Noriaki Kawaguchi
For any continuous self-map of a compact metric space, we prove a saturation of distributionally scrambled Mycielski sets under a type of shadowing and the chain transitivity.
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For continuous self-maps of compact metric spaces, we consider the notions of generic and dense chaos introduced by Lasota and Snoha and their variations for the distributional chaos, under the assumption of shadowing. We give some equivalent properties to generic uniform (distributional) chaos in terms of chains and prove their equivalence to generic (distributional) chaos under the shadowing. Several examples illustrating the main results are also given.
We explore connections among the regional proximal relation, the asymptotic relation and the distal relation for a topological dynamical system with the shadowing property, and show that if a Devaney chaotic system has the shadowing property then it is distributionally chaotic.
S-limit shadowing and a global description of Li-Yorke type chaos
该研究将紧度量空间连续自映射的链分支链邻近划分推广为相空间$G_δ$划分，在s-极限跟踪假设下实现了对应多类Furstenberg族的Li-Yorke型混沌的全局刻画。
Shadowing, transitivity and a variation of omega-chaos
针对紧 Hausdorff 空间上的链传递连续自映射，研究 DSP 跟踪性及其基本性质，并据此给出统计型 ω-混沌的充分条件、拓扑通有性及不规则点分布结果。
Some results on s-limit shadowing and Li-Yorke type chaos
本文研究紧度量空间连续自映射的 s-极限跟踪性质，通过相空间的 $G_δ$-划分将 Li-Yorke 型混沌的描述推广到一般 Furstenberg 族，补充了已有结果并给出例证。
A note on Bohr chaos and hyperbolic sets
研究动力系统中跟踪现象与玻尔混沌的关系，给出基于跟踪的玻尔混沌充分条件，结合跟踪引理得出关于玻尔混沌和双曲集的结果，突出双曲集结构性质。
In this paper we consider relations between distributional chaos in a sequence with distributional chaos, w-chaos, R-T chaos, DC 3, respectively). We give a sufficient condition and prove that the distributional chaos is equivalent to the distributional chaos in a sequence under this condition. Besides, we get that distributional chaos in a sequence and w-chaos(R-T chaos, DC 3, respectively)do not imply each other. Finally, we give a new definition of chaos, named DC 2', which is similar to DC 2, and show that for any integer N > 0, f is DC 2' if and only if f^N is also DC 2'.
In this paper, we discuss the relationship between Li-Yorke chaos and distributional chaos in a sequence. We point out the set of all distributional $δ$-scramble pairs in the sequence $Q$ is a $G_δ$ set, and prove that Li-Yorke $δ$-chaos is equivalent to distributional $δ$-chaos in a sequence, a uniformly chaotic set is a distributional scramble set in some sequence and a class of transitive system implies distributional chaos in a sequence.
We show the existence of a dynamical system without any distributionally scrambled pair which is semiconjugated to a distributionally chaotic factor.
This paper studies distributional chaos in non-autonomous discrete systems generated by given sequences of maps in metric spaces. In the case that the metric space is compact, it is shown that a system is Li-Yorkeδ-chaotic if and only if it is distributionallyδ'-chaotic in a sequence; and three criteria of distributional δ-chaos are established, which are caused by topologically weak mixing, asymptotic average shadowing property, and some expanding condition, respectively, where δ and δ' are positive constants. In a general case, a criterion of distributional chaos in a sequence induced by a X…
In this paper, we introduce the notion of distributional chaos and the measure of chaos for random dynamical systems generated by two interval maps. We give some sufficient conditions for a zero measure of chaos and examples of chaotic systems. We demonstrate that the chaoticity of the functions that generate a system does not, in general, affect the chaoticity of the system, i.e., a chaotic system can arise from two nonchaotic functions and vice versa. Finally, we show that distributional chaos for random dynamical system is, in some sense, unstable.
We prove that the set of maps which exhibit distributional chaos of type 1 (DC1) is $C^0$-dense in the space of continuous self-maps of given any compact topological manifold (possibly with boundary).
