Trajectory equivalence of optimal Morse flows on closed surfaces

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INTRODUCTION
The problem of topological classification of objects or structures is one of the main problems in the topology and theory of dynamical systems. Often the class of such objects consists of structurally stable dynamical systems. They are interesting due to the fact that their topological structure does not change under small changes of the parameters by which they are given. Morse-Smale flows on surfaces are examples of structurally stable dynamical systems. There are many papers dealing with their topological classification, for example [2], [3], [9]. An overview of different approaches to the above problem can be found in [6].
To classify Morse functions on surfaces and integrable Hamiltonian systems Bolsinov and Fomenko introduced the concept of atoms and molecules. They can also be used to classify Morse flows [1].
In fact, Morse flows are gradient flows of Morse functions. In this case the atoms specify the associated handle decompositions. If the Morse flow has one source, then the corresponding atom is a marked chord diagram.
The purpose of the present paper is to find all such chord diagrams corresponding to flows with one sink and to determine which of them correspond to reverse flows on nonorientable surfaces of small genus.

MORSE-SMALE FLOWS
We consider C 1 -smooth dynamical systems on a smooth manifold M with continuous time (flows). Since such a flow on a closed manifold is always generated by a smooth vector field, all the concepts that are introduced for dynamical systems will be applied to vector fields.
Let X t the flow generated by a vector field X. does not contain eigenvalues with zero real part (that is eigenvalues on the imaginary axis). A closed orbit γ of X is called hyperbolic whenever the differential of the Poincaré map has no eigenvalues of absolute value 1 at some point p ∈ γ. Definition 2.3. A stable manifold (denoted by W s (p) or S(p)) of a critical element p is the following set Similarly, the set is called the unstable manifold (W u (p) or U (p)) of p.
The sets of nonwandering points of a vector field X is denoted by Σ(X). 2) the set of non-wandering points Σ(X) coincides with the union of all critical elements of X; 3) stable manifolds of critical elements are transversal to unstable manifolds.
Definition 2.6. A fixed point y ∈ M is called an ω-limit (resp. an α-limit) The set of all ω-limit points (resp. α-limit points) of x is called ω-limit (α-limit) set.
Condition 2) of the definition of Morse-Smale dynamical system can be replaced by the following condition 2) ′ for each point x ∈ M its α-limit and ω-limit sets are contained in the union of critical elements.
For vector fields on two-dimensional manifolds there are three types of non-degenerated (hyperbolic) singular points: sinks, sources, and saddles. Condition 3) in this case is equivalent to the statement that there is no trajectories whose α-limit and ω-limit sets are saddle points. A well known results by Smale [11] claims that every Morse vector field is a gradient field of some Morse function with respect to a certain Riemannian metric.
Properties of Morse-Smale dynamical systems described above, as well as some other properties, can be found in the book by J. Palis and V. di Melo [7]. Trajectory equivalence of optimal flows 15 3. TOPOLOGICAL EQUIVALENCE OF DYNAMICAL SYSTEMS Definition 3.1. Let M be a C r manifold, X r (M ) be the space of C r vector fields endowed with C r topology. Two vector fields X, Y ∈ X r (M ) are called topologically (C k ) equivalent if there exists a homeomorphism (C k diffemorphism) h : M → M that sends each trajectory of X onto a trajectory of Y preserving their orientations.
Theorem 3.2. [7]. Let h be a topological equivalence between the fields X, Y ∈ X r (M ). Then 1) p ∈ M is a singular point of the field X if and only if h(p) is a singular point of the field Y ; 2) the trajectory X(p) of the field X, which passes through the point p, is closed if and only if trajectory Y (h(p)) is closed; 3) the image of the α-limit (ω -limit) set of the trajectory X(p) under h is α-limit (ω-limit) set of trajectories Y (h(p)).
Definition 3.3. [10]. A vector field X ∈ X r (M ), r > 1, is called structurally stable if there exists a neighborhood U of the field X in X r (M ) such that every Y ∈ U is topologically equivalent to X.
It is known that Morse-Smale dynamical systems are structurally stable [8]. Moreover, every structurally stable vector field on a closed twodimensional manifold is Morse-Smale and these fields form an open everywhere dense set in set of all vector fields.
In higher dimensions, there are structurally stable dynamical systems that are not Morse-Smale systems, and they do not constitute everywhere dense subsets in the spaces of all dynamical systems.

Definition 3.4.
A complete topological invariant of a dynamical system from a certain class of dynamical systems is a set of quantities (or algebraic objects) that are identical (isomorphic) for topologically equivalent dynamical systems, and distinct (not isomorphic) for non-equivalents systems.
By the topological classification of a certain class of dynamical systems, we (roughly) mean a construction of a complete topological invariant for systems of this class and such that the equivalence (isomorphism) of invariants can be easily verified, as well as the description of all possible values that the invariant can accept.

