A topological characterization of omega-limit sets on dynamical systems

跟踪性、传递性与 ω-混沌的一种变体

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。

关于非扩张动力系统ω极限集的一个注记

A remark on omega limit sets for non-expansive dynamics

本文研究非扩张动力系统的ω极限集如何依赖于范数选择，证明在严格凸范数压缩条件下，每条轨道的ω极限集可归结为线性时不变系统的ω极限集。

Polycycle omega-limit sets of flows on the compact Riemann surfaces and Eulerian path

Let $(S,Φ)$ be a pair of a closed oriented surface and $Φ$ be a real analytic flow with finitely many singularities. Let $x$ be a point of $S$ with the polycycle $ω$-limit set $ω(x)$. In this paper we give topological classification of $ω(x)$. Our main theorem says that $ω(x)$ is diffeomorphic to the boundary of a cactus in the $2$-sphere $S^{2}$. Moreover $S$ is a connected sum of the above $S^{2}$ and a closed oriented surface along finitely many embedded circles which are disjoint from $ω(x)$. This gives a natural generalization to the higher genus of the main result of \cite{JL} for the ge…

On the omega-limit sets of tent maps

For a continuous map f on a compact metric space (X,d), a subset D of X is internally chain transitive if for every x and y in D and every delta > 0 there is a sequence of points {x=x_0,x_1, ...,x_n=y} such that d(f(x_i),x_{i+1}) < delta for i=0,1, ...,n-1. It is known that every omega-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed subset D of X is internally chain transitive if and only if D is an omega-limit set for some point in X, and that the same is also true for the tent map with slope equal to 2. In this paper, we prov…

On omega limiting sets of infinite dimensional Volterra operators

In the present paper, we are aiming to study limiting behavior of infinite dimensional Volterra operators. We introduce two classes $\tilde {\mathcal{V}}^+$ and $\tilde{\mathcal{V}}^-$of infinite dimensional Volterra operators. For operators taken from the introduced classes we study their omega limiting sets $ω_V$ and $ω_V^{(w)}$ with respect to $\ell^1$-norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to $\tilde {\mathca…

Random fields on model sets with localized dependency and their diffraction

For a random field on a general discrete set, we introduce a condition that the range of the correlation from each site is within a predefined compact set D. For such a random field omega defined on the model set Lambda that satisfies a natural geometric condition, we develop a method to calculate the diffraction measure of the random field. The method partitions the random field into a finite number of random fields, each being independent and admitting the law of large numbers. The diffraction measure of omega consists almost surely of a pure-point component and an absolutely continuous comp…

Invariant measures of minimal post-critical sets of logistic maps

We construct logistic maps whose restriction to the omega-limit set of its critical point is a minimal Cantor system having a prescribed number of distinct ergodic and invariant probability measures. In fact, we show that every metrizable Choquet simplex whose set of extreme points is compact and totally disconnected can be realized as the set of invariant probability measures of a minimal Cantor system corresponding to the restriction of a logistic map to the omega-limit set of its critical point. Furthermore, we show that such a logistic map $f$ can be taken so that each such invariant measu…

A cooperative system which does not satisfy the limit set dichotomy

The fundamental property of strongly monotone systems, and strongly cooperative systems in particular, is the limit set dichotomy due to Hirsch: if x(0) < y(0), then either omega(x) < omega(y), or omega(x) = omega(y) and both sets consist of equilibria. We provide here a counterexample showing that this property need not hold for (non-strongly) cooperative systems.

Topological properties of Lorenz maps derived from unimodal maps

A symmetric Lorenz map is obtain by ``flipping'' one of the two branches of a symmetric unimodal map. We use this to derive a Sharkovsky-like theorem for symmetric Lorenz maps, and also to find cases where the unimodal map restricted to the critical omega-limit set is conjugate to a Sturmian shift. This has connections with properties of unimodal inverse limit spaces embedded as attractors of some planar homeomorphisms.

Asymptotic compactness in topological spaces

The omega limit sets plays a fundamental role to construct global attractors for topological semi-dynamical systems with continuous time or discrete time. Therefore, it is important to know when omega limit sets become nonempty compact sets. The purpose of this paper is to understand the mechanism under which a given net of subsets of topological spaces is compact in the asymptotic sense. For this purpose, we introduce the notion of asymptotic compactness for nets of subsets and study the connection with the compactness of the limit sets. In this paper, for a given net of nonempty subsets, we …

On the topological entropy of saturated sets for amenable group actions

Let $(X,ρ,G)$ be a $G-$action topological system, where $G$ is a countable infinite discrete amenable group and $X$ a compact metric space. We prove a variational principle for topological entropy of saturated sets for systems which have specification and uniform separation properties. As an application, we compute the topological entropy of level sets and irregular sets.

Minimal hyperspace actions of homeomorphism groups of h-homogeneous spaces

Let X be a h-homogeneous zero-dimensional compact Hausdorff space, i.e. X is a Stone dual of a homogeneous Boolean algebra. Using the dual Ramsey theorem and a detailed combinatorial analysis of what we call stable collections of subsets of a finite set, we obtain a complete list of the minimal sub-systems of the compact dynamical system (Exp(Exp(X)),Homeo(X)), where Exp(X) stands for the hyperspace comprising the closed subsets of X equipped with the Vietoris topology. The importance of this dynamical system stems from Uspenskij's characterization of the universal ambit of G = Homeo(X). The r…

具有多个极限集的系统的浸入性质及其对学习 Koopman 嵌入的启示

Properties of Immersions for Systems with Multiple Limit Sets with Implications to Learning Koopman Embeddings

针对含多个 ω-极限集的非线性系统，本文证明连续线性浸入必然坍缩极限集而非一对一，并指出数据驱动的近似 Koopman 嵌入在样本增多、采样变密时同样如此。