Smale Flows Research Articles

Morse–Smale systems with few non-wandering points

Let MSflow(Mn,k) and MSdiff(Mn,k) be Morse–Smale flows and diffeomorphisms respectively the non-wandering set of those consists of k fixed points on a closed n-manifold Mn. For k=3, we show that the only values of n possible are n∈{2,4,8,16}, and M2 is the projective plane. For n⩾4, Mn is simply connected and orientable. We prove that the closure of any separatrix of ft∈MSflow(Mn,3) is a locally flat n2-sphere while there is ft∈MSflow(Mn,4) such that the closure of separatrix of ft is a wildly embedded codimension two sphere. This allows us to classify flows from MSflow(M4,3). For n⩾6, one proves that the closure of any separatrix of f∈MSdiff(Mn,3) is a locally flat n2-sphere while there is f∈MSdiff(M4,3) such that the closure of any separatrix is a wildly embedded 2-sphere.

Lyapunov graphs of nonsingular Smale flows on 𝑆¹×𝑆²

In this paper, following J. Franks’ work on Lyapunov graphs of nonsingular Smale flows on S 3 S^3 , we study Lyapunov graphs of nonsingular Smale flows on S 1 × S 2 S^1 \times S^2 . More precisely, we determine necessary and sufficient conditions on an abstract Lyapunov graph to be associated with a nonsingular Smale flow on S 1 × S 2 S^1 \times S^2 . We also study the singular type vertices in Lyapunov graphs of nonsingular Smale flows on 3-manifolds.

The templates of non-singular Smale flows on three manifolds

AbstractIn this paper, we first discuss some connections between template theory and the description of basic sets of Smale flows on 3-manifolds due to F. Béguin and C. Bonatti. The main tools we use are symbolic dynamics, template moves and some combinatorial surgeries. Secondly, we obtain some relationship between the surgeries and the number of S1×S2 factors of M for a non-singular Smale flow on a given closed orientable 3-manifold M. We also prove that any template T can model a basic set Λ of a non-singular Smale flow on nS1×S2 for some positive integer n.

Regular level sets of Lyapunov graphs of nonsingular Smale flows on 3-manifolds

In this paper, we first discuss the regular level set of a nonsingular Smale flow (NSF) on a 3-manifold. The main result about this topic is that a 3-manifold $M$ admits an NSF which has a regular level set homeomorphic to $(n+1)T^{2}$ $(n\in \mathbb{Z}, n\geq 0)$ if and only if $M=M'$#$n S^{1}\times S^{2}$. Then we discuss how to realize a template as a basic set of an NSF on a 3-manifold. We focus on the connection between the genus of the template $T$ and the topological structure of the realizing 3-manifold $M$.

Non-singular Smale flows on three-dimensional manifolds and Whitehead torsion

AbstractThis paper deals with non-singular Smale flows on oriented 3-manifolds. We shall show a relation between the properties of invariant sets of a Smale flow and a kind of Whitehead torsion of the underlying manifold.

Lorenz like Smale flows on three-manifolds

In this paper, we discuss how to realize Lorenz like Smale flows (LLSF) on 3-manifolds. It is an extension of M. Sullivan's work about Lorenz Smale flows on S 3 . We focus on two questions: (1) Classify the topological conjugate classes of LLSF which can be realized on S 3 ; (2) Which 3-manifolds admit LLSF? If some 3-manifold admits LLSF, how does it admit LLSF? This paper is in some sense parallel to the work of J. Morgan and M. Wada on Morse–Smale flows on 3-manifolds.

Transverse Foliations to Nonsingular Morse–Smale Flows on the 3-Sphere and Bott-Integrable Hamiltonian Systems

In this note we apply results of Goodman, Yano and Wada to determine which nonsingular Morse–Smale flows on S3 have transverse foliations. We then observe that there is a connection to flows arising from certain Hamiltonian systems and from certain contact structures.

Round-handle decomposition of S 2×S 1

A round-handle decomposition is associated with a non-singular Morse–Smale flow on 3-manifolds prime to S 2× S 1. This decomposition has been built only for the 3-sphere S 3. In this paper we obtain the round-handle decomposition of non-singular Morse–Smale flows on S 2× S 1, in order to get all the different fattened round handles in this manifold. Some of them include non-separating boundary components that induce the topology of the links of periodic orbits.