Distributional Chaos in the Baire Space
该研究针对非紧度量动力系统，考察可数字母表上的 Baire 空间，证明有限型子移位具有稠密分布混沌，且完备、周期点稠密的有界型子移位也具有分布混沌。
Distributional Chaos in Over-Saturated Sets
本文研究非均匀规范性质下过饱和集的分布混沌，证明每个非空过饱和集具有第一类分布混沌，并给出差集混沌的充分条件及最优性例子。
A-coupled-expanding and distributional chaos
本文研究严格A-耦合扩张映射与分布混沌的关系，证明了当A为不可约转移矩阵时该映射在两种分布混沌意义下成立的充分条件。
Sufficient conditions for distributional chaos of type I
本文研究了I型分布混沌（DC1），将混合Anosov流的时间一映射表现出DC1的事实进行推广，得出了DC1的简单充分条件。
The main aim of the present paper is to study relations between $n$-scrambled tuples and their attraction-adherence properties with respect to various sequences of integers. This extends previous research on relations between chaos in the sense of Li and Yorke and distributional chaos with respect to a given sequence. Moreover, we construct a system which is $n$-distributionally chaotic but not $(n+1)$-chaotic.
In present paper we mainly focus on non-recurrent dynamical orbits with empty syndetic center and show that twelve different statistical structures over mixing expanding maps or transitive Anosov diffeomorphisms all have dynamical complexity in the sense of strongly distributional chaos.
Let $M$ be a compact smooth manifold without boundary. Based on results by Good and Meddaugh (2020), we prove that a strong distributional chaos is $C^0$-generic in the space of continuous self-maps (resp. homeomorphisms) of $M$. The results contain answers to questions by Li et al. (2016) and Moothathu (2011) in the zero-dimensional case. A related counter-example on the chain components under shadowing is also given.
The aim of the paper is to correct and improve some results concerning distributional chaos of type 3. We show that in a general compact metric space, distributional chaos of type 3, denoted DC3, even when assuming the existence of an uncountable scrambled set, is a very weak form of chaos. In particular, (i) the chaos can be unstable (it can be destroyed by conjugacy), and (ii) such an unstable system may contain no Li-Yorke pair. However, the definition can be strengthened to get DC$2\frac{1}{2}$ which is a topological invariant and implies Li-Yorke chaos, similarly as types DC1 and DC2; but…
A sufficient and necessary condition ensuring that the backward shift operator on the Köthe sequence space admits an invariant distributionally $\varepsilon$-scrambled set for some $\varepsilon>0$ is obtained, improving the main results in [F. Mart\'ınez-Giménez, P. Oprocha, A. Peris, J. Math. Anal. Appl., {\bf 351} (2009), 607--615].
Many definitions of chaos have appeared in the last decades and with them the question if they are equivalent in some more specific spaces. Our focus will be distributional chaos, first defined in 1994 and later subdivided into three major types (and even more subtypes). These versions of chaos are equivalent on a closed interval, but distinct in more complicated spaces. Since dendrites have much in common with the interval, we explore whether or not we can distinguish these kinds of chaos already on dendrites. In the end of the paper we will also briefly look at the correlation with other typ…
Mixing, Li-Yorke chaos, distributional chaos and Kato's chaos to multiple mappings
本文从集值视角定义多重映射的传递性与混合性等概念，证明多重映射的混合性蕴含三类混沌，给出Kato混沌的迭代等价性，并构造符号动力系统验证仅两个强非游荡点即可产生分布混沌。
According to the recurrent frequency, many levels of recurrent points are found, such as periodic points, almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points. In this paper, we consider symbolic dynamics and show the existence of six refined levels between Banach recurrence and general recurrence. Despite the fact that these refined levels are all null-measure under any invariant measure, we further show they carry strong topological complexity. Each refined level of recurrent points is dense in the whole space and contains an …
In this article we prove that for a diffeomorphism on a compact Riemannian manifold, if there is a nontrival homoclinic class that is not uniformly hyperbolic or the diffeomorphism is a $C^{1+α}$ and there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then we find a type of strongly distributional chaos which is stronger than usual distributional chaos and Li-Yorke chaos in the set of irregular orbits that are not uniformly hyperbolic. Meanwhile, we prove that various fractal sets are strongly distributional chaotic, such as irregular sets, level sets, several recu…
On Shadowing and Chain Recurrence in Linear Dynamics
本文研究线性动力学中的跟踪与链回归，证明巴拿赫空间上跟踪与有限跟踪等价，刻画多类空间中相关算子的混合性、分布混沌性等，得到加权移位链回归的完整刻画及广义双曲性与周期跟踪的关系。
It follows from Oseledec Multiplicative Ergodic Theorem (or Kingmans Subadditional Ergodic Theorem) that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system f with exponential specification property and a Holder continuous matrix cocycle A, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of A displays distributional chaos of type 1.
