On the Conley decomposition of Mather sets

Nonautonomous Conley Index Theory: The Connecting Homomorphism

Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and structure of connecting trajectories of the underlying dynamical system. To give a meaningful generalization of this general principle to nonautonomous problems, the nonautonomous homology Conley index is expressed as a direct limit. Moreover, it is shown that a nontrivial connecting homomorphism implies, on the dynamical systems level, a sort of uniform connectedness of the attractor-repeller decomposition.

Shift Equivalence and the Conley index

In this paper we introduce filtration pairs for isolated invariant sets of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Lastly, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.

Conley 指数理论中无延续性的转移矩阵

Transition Matrix without Continuation in the Conley Index Theory

本文扩展了无 Morse 分解延续性情况下的奇异转移矩阵定义，以从拓扑动力学视角研究伴随 Morse 分解变化的分岔问题。

Conley index and stable sets for flows on flag bundles

Consider a continuous flow of automorphisms of a G-principal bundle which is chain transitive on its compact Hausdorff base. Here G is a connected noncompact semi-simple Lie group with finite center. The finest Morse decomposition of the induced flows on the associated flag bundles were obtained in previous articles. Here we describe the stable sets of these Morse components and, under an additional assumption, their Conley indices.

吸引子-排斥子分解的连接映射的动力学解释

A dynamical interpretation of the connection map of an attractor-repeller decomposition

本文在 Conley index theory 框架下，为吸引子-排斥子分解中正合序列的连接同态 $Γ$ 提供了动力学解释，阐明了排斥子发射波前经连接轨道演化至吸引子并被接收的过程。

The random case of Conley's theorem: III. Random semiflow case and Morse decomposition

In the first part of this paper, we generalize the results of the author \cite{Liu,Liu2} from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose invariant random compact set (e.g. global random attractor) into random Morse sets and connecting orbits between them, which generalizes the Morse decomposition of invariant sets originated from Conley \cite{Con} to the random semiflow setting and gives the pos…

Persistence of Conley-Morse Graphs in Combinatorial Dynamical Systems

Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invaria…

通过更新矩阵分解高效计算Conley-Morse持续同调条形码

Computing Conley-Morse Persistence Barcode Efficiently by Updating Matrix Decompositions

本文针对Conley-Morse持续同调条形码计算成本高的问题，提出用更简单的块取代指数对，通过更新矩阵分解的方法实现高效计算，优化了现有算法。

Covering Action on Conley Theory

In this paper, we apply Conley index theory in a covering space of an invariant set $S$, possibly not isolated, in order to describe the dynamics in $S$. More specifically, we consider the action of the covering translation group in order to define a topological separation of $S$ which distinguish the connections between the Morse sets within a Morse decomposition of $S$. The theory developed herein generalizes the classical connection matrix theory, since one obtains enriched information on the connection maps for non isolated invariant sets, as well as, for isolated invariant sets. Moreover,…

Random Chain Recurrent Sets for Random Dynamical Systems

It is known by the Conley's theorem that the chain recurrent set $CR(ϕ)$ of a deterministic flow $ϕ$ on a compact metric space is the complement of the union of sets $B(A)-A$, where $A$ varies over the collection of attractors and $B(A)$ is the basin of attraction of $A$. It has recently been shown that a similar decomposition result holds for random dynamical systems on noncompact separable complete metric spaces, but under a so-called \emph{absorbing condition}. In the present paper, the authors introduce a notion of random chain recurrent sets for random dynamical systems, and then prove th…

基于组合多向量场的马尔可夫链中Morse分解的持续同调

Persistent homology of Morse decomposition in Markov chains based on combinatorial multivector fields

本文提出一种基于组合多向量场的持续框架，用于马尔可夫链中的Morse分解，通过转移概率定义过滤，构建多向量场并应用Morse分解，得到不变集的层次结构及持续图，以分析有限状态随机过程的复发性和稳定性。

Symplectomorphisms and discrete braid invariants

Area and orientation preserving diffeomorphisms of the standard 2-disc, referred to as symplectomorphisms of $\mathbb{D}^{2}$, allow decompositions in terms of positive twist diffeomorphisms. Using the latter decomposition we utilize the Conley index theory of discrete braid classes as introduced in [Ghrist et al., C. R. Acad. Sci. Paris Sér. I Math., 331(11), 2000, Invent. Math., 152(2), 2003] in order to obtain a Morse type forcing theory of periodic points: a priori information about periodic points determines a mapping class which may force additional periodic points.

Attractors and Attracting Neighborhoods for Multiflows

We already know a great deal about dynamical systems with uniqueness in forward time. Indeed, flows, semiflows, and maps (both invertible and not) have been studied at length. A view that has proven particularly fruitful is topological: consider invariant sets (attractors, repellers, periodic orbits, etc.) as topological objects, and the connecting sets between them form gradient like flows. In the case of systems with uniqueness in forward time, an attractor in one system is related to nearby attractors in a family of other, "close enough" systems. One way of seeing that connection is through…

关于链组件的线性序集

On linearly ordered sets of chain components

本文研究了动力系统中链组件的偏序结构，证明了连续映射下链组件偏序不能线性稠密有序，且区间映射的链组件偏序可为可数well-order。