Round Handles and Non-Singular Morse-Smale Flows

Asimov, Daniel

doi:10.2307/1970972

Cited by 112 publications

(78 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The notion of the round handle decomposition was introduced by Asimov [1] and modified by Morgan [6]; it establish a correspondence between NMS flows and round handle decompositions. , and an embedding ϕ : (∂D…”

Section: Round Handle Decompositionsmentioning

confidence: 99%

“…In dimension three, only the set of periodic orbits of non singular Morse-Smale systems (NMS) on S 3 (M. Wada [8], K. Yano [9]) and S 2 × S 1 (A. Cordero, J. Martínez Alfaro and P. Vindel [5]) have been characterized in terms of links. These characterizations are based on the round handle decomposition (RHD) introduced by Asimov [1] and Morgan [6].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

NMS Flows on S 3 with no Heteroclinic Trajectories Connecting Saddle Orbits

Campos

Vindel

2012

J Dyn Diff Equat

View full text Add to dashboard Cite

Abstract.In this paper we find topological conditions for the non existence of heteroclinic trajectories connecting saddle orbits in non singular Morse-Smale flows on S 3 .We obtain the non singular Morse-Smale flows that can be decomposed as connected sum of flows and we show that these flows are those who have no heteroclinic trajectories connecting saddle orbits. Moreover, we characterize these flows in terms of links of periodic orbits.MSC: 37D15 Keywords: NMS systems, links of periodic orbits, round handle decomposition, connected sum. M. Wada [8, Theorem 1] characterizes the links of periodic orbits of NMS flows on S 3 in terms of six operations and a generator, the hopf link. He states that every link obtained by applying these operations corresponds to the set of periodic orbits of a NMS flow on S 3 . The link of periodic orbits of a NMS flow on S 3 is defined by the cores of the round-handles in the round handle decomposition of the manifold. Although there is a 1-1 correspondence between the flow and the round handle decomposition, this is not the case for the link of periodic orbits. Different round handle decompositions can yield the same link (B. Campos, J. Martínez Alfaro and P. Vindel [2]), but the corresponding flows are not topologically equivalent (B. Campos and P. Vindel [3]). So, the link of periodic orbits does not describe completely the flow. Nevertheless,

show abstract

Section: Round Handle Decompositionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

NMS Flows on S 3 with no Heteroclinic Trajectories Connecting Saddle Orbits

Campos

Vindel

2012

J Dyn Diff Equat

View full text Add to dashboard Cite

show abstract

“…It should be noted that unlike to the round Morse functions which gained a lot of attention [1], [30], [24], [8], round functions with degenerate critical loops (called below degenerate round functions) are rather poorly understood. For example, it is still unclear how to describe the class of compact closed manifolds which possess round functions.…”

Section: Introductionmentioning

confidence: 99%

Remarks on minimal round functions

Khimshiashvili

Siersma

2003

Geometry and Topology of Caustics – Caustics '02

View full text Add to dashboard Cite

Abstract. We describe the structure of minimal round functions on compact closed surfaces and three-dimensional manifolds. The minimal possible number of critical loops is determined and typical non-equisingular round function germs are interpreted in the spirit of isolated line singularities. We also discuss a version of Lusternik-Schnirelmann theory suitable for round functions. Introduction.A differentiable function on a differentiable manifold is called a round function if its critical set is a union of embedded one-dimensional submanifolds. In general it is not assumed that the critical submanifolds are non-degenerate critical submanifolds in the sense of R. Bott [3] so this is a straightforward extension of the notion of round Morse function introduced by W. Thurston [30].Our main concern in this paper are round functions on a given compact closed (i.e., without boundary) smooth manifold with the minimal possible number of critical submanifolds (critical loops). By analogy with the terminology of [29] they will be called minimal round functions as in [21]. We are especially interested in describing possible changes of the topology of their Lebesgue sets (preimages of semi-infinite intervals) and typical local models of their singular behaviour near critical loops. Our setting and approach are much in the spirit of F. Takens' paper [29] which contains a comprehensive

show abstract

“…According to [1], M can be decomposed as a union of round handles Rk = Sx X Dk X D"~k~x. Each round fc-handle Rk is supplied with a vector field V = d/dt -2 x,a/3x,.…”

mentioning

confidence: 99%

“…Modifying the vector field V on each Rk, we will get a nonsingular vector field W such that for k > 0 (respectively k = 0) any trajectory of W approaching 5'xOxO (respectively any trajectory of W) meets Sx X (x,-axis) (respectively Sx X (^,-axis)) in Rk. By patching up the vector fields Won the Rks as in [1], we construct a nonsingular vector field Y on M with finitely many closed orbits {C,}, where C¡ corresponds to Sx X 0 X 0 on each Rk. By using the standard transversality argument, we may assume that near C, for each / E Sx the x,-axis (or^-axis) forms part of an orbit of X.…”

mentioning

confidence: 99%

On Autonomous Control Systems on Certain Manifolds

Liang¹

1980

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

Abstract. Let M" be a compact C°° manifold, n > 4, admitting a vector field with every orbit a circle. Then there exists a completely controllable set S consisting of two nonsingular C°° vectors X and Y such that every orbit of A-is a circle.An autonomous control system on a smooth manifold M is the same as a set of vector fields on M. A set S of vector fields on a smooth manifold M is said to be controllable if for every pair (m, m') of points of M there exists a trajectory of S from m to m'. Here a trajectory of S is a curve which is an integral curve (orbit) of some X G S or a finite concatenation of such curves such that a trajectory of S run in reverse is not allowed. (We refer the readers to [2] for details.)In [2], N. Levitt and H. J. Sussmann showed that on every connected paracompact manifold of class Ck, 2 < k < oo, or k = w, there exists a completely controllable set S consisting of two vector fields of class Ck~x.For simplicity, we assume that all the manifolds, vector fields, etc., considered here are of class C°°.A manifold M is called closed if it is compact and without boundary. Let Dk denote the A>dimensional disk and Sk~x its boundary.The purpose of this paper is to prove the following theorem:Theorem. // a connected closed n-dimensional manifold M", n > 4, admits a vector field X0 with every orbit a (nondegenerate) circle, then there exists a completely controllable set S consisting of two nonsingular vectors X and Y such that every orbit of X is a circle and Y has finitely many closed orbits.We first give a brief sketch of the proof. Key words and phrases. Completely controllable set of vector fields, orbits, trajectory, round handle decomposition of a manifold.'Supported by the University of Kansas General Research Fund.

show abstract

Round Handles and Non-Singular Morse-Smale Flows

Cited by 112 publications

References 0 publications

NMS Flows on S 3 with no Heteroclinic Trajectories Connecting Saddle Orbits

NMS Flows on S 3 with no Heteroclinic Trajectories Connecting Saddle Orbits

Remarks on minimal round functions

On Autonomous Control Systems on Certain Manifolds

Contact Info

Product

Resources

About