Backpropagation in hyperbolic chaos via adjoint shadowing
本文提出伴随跟踪算子 S，将反向传播推广到离散与连续时间双曲混沌，并统一刻画线性响应中的跟踪与不稳定贡献。
The main aim of this article is to show that maps with specification property have invariant distributionally scrambled sets and that this kind of scrambled set can be transferred from factor to extension under finite-to-one factor maps. This solves some open questions in the literature of the topic. We also show how our method can be applied in practice, taking as example Poincaré map of time-periodic nonautonomous planar differential equation $$ \dot{z}=v(t,z)=(1+e^{iκt}\abs{z}^2)\z^3 -N $$ where $κ$ and $N$ are sufficiently small positive real numbers.
This paper proves that shadowing solutions can be almost surely nonphysical. This finding invalidates the argument that small perturbations in a chaotic system can only have a small impact on its statistical behavior. This theoretical finding has implications for many applications in which chaotic mechanics plays an important role. It suggests, for example, that we can control the climate through subtle perturbations. It also suggests that numerical simulations of chaotic dynamics, such as turbulent flows, may fail to predict the true long-term or statistical behavior.
Distributional chaotic generalized shifts
本文刻画有限离散空间乘积上广义移位的分布混沌、稠密分布混沌与传递分布混沌条件，并用反例补充说明。
In discrete dynamical system $(X, f)$ where $X$ is a topological space and $f \in C(X,X)$, three notions of distributional chaos were defined. They were denoted by $DC1, DC2$ and $DC3$. For interval systems such three notions coincide and they will be denoted by DC-chaos. Generally speaking we have $DC1 \subseteq DC2 \subseteq DC3$-chaos. We wonder if it is possible that chaos can concentrate in some points and develop a local idea of the distributional chaos. Answering to this question, in [12] is introduced the new notion of $DCi$-points for $i = 1,2,3$. Such special points are those in whic…
We disprove the conjecture that distributional chaos of type 3 (briefly, DC3) is iteration invariant and show that a slightly strengthened definition, denoted by DC2$\frac{1}{2}$, is preserved under iteration, i.e. $f^n$ is DC2$\frac{1}{2}$ if and only if $f$ is too. Unlike DC3, DC2$\frac{1}{2}$ is also conjugacy invariant and implies Li-Yorke chaos. The definition of DC2$\frac{1}{2}$ is the following: a pair $(x,y)$ is DC2$\frac{1}{2}$ iff $Φ_{(x,y)}(0)<Φ^*_{(x,y)}(0)$, where $Φ_{(x,y)}(δ)$ (resp. $Φ^*_{(x,y)}(δ)$) is lower (resp. upper) density of times $k$ when $d(f^k(x),f^k(y))<δ$ and both…
Positive topological entropy and distributional chaos are characterized for hereditary shifts. A hereditary shift has positive topological entropy if and only if it is DC2-chaotic (or equivalently, DC3-chaotic) if and only if it is not uniquely ergodic. A hereditary shift is DC1-chaotic if and only if it is not proximal (has more than one minimal set). As every spacing shift and every beta shift is hereditary the results apply to those classes of shifts. Two open problems on topological entropy and distributional chaos of spacing shifts from an article of Banks et al. are solved thanks to this…
Genericity of distributional chaos in non-autonomous dynamical systems
本文解决了非自治离散动力系统中关于分布混沌的两个开问题，证明具有正拓扑熵系统的极限函数是否为DC2取决于度量，且成为DC1并非逐点收敛系统的通有性质。
Distributionally chaotic $C_0$-semigroups on complex sectors
该研究针对复扇区指标的巴拿赫空间$C_0$-半群的分布混沌展开，建立分布敏感性与混沌的关联，给出分布不规则向量线性流形存在条件，推导加权$L_p$空间平移半群稠密分布混沌的权条件，并构造出Devaney混沌但非分布混沌的实例。
