The Classification of Three-Dimensional Gradient Like Morse-Smale Dynamic Systems

Classes d'homotopie de champs de vecteurs Morse-Smale sans singularité sur les fibrés de Seifert

In the first part of this paper, we consider smooth maps from a compact orientable 3-manifold without boundary to the 2-sphere. We give a geometric criterion to decide whether two given maps are homotopic, based on the sets of points where the maps are equal or antipodal. We extend this criterion to non-singular vector fields (and co-oriented plane fields) on 3-manifolds. In the second part, we make use of this criterion to study non-singular Morse-Smale vector fields on Seifert manifolds, giving a simple proof of a result of Yano.

Horizon saddle connections and Morse-Smale dynamics of dilation surfaces

Dilation surfaces are generalizations of translation surfaces where the transition maps of the atlas are translations and homotheties with a positive ratio. In contrast with translation surfaces, the directional flow on dilation surfaces may contain trajectories accumulating on a limit cycle. Such a limit cycle is called hyperbolic because it induces a nontrivial homothety. It has been conjectured that a dilation surface with no actual hyperbolic closed geodesic is in fact a translation surface. Assuming that a dilation surface contains a horizon saddle connection, we prove that the directions…

Morse-Smale surfaced diffeomorphisms with orientable heteroclinic

In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable heteroclinic orbits.

Morse-Smale systems and horseshoes for three dimensional singular flows

We prove that for every three-dimensional vector field, either it can be accumulated by Morse-Smale ones, or it can be accumulated by ones with a transverse homoclinic intersection of some hyperbolic periodic orbit in the $C^1$ topology.

Depth $0$ Nonsingular Morse Smale flows on $S^3$

In this paper, we first develope the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits (abbreviated as depth $0$ NMS flows). It mainly contains the following several directions: \begin{enumerate} \item we use WLG to list depth $0$ NMS flows on $S^3$; \item with th…

Supporting manifolds for high-dimensional Morse-Smale diffeomorphisms with few saddles

We describe the topological structure of closed manifolds of dimension no less than four which admit Morse-Smale diffeomorphisms such that its non-wandering set contains any number of sink periodic points, and any number of source periodic points, and few saddle periodic points.

Complete invariant for two-dimentional Morse-Smale diffeomorphisms

We consider general Morse-Smale diffeomorphisms on a closed orientable two-dimentional surface. In this paper it is proved that the complete topological invariant of Morse-Smale diffeomorphisms is finite, the algorithm of the construction of the complete topological invariant in explicit form is given and necessary and sufficient conditions of topological conjugacy of Morse-Smale diffeomorphisms is obtained.

Morse-Bott homology

We give a new proof of the Morse Homology Theorem by constructing a chain complex associated to a Morse-Bott-Smale function that reduces to the Morse-Smale-Witten chain complex when the function is Morse-Smale and to the chain complex of smooth singular $N$-cube chains when the function is constant. We show that the homology of the chain complex is independent of the Morse-Bott-Smale function by using compactified moduli spaces of time dependent gradient flow lines to prove a Floer-type continuation theorem.

On Embedding of Multidimensional Morse-Smale Diffeomorphisms in Topological Flows

J.~Palis found necessary conditions for a Morse-Smale diffeomorphism on a closed $n$-dimensional manifold $M^n$ to embed into a topological flow and proved that these conditions are also sufficient for $n=2$. For the case $n=3$ a possibility of wild embedding of closures of separatrices of saddles is an additional obstacle for Morse-Smale cascades to embed into topological flows. In this paper we show that there are no such obstructions for Morse-Smale diffeomorphisms without heteroclinic intersection given on the sphere $S^n, \,n\geq 4$, and Palis's conditions again are sufficient for such di…

Topological classification of Morse-Smale flows on 3-manifolds

We construct a topological invariant for a Morse-Smale flow on a 3-manifold and prove that the flows are topologically equivalent iff their invariants are same.

Polynomial entropy of Morse-Smale diffeomorphisms on surfaces

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.

