许构造的基本集合具有复杂的拓扑结构，例如具有分形边界。使用技术从热力学形式主义，我们计算的分形维数的建设的极限集。作为一个主要应用，我们考虑随机非齐次莫兰集构造的维度结果，其中混沌动力系统用于控制构造每个步骤的收缩因子。
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