There are lots of results to study dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure) etc.. However, it is unknown from the viewpoint of chaos. There are lots of results on the relationship of positive topological entropy and various chaos but it is known that positive topological entropy does not imply a strong version of chaos called DC1 so that it is non-trivial to study DC1 on irregular sets and level sets. In t…
This paper establishes some criteria of chaos in non-autonomous discrete systems. Several criteria of strong Li-Yorke chaos are given. Based on these results, some criteria of distributional chaos in a sequence are established. Moreover, several criteria of distributional chaos induced by coupled-expansion for an irreducible transition matrix are obtained. Some of these results not only extend the existing related results for autonomous discrete systems to non-autonomous discrete systems, but also relax the assumptions of the counterparts. One example is provided for illustration.
Distributional and mean Li-Yorke chaos for weighted shifts on Fréchet sequence spaces
本文研究了加权后向移位在一般弗雷歇序列空间中分布性混沌和均值Li-Yorke混沌的特征，并在Köthe序列空间λ_p(A,J)中推导了这两种混沌类型的判据。
Distributional chaos for composition operators on $L^{p}$-spaces
本文研究了 $L^{p}$ 空间上复合算子的分布混沌，给出了 $\varphi$ 的刻画与充分条件，证明了分布不规则向量蕴含稠密分布混沌，并刻画了加权 $\ell^{p}$ 空间上各类移位算子的权序列及单位圆盘自同构诱导复合算子的混沌条件。
Expansivity and Shadowing in Linear Dynamics
本文研究Banach空间上线性算子的扩张性、混沌性质与跟踪性，刻画加权移位并证明存在具有跟踪性的非双曲可逆算子。
Motivated by C*-algebra theory, ultragraph edge shift spaces generalize shifts of finite type to the infinite alphabet case. In this paper we study several notions of chaos for ultragraph shift spaces. More specifically, we show that Li-Yorke, Devaney and distributional chaos are equivalent conditions for ultragraph shift spaces, and characterize this condition in terms of a combinatorial property of the underlying ultragraph. Furthermore, we prove that such properties imply the existence of a compact, perfect set which is distributionally scrambled of type 1 in the ultragraph shift space (a r…
Fast differentiation of hyperbolic chaos
本文提出了一种用于双曲混沌系统快速求导的算法，通过非侵入性阴影算法和归一化二阶切线方程计算参数导数，并在21维例子中验证了其有效性。
Chaos on Peano continua
本研究推广Agronsky和Ceder的结果，证明Peano连续统存在精确Devaney混沌的连续变换，界定含一维连续统、绝对邻域收缩核的Peano连续统类，通过局部小扰动证明该类中混沌系统的稠密性与相关通有性结论。
Generalized Hyperbolicity, Stability and Expansivity for Operators on Locally Convex Spaces
本文研究局部凸空间上算子的广义双曲性、拓扑稳定性与拓扑扩张性，建立其与有限跟踪、严格周期跟踪、混沌及传递性之间的关系，并刻画加权移位情形。
In our another recent article, we introduce a new dynamical property for linear operators called norm-unimodality which implies distributional chaos. In the present paper, we'll give a further discussion of norm-unimodality. It is showed that norm-unimodality is similar invariant and the spectra of norm-unimodal operator is referred to. As an application, in each nest algebra there exist distributional chaotic operators. Moreover, normal operators and compact operators with regard to norm-unimodality and Li-Yorke chaos are also be considered. Specially, a small compact perturbation of the unit…
We consider nonautonomous discrete dynamical systems $\{ f_n\}_{n\ge 1}$, where every $f_n$ is a surjective continuous map $[0,1]\to [0,1]$ such that $f_n$ converges uniformly to a map $f$. We show, among others, that if $f$ is chaotic in the sense of Li and Yorke then the nonautonomous system $ \{ f_n\}_{n\ge 1}$ is Li-Yorke chaotic as well, and that the same is true for distributional chaos. If $f$ has zero topological entropy then the nonautonomous system inherits its infinite $ω$-limit sets.
