Huijun Fan, Juergen Jost
The present paper contains an interpretation and generalization of Novikov's theory of Morse type inequalities for 1-forms in terms of Conley's theory for dynamical systems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We present a new approach to Morse and Novikov theories, based on the deRham Federer theory of currents, using the finite volume flow technique of Harvey and Lawson. In the Morse case, we construct a noncompact analogue of the Morse complex, relating a Morse function to the cohomology with compact forward supports of the manifold. This complex is then used in Novikov theory, to obtain a geometric realization of the Novikov Complex as a complex of currents and a new characterization of Novikov Homology as cohomology with compact forward supports. Two natural ``backward-forward'' dualities are a…
We introduce a Milnor metric on the determinant line of the cohomology of the underlying closed manifold with coefficients in a flat vector bundle, by means of interactions between the fixed points and the closed orbits of a Morse-Smale flow. This allows us to generalise the notion of the absolute value at zero point of the Ruelle dynamical zeta function, even in the case where this value is not well defined in the classical sense. We give a formula relating the Milnor metric and the Ray-Singer metric. An essential ingredient of our proof is Bismut-Zhang's Theorem.
Dynamical Morse entropy
本文针对可平铺顺从群作用下的拓扑空间连续函数，定义了同调可检测临界点数量的熵，证明其下界为贝蒂数熵之和，推广了微分几何中的标准莫尔斯不等式。
We employ the relative category to develop relations between the Ważewski pair $(N,E)$ and the Morse decomposition of the maximal invariant set in $\ol{N\sm E}$ for infinite-dimensional dynamical systems. Via these relations, we can detect connecting trajectories between Morse sets and obtain a dynamical-system version of critical point theorem with relative category.
The Morse spectrum for linear random dynamical systems
本文证明线性随机动力系统的射影化存在唯一最细弱 Morse 分解，据此定义 Morse 谱，并证明其为有限个闭区间之且在有界增长条件下等于非均匀二分谱。
Let $f:M \to \mathbb{R}$ be a Morse-Bott function on a compact smooth finite dimensional manifold $M$. The polynomial Morse inequalities and an explicit perturbation of $f$ defined using Morse functions $f_j$ on the critical submanifolds $C_j$ of $f$ show immediately that $MB_t(f) = P_t(M) + (1+t)R(t)$, where $MB_t(f)$ is the Morse-Bott polynomial of $f$ and $P_t(M)$ is the Poincaré polynomial of $M$. We prove that $R(t)$ is a polynomial with nonnegative integer coefficients by showing that the number of gradient flow lines of the perturbation of $f$ between two critical points $p,q \in C_j$ c…
In the stability theory of dynamical systems, Lyapunov functions play a fundamental role. In this paper, we study the attractor-repeller pair decomposition and Morse decomposition for compact metric space in the random setting. In contrast to [8], by introducing slightly stronger definitions of random attractor and repeller, we characterize attractor-repeller pair decompositions and Morse decompositions for random dynamical systems through the existence of Lyapunov functions. These characterizations, we think, deserve to be known widely.
In this paper we first construct smooth Morse-Lyapunov functions of attractors for nonsmooth dynamical systems. Then we prove that all open attractor neighborhoods of an attractor have the same homotopy type. Based on this basic fact we finally introduce the concept of critical group for Morse sets of an attractor and establish Morse inequalities and equations.
In this article, a spectral sequence analysis of a filtered Novikov complex $(\mathcal{N}_{\ast}(f),Δ)$ over $\mathbb{Z}((t))$ is developed with the goal of obtaining results relating the algebraic and dynamical settings. Specifically, the unfolding of a spectral sequence of $(\mathcal{N}_{\ast}(f),Δ)$ and the cancellation of its modules is associated to a one parameter family of circle valued Morse functions on a surface and the dynamical cancellations of its critical points. The data of a spectral sequence computed for $(\mathcal{N}_{\ast}(f),Δ)$ is encoded in a family of matrices $Δ^r$ prod…
The complete invariant for gradient like Morse-Smale dynamical systems (vector fields and diffeomorphisms) on closed 4-manifolds are constructed. It is same as Kirby diagram in a case of polar vector field without fixed points of index 3.