OPTIMAL FLOWS ON SURFACES
Hyperbolicity condition on critical elements means that for vector fields on two-dimensional manifolds there possible three types of non-degenerate singular points: sinks, sources and saddles.
A Morse flow on a surface is called optimal if it has the least number of singularities among all Morse flows on this surface. Proof. To prove the theorem, we should show the following statements: 1) a Morse flow on a surface contains at least one sink and one source; 2) there exists Morse flow on a given surface, which contains only one sink and one source; 3) if the flow has more than one source or sink, than it is not optimal. Firstly, It follows from the definition of Morse flow that every trajectory starts at a singular point. As there are only finitely many saddle points, and no more than 2 trajectories could be obtained from each of them, we see that only a finite number of trajectories starts at saddle points. Consequently, the remaining trajectories (it is infinite number) start at the source. By similar arguments there must exist at least one sink.
Secondly. As is well known (see for example [5]), each closed surface is diffeomorphic to a sphere with handles or a sphere with Mobius strips pasted into it. Therefore, it is sufficient to construct a vector field with one sink and one source on these surfaces. As an example of this field, one can take the gradient field of an optimal Morse function (that is, a function with one local minimum and maximum whose existence in is proved in Morse's theory [5]).
Thirdly. By Poincare-Hopf theorem, the Euler characteristic of the surface M is equal to the number of sources plus the number of sinks minus the number of saddle points. This formula shows that increasing the number of sources or sinks leads to increasing of the number of saddle points on a given surface, and, consequently, implies an increasing of the total number of singular points. Hence a flow with one source and one sink is optimal. □ Recall that a flow with one source and one sink, also, is called polar. So, optimal flows are precisely polar flows.

CHORD DIAGRAMS
Chord diagrams are widely used in modern low-dimension topology, especially in knot theory.
Definition 5.1. A configuration consisting of a circle, 2n distinct points on it, and n chords, which specify a partition of these points into pairs, will be called a chord diagram with n chords, or n-diagram.
Thus, a chord diagram is a graph of valency 3 with a distinguished cycle containing each vertex exactly once. In this case, that cycle is called a circle, and the edges that do not belong to it are called chords.
Definition 5.2. Two chord diagrams are said to be equivalent if there is a homeomorphism between them that maps the cycle into a cycle, and chords into chords.
Let D be an arbitrary chord diagram with 2n points which we enumerate with numbers from 1 to 2n in the clockwise direction starting from a certain fixed point on the dual graph. We construct a substitution α obtained as a result of the transposition multiplication (a j , b j ), j = 1, . . . , n, where a j , b j are the numbers of points connected by chord on the diagram D: Let Q be a regular polygon in a complex plane C with 2n vertices being roots of unit of degree 2n.Then one can draw (with self intersections) each 2n-diagram D(α) in C by connecting the corresponding pairs of vertices by straight segments and regarding Q as a circle of a diagram. Such a representation of a 2n-diagram will be called good.
Obviously, in this approach, each diagram D(α) unequivocally determines the substitution of α and vice versa. There are two non-isomorphic and non-equivalent chord diagrams with two chords: 1) a diagram with intersecting chords; it corresponds to the substitution (1, 3)(2, 4); 2) a diagram with non-intersecting chords; it corresponds to the substitution (1, 2)(3, 4).

CHORD DIAGRAM OF AN OPTIMAL MORSE VECTOR FIELD
Let X be an optimal Morse vector field. Consider some small neighborhood U for the source x 0 such that its boundary ∂U (x 0 ) is a smooth curve and transversal to each flow trajectory. For each saddle point we consider its stable manifold consisting of two trajectories converging to that saddle. It follows from conditions 2) and 3) of the definition of Morse-Smale vector field that these trajectories start at x 0 . Consequently, each of them intersects the boundary ∂U at one point. Thus we get a couple of points on the boundary. The boundary ∂U is a closed curve, and so it is homeomorphic to a circle S 1 = {(x, y)|x 2 +y 2 = 1}.
Via this homeomorphism the couple of points maps into a couple of points at the circle S 1 . For another saddle point we will get another couple of points on S 1 . Connecting the points of the same couple with the chord we will obtain chords at the circle, which do not have common ends, since the stable manifolds of the saddle points do not intersect with each other. Thus we obtain a chord diagram. If the union of U with a regular neighborhood of the stable manifold is an oriented surface (homeomorphic to S 1 ×[0, 1]), then we leave chord unmarked (or formally associate to the chord the number 0). Otherwise (in the nonoriented case of Möbius strip), and we mark the chord by X (or formally associate to the chord the number 1). Thus, each chord has a marking 0 or 1.