Isolating Blocks for Periodic Orbits

In this article, we prove that the Lyapunov semi-graphs associated with periodic orbits are realizable by constructing isolating blocks N n for periodic orbits of Morse---Smale flows. We analyze the effects on the Betti numbers of a manifold after performing a round handle operation and considering a variety of situations. Since we are concerned in showing the existence of certain blocks, we keep the complexity of the manifolds in consideration under control by considering mainly manifolds with free homology groups, in particular, we consider connected sums of tori manifolds.

The simplest branched surfaces for a foliation

Given a foliation F of a closed 3-manifold and a Smale flow f transverse to F, we associate a branched surface with the pair (F,f), which is unique up to two combinatorial moves. We show that all branched surfaces constructed from F and f can be obtained from the simplest model by applying a finite sequence of these moves chosen so that each intermediate branched surface also carries F. This is used to partition foliations transverse to the same flow into countably many equivalence classes.

Flots de Smale en dimension 3: présentations finies de voisinages invariants d'ensembles selles

Given a vector field X on a compact 3-manifold, and a hyperbolic saddle-like set K of that vector field, we consider all the filtering neighbourhood of K: by such, we mean any submanifold which boundary is tranverse to X, the maximal invariant of which is equal to K and which intersection with every orbit of X is connected. Up to topological equivalence, there is only a finite number of such neighbourhoods. We give a finite combinatorial presentation of the global dynamics on any such neighbourhood. A key step is the construction of a unique model of the germ of X along K; this model is, roughly speaking, the simplest three-dimensional manifold and the simplest Smale flow exhibiting the germ considered above. Then, we give a combinatorial description of the surgeries leading back to the initial filtering neighbourhoods.

Visually building Smale flows in [formula omitted

A Smale flow is a structurally stable flow with one-dimensional invariant sets. We use information from homology and template theory to construct, visualize and in some cases, classify, nonsingular Smale flows in the 3-sphere.

Gradient-like flows on high-dimensional manifolds

The main purpose of this paper is to study the implications that the homology index of critical sets of smooth flows on closed manifolds M have on both the homology of level sets of M and the homology of M itself. The bookkeeping of the data containing the critical set information of the flow and topological information of M is done through the use of Lyapunov graphs. Our main result characterizes the necessary conditions that a Lyapunov graph must possess in order to be associated to a Morse–Smale flow. With additional restrictions on an abstract Lyapunov graph L we determine sufficient conditions for L to be associated to a flow on M.

Errata to 'An invariant of basic sets of Smale flows'

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From Morse-Smale to all knots and links

We analyse the topological (knot-theoretic) features of a certain codimension-one bifurcation of a partially hyperbolic fixed point in a flow on originally described by Shil'nikov. By modifying how the invariant manifolds wrap around themselves, or `pleat', we may apply the theory of templates, or branched two-manifolds, to capture the topology of the flow. This analysis yields a class of flows which bifurcate from a Morse-Smale flow to a Smale flow containing periodic orbits of all knot and link types.

An invariant of basic sets of Smale flows

We consider one-dimensional flows which arise as hyperbolic invariant sets of a smooth flow on a manifold. Included in our data is the twisting in the local stable and unstable manifolds. A topological invariant sensitive to this twisting is obtained.

Bifurcations of links of periodic orbits in non-singular Morse - Smale systems on

The set of periodic orbits of a non-singular Morse - Smale (NMS) flow on defines a link; a characterization of all possible links of NMS flows on has been developed by Wada. In the frame of codimension-one bifurcations, this characterization allows us to study the restrictions a link requires for suffering a given bifurcation. We also derive the topological description of the new link and the possibility of relating links by a chain of this type of bifurcation.

Reidemeister torsion and Morse–Smale flows

AbstractFor several types of stable flows φ and representations ρ of the fundamental group of the underlying manifold, R-torsion for ρ can be computed from the periodic orbits of φ. However, there are counterexamples and the purpose of this paper is to describe the role of heteroclinic orbits in such counterexamples.

The C*-algebras of Morse—Smale flows on two-manifolds

AbstractWe give a classification of all the C*-algebras of Morse-Smale flows on closed two-manifolds, and determine the relation between the invariants of dynamical systems and the topological invariants of the C*-algebras.

Notes on Lyapunov graphs and non-singular Smale flows on three manifolds

In the 1980s, Franks, Pugh and Shub raised the question “Given any subshift of finite typeσA: ΣA→ ΣAis there a non-singular Smale flow (or an NS flow for short) on S3with the suspension ofσAas a basic set?” (See [5] and [12]).