Morse decomposition for semi-dynamical systems with an application to systems of state-dependent delay differential equations
本文提出统一框架，基于离散Lyapunov函数在任意度量空间的半动力系统上构造全局吸引子的Morse分解，推广了已有结论并应用于阈值型状态依赖时滞微分方程循环系统。
We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions, and Conley-Morse graphs of the two dynamical systems are in one-to-one correspondence.
A recent generalization of the Conley index to discrete multivalued dynamical systems without a continuous selector is motivated by applications to data-driven dynamics. In the present paper we continue the program by studying attractor-repeller pairs and Morse decompositions in this setting. In particular, we prove Morse equation and Morse inequalities.
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…
The term "hybrid system" refers to a continuous time dynamical system that undergoes Markovian perturbations at discrete time intervals. In this paper, we find that under the right formulation, a hybrid system can be treated as a dynamical system on a compact space. This allows one to study its limit sets. We examine the Morse decompositions of hybrid systems, find a sufficient condition for the existence of a non-trivial Morse decomposition, and study the Morse sets of such a decomposition. Finally, we consider the case in which the Markovian perturbations are small, showing that trajectories…
A classical problem in dynamical systems is to measure the complexity of a map in terms of their orbits. One of the main tools we have to achieve this goal is entropy. However, many interesting families of dynamical systems have every element with zero-entropy. One of said families are the Morse-Smale diffeomorphisms. The first author and E. Pujals introduced the concept of generalized entropy that extends the classical notion of entropy. In this work, we compute the generalized entropy of Morse-Smale diffeomorphisms on surfaces.
Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields
本文通过计数闭轨道与孤立临界点间的流线，构造广义Morse-Smale系统的Floer型边缘算子，并推广到Forman组合向量场，从而直接恢复光滑与离散结构的$\mathbb{Z}_2$同调。
We establish necessary and sufficient conditions for a dynamical system to be topologically conjugate to the Morse minimal set, the shift orbit closure of the Morse sequence, and conditions for topological conjugacy to the closely related Teoplitz minimal set.
Given a smooth closed manifold M, the Morse-Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in [We-93] and is based on tools from hyperbolic dynamical systems. For instance, we apply the Grobman-Hartman theorem and the Lambda-Lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow …
We show that the images via $z\mapsto z^m$ of the continuous part of the spectral measures of the dynamical systems generated by the 0-1 sequences of the Thue-Morse type are pairwise mutually singular for different odd numbers $m\in\N$. Sarnak's conjecture on orthogonality with the Möbius function is shown to hold for such dynamical systems. The same conjecture is shown to hold for all systems induced by regular Toeplitz sequences. A non-regular Toeplitz sequence for which Sarnak's conjecture fails is constructed.
Coarse chain recurrence, Morse graphs with finite errors, and persistence of circulations
本文提出了一种参数化的粗链复发框架，用于研究动力系统在有限扰动下的持续性及循环行为，通过粗莫尔斯图和过滤器揭示了系统在扰动消失时的奇异极限行为。
Call a group action on a topological space \emph{biminimal} if for any points $x,y\in X$ there exists a group element taking $x$ arbitrarily close to $y$ and whose inverse takes $y$ arbitrarily close to $x$. A symbolic encoding of the Thue-Morse dynamical system is given, in terms of $ω$-automata. It is used to prove that the Thue-Morse dynamical system is minimal but not biminimal. The $ω$-automata also establish a link between Nekrashevych's presentation of limit spaces and solenoids with a construction described by Vershik and Solomyak.