Definition 6.1. A marked chord diagram is a chord diagram with marking of its chord.
A marked chord diagram constructed in such a way is a complete topological invariant of the vector field [4]. We call it a chord diagram of a vector field.

INVERSE CHORD DIAGRAM
Let's describe a chord cycle on a chord diagram. Choose a point on a certain arc that is not the end of a chord and the direction of circumvention of a circle, for example, counterclockwise. We move from the chosen point in this direction to the nearest end of the chord. At every such point the movement on the circle changes on the chord movement. After passing the chord we continue the circular motion in counterclockwise direction if the chord is unmarked and change the direction along to clockwise if the chord is marked. We move to the nearest point that is the end of the chord. In it again change the movement on the movement of the chord. After marked chords we change the direction and after unmarked one we preserve it. We then move along the higher horizontal chord 1 to its right endpoint. This chord is marked, so from its endpoint we change circle direction and move clockwise. We will continue this process until we return to the starting point.

Definition 7.1.
A chord cycle is a cycle in chord diagram that obtained in a such way.
If chord cycle covers all chords (two times) and arcs we have a diagram with one chord cycle (a one-cycle diagram). Now we construct an inverse diagram for a one-cycle diagram. On each chord we mark one point (which does not belong to other chords). When moving along the cycle we count points consecutively. Hence each point will receive two numbers, because each chord in the diagram is passed twice. We obtain a substitution of 2n elements and associate to it a 2n-diagram in which points are connected, if their numbers correspond to the same couple in substitution, as above. By construction, each chord in the initial diagram corresponds to a chord in the inverse diagram and corresponds to a single saddle point. The chord cycle induces on each chord two directions. If these directions are opposite, then the new chord is unmarked (of marking 0). If these directions are same, then it is marked (of marking 1). For example, in Figure 7.1 the direction on high horizontal are same, thus, the chord 1 − 3 is marked in inverse diagram.
Let us prove that the inverse chord diagram of a field X is the chord diagram for the field −X, that is, the field has the same trajectories, but the motion behind them is changed to the opposite. Indeed, consider a regular neighborhood W of the union of sources and stable manifolds of saddles. It follows from the definition of nondegenerate singular points that we can choose this neighborhood so that X is transversal to the boundary ∂W and directed outside it. In this case, bypassing the cycle of the chord diagram corresponds to bypassing ∂W . The intersections of the unstable manifolds of saddle points with ∂W correspond to the chosen points on the chords and the pairs of numbers that determine the substitution. In addition, M \W contains one singular point, a sink. It is the source for −X and ∂W is the circle of the chord diagram for −X. The unstable manifolds of the saddles of X are stable manifolds of saddles of −X. It completes the proof.
Consequently, the circle of the chord diagram is homeomorphic to the boundary of a regular neighborhood of the source.
From the construction of the chord diagram and the considerations above we get the following theorem. We consider the case of nonorientable surfaces. The number of the chords is equal to the genus of the surface.
On the surface of genus 1 (the projective plane) there is unique diagram. It corresponds atom B from Bolsinov and Fomenko [1]. Proof. The proof consists of three parts. At first we find all chord diagrams with 4 chords. Further, we set marks on them and determine which labeled diagrams have one cycle. For every such diagram we will also find its inverse.
To list the diagrams we use lengths of their chords. The length of the chord is the minimum number of arcs of a circle necessary to pass from one end of the chord to the other along the circumference. It is one more than the number of ends of the chords being traversed in this case. For example, the diameter has a length of 4, and the smallest chord length can be equal to 1.
If all the chords have the maximum length, then all of them are diameters. The corresponding diagram is shown in Fig. 8.5-1a (Case 1). If three chords have a length of 4, then the fourth one will also be of length 4, and this is the case 1.
Assume that two chords have length 4, and the lengths of other chords are less than 4. Then there are two options: 1) chords of length 4 perpendicular, or 2) neighboring. In the first variant the remaining two chords either have length 4, as in the case 1, or they have length 2 (case 2) as in Figures 8.5

1)
2 In the case of four chords of length 3, one diagram is possible (case 8, Figures 8.5.11a and 8.6.11).

1)
2 . In some diagrams, where there are symmetries, the number of markings will be smaller. Next, for each marked diagram, we find the number of cycles and leave the diagrams with one cycle. We obtained 47 marked one-cycle chord diagrams on N 4 (see Figures 8.5 and 8.6).
For them, as above, we find the inverse. As a result, we obtain 12 pairs of mutually inverse diagrams (Figures 8.5) and 23 diagrams, which are inverses to themselves (Figure 8.6).