Stable foliations and CW-structure induced by a Morse-Smale gradient-like flow

We prove that a Morse-Smale gradient-like flow on a closed manifold has a "system of compatible invariant stable foliations" that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse-Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse-Smale gradient-like flow on a closed manifold $M$ are the open cells of a $CW$-decomposition …

S¹×S²上非奇异Morse-Smale流的李雅普诺夫图

Lyapunov graphs of Non-singular Morse-Smale flows on $S^1 \times S^2$

本文在B. Yu的研究基础上，以过滤邻域为局部数据，刻画S¹×S²上非奇异Morse-Smale流的李雅普诺夫图，并明确了可实现为该类图的有向图类型。

Spectral analysis of morse-smale gradient flows

On a smooth, compact and oriented manifold without boundary, we give a complete description of the correlation function of a Morse-Smale gradient flow satisfying a certain nonresonance assumption. This is done by analyzing precisely the spectrum of the generator of such a flow acting on certain anisotropic spaces of currents. In particular, we prove that this dynamical spectrum is given by linear combinations with integer coefficients of the Lyapunov exponents at the critical points of the Morse function. Via this spectral analysis and in analogy with Hodge-de Rham theory, we give an interpret…

Periods of Morse--Smale diffeomorphisms on $\mathbb{S}^n$, $\mathbb{S}^m \times \mathbb{S}^n$, $\mathbb{C}P^n}$ and $\mathbb{H}P^n}$

We study the set of periods of the Morse--Smale diffeomorphisms on the $n$-dimensional sphere $\mathbb{S}^n$, on products of two spheres of arbitrary dimension $\mathbb{S}^m \times \mathbb{S}^n$ with $m \neq n$, on the $n$-dimensional complex projective space $\mathbb{C}\text{\textbf{P}}^n}$ and on the $n$-dimensional quaternion projective space $\mathbb{H}\text{\textbf{P}}^n}$. We classify the minimal sets of Lefschetz periods for such Morse--Smale diffeomorphisms. This characterization is done using the induced maps on the homology. The main tool used is the Lefschetz zeta function.

Fat handles and phase portraits of Non Singular Morse-Smale flows on S^3 with unknotted saddle orbits

In this paper we build Non-singular Morse-Smale flows on S^3 with unknotted and unlinked saddle orbits by identifying fat round handles along their boundaries. This way of building the flows enables to get their phase portraits. We also show that the presence of heteroclinic trajectories imposes an order in the round handle decomposition of these flows; this order is total for NMS flows composed of one repulsive, one attractive and n unknotted saddle orbits, for n >1.

图流形上非奇异 Morse-Smale 流的指标链环

The indexed links of Non-singular Morse-Smale flows on graph manifolds

本文研究图流形上非奇异 Morse-Smale 流的闭轨道指标链环，证明其可由含奇异与特定正则 Seifert 纤维的特殊链环经有限操作构造，从而完成分类。

Morse-Smale systems without heteroclinic submanifolds on codimension one separatrices

We study a topological structure of a closed $n$-manifold $M^n$ ($n\geq 3$) which admits a Morse-Smale diffeomorphism such that codimension one separatrices of saddles periodic points have no heteroclinic intersections different from heteroclinic points. Also we consider gradient like flow on $M^n$ such that codimension one separatices of saddle singularities have no intersection at all. We show that $M^n$ is either an $n$-sphere $S^n$, or the connected sum of a finite number of copies of $S^{n-1}\otimes S^1$ and a finite number of special manifolds $N^n_i$ admitting polar Morse-Smale systems.…

Cascades and perturbed Morse-Bott functions

Let $f:M \rightarrow \mathbb{R}$ be a Morse-Bott function on a finite dimensional closed smooth manifold $M$. Choosing an appropriate Riemannian metric on $M$ and Morse-Smale functions $f_j:C_j \rightarrow \mathbb{R}$ on the critical submanifolds $C_j$, one can construct a Morse chain complex whose boundary operator is defined by counting cascades \cite{FraTheA}. Similar data, which also includes a parameter $ε> 0$ that scales the Morse-Smale functions $f_j$, can be used to define an explicit perturbation of the Morse-Bott function $f$ to a Morse-Smale function $h_ε:M \rightarrow \mathbb{R}$ \…