On the Novikov problem with a large number of quasiperiods and its generalizations
本文研究准周期数不受限的平面Novikov问题，分析大尺寸开放与闭合水平线的产生，并将结果推广到任意维空间的广义Novikov问题，为含准周期调制的系统研究提供支撑。
In this mainly expository paper we present a detailed proof of several results contained in a paper by M. Bertelson and M. Gromov on Dynamical Morse Entropy. This is an introduction to the ideas presented in that work. Suppose $M$ is compact oriented connected $C^\infty$ manifold of finite dimension. Assume that $f_0 :M \to [0,1]$ is a surjective Morse function. For a given natural number $n$, consider the set $M^n$ and for $x=(x_0,x_1,...,x_{n-1}) \in M^n$, denote $ f_n (x) = \frac{1}{n} \, \sum_{j=0}^{n-1} f_0 (x_j).$ The Dynamical Morse Entropy describes for a fixed interval $I\subset [0,1]…
We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs $(N,L)$, established in [2,3], as a dynamical thickening of the stable manifold. As a first application and to illustrate efficiency of the concept we reprove a fundamental theorem of classical Morse theory, Milnor's homotopical cell attachment theorem [1]. Dynamical thickening leads to a conceptually simple and short proof.
Conley-Morse persistence barcode: a homological signature of combinatorial bifurcations
本文提出Conley-Morse持续条形码，用于描述组合分岔的同调特征，基于拓扑持续性理论，利用Conley指数构建持续模块，并通过gentle代数证明其分解为简单区间。
We analyze the chain recurrent set of skew product semiflows obtained from nonautonomous differential equations -- ordinary differential equations or semilinear parabolic differential equations. For many gradient-like dynamical systems, Morse-Smale dynamical systems e.g., the chain recurrent set contains only isolated equilibria. The structure in the asymptotically autonomous setting is richer but still close to to the structure of a Morse-Smale dynamical system. The main tool used in this paper is a nonautonomous flavour of Conley index theory developed by the author. We will see that for a c…
The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other trajectories of the system. If there is a way to choose this pair so that the space orbits in its complement (the characteristic space of orbits) is connected, this creates prerequisites for finding complete topological invariants of the dynamical system. It is known that such a pair always exists for arbitrary Morse-Smale diffeomorphisms given on any manif…
At the beggining of the 80's, H.Masur and W.Veech started the study of generic properties of interval exchange transformations proving that almost every such transformation is uniquely ergodic. About the same time, S.Novikov's school and French mathematicians independently discovered very intriguing phenomena for classes of measured foliations on surfaces and respective IETs. For instance, minimality is exceptional in these families. A precise version of this statement is a conjecture by Novikov. The French and Russian constructions are very different ones. Nevertheless, in the most simple sit…
Primitive constant length substitutions generate minimal symbolic dynamical systems. In this article we present an algorithm which can produce the list of injective substitutions of the same length that generate topologically conjugate systems. We show that each conjugacy class contains infinitely substitutions which are not injective. As examples, the Toeplitz conjugacy class contains three injective substitutions (two on two symbols and one on three symbols), and the length two Thue-Morse conjugacy class contains twelve substitutions, among which are two on six symbols. Together, they consti…
On global Lyapunov functions being Morse
本文研究全局渐近稳定系统是否存在Morse Lyapunov函数，证明双曲平衡点情形成立，但一般情形与四维庞加莱猜想相关，并给出反例否定一般性。
We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics directly from the sample. We study the homological persistence of {\em Morse decompositions} of such systems, an important descriptor of the dynamics, as a tool for validating the reconstruction. Our framework can be viewed as a step toward extending the classical persistence theory to "vector cloud" data. We present experimental results on two numerical examp…
Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words
本文研究 Fibonacci 词中固定公差最长单色等差数列的长度与位置，结合圆周旋转动力学与 Walnut 自动定理证明给出长度公式，并推广到 Thue-Morse 和 Rudin-Shapiro 词。
WE trace the beginning of symbolic dynamics--the study of the shift dynamical system--as it arose from the use of coding to study recurrence and transitivity properties of geodesics. The normal citations for the first appearance of symbolic dynamics, Hadamard's 1898 paper and the 1938 and 1940 papers of Morse and Hedlund, don't truly represent the abstract point of view associated with the subject today. Based in part on the evidence of a 1941 letter from Hedlund to Morse, we place the beginning of symbolic dynamics in a 1944 paper by Hedlund.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。