We deal with a mechanism of generating distributional chaos in planar nonautonomous ODEs and try to measure chaosity in terms of topological entropy. It is based on the interplay between simple periodic solutions. We prove the existence of infinitely many heteroclinic solutions betwen the periodic ones.
We survey the connections between entropy, chaos, and independence in topological dynamics. We present extensions of two classical results placing the following notions in the context of symbolic dynamics: 1. Equivalence of positive entropy and the existence of a large (in terms of asymptotic and Shnirelman densities) set of combinatorial independence for shift spaces. 2. Existence of a mixing shift space with a dense set of periodic points with topological entropy zero and without ergodic measure with full support, nor any distributionally chaotic pair. Our proofs are new and yield conclusion…
Open cover specification property and Chaos
本文研究开覆盖 specification 性质，证明其为拓扑不变量，是经典拓扑 specification 性质的推广，还得出其在正则空间蕴含 Devaney 混沌、在特定空间保证两类混沌的结论，凸显其关联拓扑结构与动力学复杂性的意义。
Chaos for endomorphisms of completely metrizable groups and linear operators on Fréchet spaces
本文利用拓扑动力学技巧统一处理完全可度量化群连续自同态的三种混沌，通过半非正则点存在性进行刻画，并应用于Fréchet空间线性算子改进现有结果。
For a continuous self-map $f$ on a compact interval $I$ and the induced map $\hat f$ on the space $\mathcal{M}(I)$ of probability measures, we obtain a sharp condition to guarantee that $(I,f)$ is transitive if and only if $(\mathcal{M}(I),\hat f)$ is transitive. We also show that the sensitivity of $(I,f)$ is equivalent to that of $(\mathcal{M}(I),\hat f)$. We prove that $(\mathcal{M}(I),\hat f)$ must have infinite topological entropy for any transitive system $(I,f)$, while there exists a transitive non-autonomous system $(I,f_{0,\infty})$ such that $(\mathcal{M}(I),\hat f_{0,\infty})$ has z…
On the Dynamics of Induced Maps on the Space of Probability Measures
该研究考察Cantor空间上通有连续映射与同胚在概率测度空间诱导映射的动力学性质，涵盖Li-Yorke混沌等多类核心概念，并推导了任意紧度量空间下诱导映射的相关结果。
Dynamical properties for composition operators on $H^{2}(\mathbb{C}_{+})$
本文基于线性动力学中扩张性、Li-Yorke混沌与跟踪的刻画，研究了右半平面仿射自映射在 Hardy-Hilbert 空间上诱导的复合算子的这三种动力学性质。
On the Dynamics of Weighted Composition Operators
本文研究了加权复合算子在不同空间上的动力学性质，并通过具体应用展示了其在序列空间和函数空间中的表现。
The Specification Property for $C_0$-Semigroups
本研究将巴拿赫空间算子的 specification 性质推广到 $C_0$-半群，探究其与混合性、Devaney 混沌等动力学性质的关系，并给出包括双曲热方程、Black-Scholes 方程解半群在内的多个满足该性质的实例。
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