Topological classification of Morse-Smale diffeomorphisms on 3-manifolds

Topological classification of even the simplest Morse-Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possible "wild" behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may be not only a point as in the two-dimensional case, but also a curve, compact or non-c…

Morse-Smale向量场的规范链复形

Canonical chain complexes for Morse-Smale vector fields

本文提出Morse-Smale向量场的规范链复形，通过Čech同调谱序列技术，证明其在拓扑等价下不变，并代数推导Morse不等式。

On Topological Classification of Morse-Smale Diffeomorphisms on the Sphere $S^n$

We consider a class $G(S^n)$ of orientation preserving Morse-Smale diffeomorphisms of the sphere $S^{n}$ of dimension $n>3$ in assumption that invariant manifolds of different saddle periodic points have no intersection. We put in a correspondence for every diffeomorphism $f\in G(S^n)$ a colored graph $Γ_f$ enriched by an automorphism $P_f$. Then we define the notion of isomorphism between two colored graphs and prove that two diffeomorphisms $f, f'\in G(S^n)$ are topologically conjugated iff the graphs $Γ_f$, $Γ_f'$ are isomorphic. Moreover we establish the existence of a linear-time algorith…

A complete topological classification of Morse-Smale diffeomorphisms on surfaces: a kind of kneading theory in dimension two

In this paper we give a complete topological classification of orientation preserving Morse-Smale diffeomorphisms on orientable closed surfaces. For MS diffeomorphisms with relatively simple behaviour it was known that such a classification can be given through a directed graph, a three-colour directed graph or by a certain topological object, called a scheme. Here we will assign to any MS surface diffeomorphism a finite amount of data which completely determines its topological conjugacy class. Moreover, we show that associated to any abstract version of this data, there exists a unique conju…

具有相同不稳定流形维数鞍点的Morse-Smale三维微分同胚

Morse-Smale 3-diffeomorphisms with saddles of the same unstable manifold dimension

本文研究闭三维流形上所有鞍点具有相同不稳定流形维数的Morse-Smale微分同胚，推广Reeb的结果，证明此类微分同胚仅能在三维球面上定义。

Knot as a complete invariant of a Morse-Smale 3-diffeomorphism with four fixed points

Lens spaces are the only 3-manifolds that admit gradient-like flows with four fixed points. This is an immediate corollary of Morse inequality and of the Morse function with four critical points existence. A similar question for gradient-like diffeomorphisms is open. Solution can be approached by describing a complete topological conjugacy invariant of the class of considered diffeomorphisms and constructing of representative diffeomorphism for every conjugacy class by the abstract invariant. Ch. Bonnati and V. Z. Grines proved that the topological conjugacy class of Morse-Smale flows with uni…

Characteristic space of orbits of Morse-Smale diffeomorphisms on surfaces

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…

Morse-Smale flow, Milnor metric, and dynamical zeta function

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.

A Morse complex for Axiom A flows

On a smooth compact Riemannian manifold without boundary, we construct a finite dimensional cohomological complex of currents that are invariant by an Axiom A flow verifying Smale's transversality assumptions. The cohomology of that complex is isomorphic to the De Rham cohomology via certain spectral projectors. This construction is achieved by defining anisotropic Sobolev spaces adapted to the global dynamics of Axiom A flows. In the particular case of Morse-Smale gradient flows, this complex coincides with the classical Morse complex.

Shadowing, topological entropy and recurrence of induced Morse-Smale diffeomorphisms

Let $f : M \rightarrow M$ be a Morse-Smale diffeomorphism defined on a compact and connected manifold without boundary. Let $C(M)$ denote the hyperspace of all subcontinua of M endowed with the Hausdorff metric and $C(f) : C(M) \rightarrow C(M)$ denote the induced homeomorphism of $f$. We show in this paper that if $M$ is the unit circle $S^1$ then the induced map $C(f)$ has not the shadowing property. Also we show that the topological entropy of $C(f)$ has only two possible values: $0$ or $\infty$. In particular, we show that the entropy of $C(f)$ is $0$ when $M$ is the unit circle $S^1$ and …

A Morse-Smale index theorem for indefinite elliptic systems and bifurcation

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. This extends recent results of the authors for scalar equations.

