S. Aranson, E. Zhuzhoma
In 1946, M. Morse proposed a conjecture that an analytic topologically transitive systems is metrically transitive. We prove this Morse conjecture for flows on a closed orientable surface of negative Euler characteristic. As a consequence, the Morse conjecture is true for highly transitive flows on non-orientable surfaces.
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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.
It is known that $C^r$ Morse-Smale vector fields form an open dense subset in the space of vector fields on orientable closed surfaces and are structurally stable for any $r \in \mathbb{Z}_{>0}$. In particular, $C^r$ Morse vector fields (i.e. Morse-Smale vector fields without limit cycles) form an open dense subset in the space of $C^r$ gradient vector fields on orientable closed surfaces and are structurally stable. Therefore generic time evaluations of gradient flows on orientable closed surfaces (e.g. solutions of differential equations) are described by alternating sequences of Morse flows…
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.…
Non-singular flows with twisted saddle orbit on orientable 3-manifolds
本文分类了闭可定向3-流形上仅含一个扭转鞍形轨道的非奇异Morse-Smale流，确定了容许流形类型并计算了等价类数量。
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…
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。
For a closed oriented 3-manifold $Y$ we define $n(Y)$ to be the minimal non-negative number such that in each homotopy class of non-singular vector fields of $Y$ there is a Morse-Smale vector field with less or equal to $n(Y)$ periodic orbits. We combine the construction process of Morse-Smale flows given in [2] with handle decompositions of compact orientable surfaces to provide an upper bound to the number $n(Y)$ for oriented Seifert manifolds and oriented graph manifolds prime to $\stwo\times\sone$.
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…
Classification of NMS-flows with unique twisted saddle orbit on orientable 4-manifolds
本文研究闭可定向4-流形上带唯一扭鞍点轨道的NMS流分类，证明仅$\u2118^3\times\u2118^1$容许这类流，且其恰好分为八个等价类并给出代表流构造，完善了NMS流拓扑等价分类研究。
Gradient Flow in Morse and Analytic Settings
本报告研究 Morse 型与解析函数的梯度流，聚焦其收敛性、极限速率与方向，并包含若干自研小结果。
Enumeration of one-cycle signed chord diagrams and their duals for optimal Morse flows on closed surfaces
该研究将闭曲面上最优Morse流的分类构造算法化，提出枚举单圈带符号弦图及对偶的算法，经多类数据验证，可分组等价类并识别自对偶类。
Let $Ω$ in $R^M$ be a compact connected $M$-dimensional real analytic domain with boundary and $ϕ$ be a primal navigation function; i.e. a real analytic Morse function on $Ω$ with a unique minimum and with minus gradient vector field $G$ of $ϕ$ on the boundary of $Ω$ pointed inwards along each coordinate. Related to a robotics problem, we define a sequential hybrid process on $Ω$ for $G$ starting from any initial point $q_0$ in the interior of $Ω$ as follows: at each step we restrict ourselves to an affine subspace where a collection of coordinates are fixed and allow the other coordinates cha…
By a gradient-like flow on a closed orientable surface $M$, we mean a closed 1-form $β$ defined on $M$ punctured at a finite set of points (sources and sinks of $β$) such that there exists a Morse function $f$ on $M$, called an energy function of $β$, whose critical points coincide with equilibria of $β$, and the pair $(f,β)$ has a canonical form near each critical point of $f$. Let $\mathcal{B}=\mathcal{B}(β_0)$ be the space of all gradient-like flows on $M$ having the same types of local singularities as a flow $β_0$, and $\mathcal{F}=\mathcal{F}(f_0)$ the space of all Morse functions on $M$…
We elaborate on an idea of M. Abouzaid of equipping the Morse cochain complex of a smooth Morse function on a closed oriented manifold with the structure of an $A_\infty$-algebra. This is a variation on K. Fukaya's definition of Morse-$A_\infty$-categories for closed oriented manifolds involving families of Morse functions. The purpose of this article is to provide a coherent and detailed treatment of Abouzaid's approach including a discussion of all relevant analytic notions and results.
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。
Deformational symmetries of smooth functions on non-orientable surfaces
本文研究非可定向曲面（Möbius带等）上光滑函数的形变对称性，计算稳定子边界固定子群的π₀，进而给出所有非可定向曲面（除Klein瓶和射影平面）轨道基本群的显式代数描述。
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