关于容许具有四个两两不同 Morse 指标不动点的 Morse-Smale 微分同胚的 3-流形拓扑

On the topology of 3-manifolds admitting Morse-Smale diffeomorphisms with four fixed points of pairwise different Morse indices

本文研究含四个异 Morse 指标不动点的定向 Morse-Smale 3-流形，证明任意此类微分同胚的支撑流形均为透镜空间，并在所有透镜空间上构造野性分界线例子。

离散惠更斯网络中的莫尔斯-斯梅尔动力学

Morse-Smale dynamics in discrete Huygens networks

本文研究离散惠更斯网络的相位振子动力学，推导映射z↦z+hK sin z的莫尔斯-斯梅尔准则，构造含非横截异宿交点的树，拓展了同步网络的动力学分析。

Teichmüller dynamics, dilation tori and piecewise affine circle homeomorphisms

We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic piecewise affine circle homeomorphism with two break points -with respect to the Lebesgue measure- is Morse-Smale.

Floer同调：从广义Morse-Smale动力系统到Forman组合向量场

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。

自然数的任意有限集合均可实现为 Morse–Smale 微分同胚的代数周期

Every finite set of natural numbers is realizable as algebraic periods of a Morse$\unicode{x2013}$Smale diffeomorphism

本文解决了代数周期实现问题，证明了自然数的任意有限集合均可作为某可定向或不可定向曲面上 Morse–Smale 微分同胚的代数周期，并给出了反转定向情形的限制及映射类共轭类数目的指数估计。

Generic Morse-Smale property for the parabolic equation on the circle

In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showi…

Morse-Conley-Floer Homology

For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse functions. A similar approach can be used to define homological invariants for isolated invariant sets of flows on a smooth manifold, which gives an analogue of the Conley index and the Morse-Conley relations. The approach will be referred to as Morse-Conley-Floer homology.

圆周上时间周期标量反应扩散方程的Morse-Smale性质

The Morse Smale property for time-periodic scalar reaction-diffusion equation on the circle

本文研究圆周上时间周期标量半线性抛物方程的Morse-Smale性质，证明其Poincaré映射双曲不动点无同宿连接、异宿不动点稳定与不稳定流形横截相交，且在所有ω-极限集双曲时方程具备Morse-Smale性质。

由Morse梯度系统诱导的连续体超空间的影子性质

Shadowing property on hyperspace of continua induced by Morse gradient system

本文研究了由Morse梯度系统诱导的连续体超空间的影子性质，证明了此类映射不具有影子性质。

Nonsingular Morse-Smale flows of n-manifolds with attractor-repeller dynamics

In the present paper the exhaustive topological classification of nonsingular Morse-Smale flows of $n$-manifolds with two limit cycles is presented. Hyperbolicity of periodic orbits implies that among them one is attracting and another is repelling. Due to Poincare-Hopf theorem Euler characteristic of closed manifold $M^n$ which admits the considered flows is equal to zero. Only torus and Klein bottle can be ambient manifolds for such flows in case of $n=2$. Authors established that there exist exactly two classes of topological equivalence of such flows of torus and three of the Klein bottle.…

A topological classification of convex bodies

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from…

具有球锥奇点的伪流形上的奇异Morse-Smale流：Conley理论与交合同调

Singular Morse-Smale Flows on Pseudomanifolds with Spherical-Cone Singularities: Conley Theory and Intersection Homology

将Morse-Conley理论推广到具有球锥奇点的伪流形，通过引入奇异Morse-Smale流，建立了奇点Conley指数的局部动力学公式，证明了全局Lyapunov函数的存在性，并利用磨光化方法关联奇异与光滑情形，最后通过动力系统方法计算交合同调。

Morse-Smale同步微分同胚的Lyapunov函数

Lyapunov functions for Morse-Smale synchronisation diffeomorphisms

针对三对称耦合振荡器的相位差系统，构造离散Lyapunov函数证明T^2上两个Huygens同步不动点渐近稳定，且其吸引域并集闭包覆盖全环面。

Homotopical Dynamics II: Hopf invariants, smoothings and the Morse complex

The ambient framed bordism class of the connecting manifold of two consecutive critical points of a Morse-Smale function is estimated by means of a certain Hopf invariant. Applications include new examples of non-smoothable Poincare duality spaces as well as an extension of the Morse complex.

Equilibrium-Like Solutions of Asymptotically Autonomous Differential Equations

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…

On the ergodic theory of certain non-discrete actions and topological orbit equivalences

Quasi-invariant measures for non-discrete groups of diffeomorphisms containing a Morse-Smale dynamics are studied. The assumption concerning the presence of a Morse-Smale dynamics allows us to extend to higher dimensions a number of recently established results for non-discrete groups acting on the circle. These results are also applied to show that, for many groups as above, every continuous orbit equivalence must coincide almost everywhere with a diffeomorphism of the corresponding manifold.

瞬态反叛在库莫托振子中的研究：莫尔斯-塞male结构稳定性与有限2-移型连接图

Transient rebellions in the Kuramoto oscillator: Morse-Smale structural stability and connection graphs of finite 2-shift type

本文研究了库莫托振子模型中瞬态反叛现象，揭示其结构稳定性及有限2-移型连接图特性。

Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems

We prove a conjecture of J. Palis: any diffeomorphism of a compact manifold can be C1-approximated by a Morse-Smale diffeomorphism or by a diffeomorphism having a transverse homoclinic intersection. ----- Cr'eation d'intersection homoclines : un mod`ele pour la dynamique centrale des syst`emes partiellement hyperboliques. Nous montrons une conjecture de J. Palis : tout diff'eomorphisme d'une vari'et'e compacte peut ^etre approch'e en topologie C1 par un diff'eomorphisme Morse-Smale ou par un diff'eomorphisme ayant une intersection homocline transverse.

On the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with compact center leaves

In this paper, we study the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms defined on $3-$torus with compact center leaves. Assuming the existence of a periodic leaf with Morse-Smale dynamics we prove a sharp upper bound for the number of maximal measures in terms of the number of sources and sinks of Morse-Smale dynamics. A well-known class of examples for which our results apply are the so-called Kan-type diffeomorphisms admitting physical measures with intermingled basins.

MVF Designer: Design and Visualization of Morse Vector Fields

Vector field design on surfaces was originally motivated by applications in graphics such as texture synthesis and rendering. In this paper, we consider the idea of vector field design with a new motivation from computational topology. We are interested in designing and visualizing vector fields to aid the study of Morse functions, Morse vector fields, and Morse-Smale complexes. To achieve such a goal, we present MVF Designer, a new interactive design system that provides fine-grained control over vector field geometry, enables the editing of vector field topology, and supports a design proces…

不变Thom-Smale复形的解析与拓扑实现

Analytic and topological realizations of the invariant Thom-Smale complex

在连通紧李群G的光滑作用下，借助G不变Witten瞬子复形解析实现G不变Thom-Smale复形，二者关联闭定向G流形上的特定Morse-Bott函数，纳入临界集水平方向影响并推广严格Morse情形。

Dynamics of the Morse Oscillator: Analytical Expressions for Trajectories, Action-Angle Variables, and Chaotic Dynamics

We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the "Morse oscillator"). We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for the bounded trajectories, and action-angle variables. We use these trajectories to prove sufficient conditions for chaotic dynamics, in the sense of Smale horseshoes, for the time-periodically perturbed Morse oscillator using a Melnikov type approach.

Topological structure of optimal flows on the Girl's surface

We investigate the topological structure of flows on the Girl's surfaces which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of the selfintersection. First, we describe the cellular structure of the Boy's and Girl's surfaces and prove that there are unique images of the project plane in the form of a 2-disc, in which the opposite points of the boundary are identified and this boundary belongs to the preimage of the 1-skeleton of the surface. Second, we described three structures of flows with one fixed point and no separatrix on the …

Quotient spaces and topological invariants of flows

We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces, to describe both gradient dynamics and recurrent dynamics. In particular, the abstract weak orbit spaces of flows on topological spaces are generalizations of both Morse graphs of flows on compact metric spaces and Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces. Moreover, we show that the abstract weak orbit spaces are complete and finite for several kinds of flows on manifolds, and we state several examples whose Morse graphs are singleton…