Morse – Smale 3-Diffeomorphisms with Saddles of the Same Unstable Manifold Dimension Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 1, pp. 167-178. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd240301

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 34C45, 37D15, 37C05, 70K44

Morse -Smale 3-Diffeomorphisms with Saddles of the Same Unstable Manifold Dimension

E. M. Osenkov, O. V. Pochinka

In this paper, we consider a class of Morse-Smale diffeomorphisms defined on a closed 3-manifold (not necessarily orientable) under the assumption that all their saddle points have the same dimension of the unstable manifolds. The simplest example of such diffeomorphisms is the well-known "source-sink" or "north pole - south pole" diffeomorphism, whose non-wandering set consists of exactly one source and one sink. As Reeb showed back in 1946, such systems can only be realized on the sphere. We generalize his result, namely, we show that diffeomorphisms from the considered class also can be defined only on the 3-sphere.

Keywords: Morse-Smale diffeomorphisms, ambient manifold topology, invariant manifolds, heteroclinic orbits, hyperbolic dynamics

Introduction and formulation of the results

The class of dynamical systems, introduced by S. Smale in 1960 [15] and known today as Morse-Smale systems, played not the least role in the formation of the modern dynamical systems theory. The study of these systems remains an important part of it because they form a class of structurally stable systems which, in addition, have zero topological entropy [3, 11, 13], which makes them in this sense "the simplest" structurally stable systems.

A close relation of the Morse-Smale diffeomorphisms (MS-diffeomorphisms) with the topology of the ambient manifold allows us to realize various topological effects in the dynamics of such systems. Systems with exactly two points of extreme Morse indices are a classical example demonstrating such a relation. In this case, it follows from the Reeb's theorem [12] that the ambient manifold is homeomorphic to the sphere.

Received September 13, 2023 Accepted February 13, 2024

The work was supported by the Russian Science Foundation (Project No. 23-71-30008).

Eugene M. Osenkov eosenkov@hse.ru Olga V. Pochinka olga-pochinka@yandex.ru

National Research University "Higher School of Economics" Bol'shaya Pecherskaya ul. 25/12, Nizhny Novgorod, 603155 Russia

Another brilliant illustration of the relations under study is the decomposition of an orientable 3-manifold into a connected sum of S2 x S1 whose number of summands is completely determined by a structure of the non-wandering set of an MS-diffeomorphism without hetero-clinic curves defined on it. This result was obtained in papers by C.Bonatti, V. Grines, and V. Medvedev [5, 6] and is based on the breakthrough result on the existence of a tame neighborhood of a 2-sphere with one point of wildness. The ideas that the authors put into their proofs have been extremely helpful in our research.

The present paper is a straightforward generalization of the Reeb's theorem on the following class of difeomorphisms.

Let f be an MS-diffeomorphism defined on a closed connected 3-manifold M3 and let all its saddle points have the same dimension of their unstable manifolds. Denote the class of such diffeomorphisms as G.

Theorem 1. Any closed connected 3-manifold M3, admitting a diffeomorphism f G G, is homeomorphic to the 3-sphere.

It is worth noting that in [8] V. Medvedev and E. Zhuzhoma considered a similar class of diffeomorphisms in the case when the ambient manifold has dimension greater than three. Actually, Theorem 1 complements their result for the last unexplored dimension.

1. Auxiliary information and facts

This section introduces basic concepts and facts from topology and dynamical systems theory.

1.1. Some topological facts

Let X, Y be topological spaces, A c X and B c Y be their subsets and g: A — B be a homeomorphism. Let ~ be the minimal equivalence relation on X U Y for which a ~ g(a) for all a G A. The factor space for this equivalence relation is said to be obtained by gluing the

space Y to the space X by the map g, written X U Y.

g

Let X, Y be compact n-manifolds, D1 c X, D2 c Y be subsets homeomorphic to Dra, h1: Bra — D1, h2: Dra — D2 be the corresponding homeomorphisms and g: dD1 — dD2 be a homeomorphism such that the map h-1gh1\do„ : Sra_1 — Sra_1 reverses orientation. Then the

space X#Y = (X \ int D1) U(Y \ int D2) is called the connected sum of X and Y.

g

If X c Y, then the map iX : X — Y such that iX (x) = x for all x G X is called the inclusion map of X into Y.

Let X and Y be Cr-manifolds. Denote by Cr(X, Y) the set of all Cr-maps A: X — Y. A map A: X — Y is said to be a Cr-embedding if it is a Cr-diffeomorphism onto the subspace A(X). C0-embedding is also called a topological embedding.

A topological embedding A: X — Y of an m-manifold X into an n-manifold Y (m ^ n) is said to be locally flat at the point A(x), x G X, if the point A(x) is in the domain of such a chart (U, of the manifold Y that ^(U n A(X)) = Rm, where Rm c Rra is the set of points for which the last n — m coordinates equal to 0 or ^(U n A(X)) = R™, where R™ c Rm is the set of points with the nonnegative last coordinate.

An embedding A is said to be tame and the manifold X is said to be tamely embedded if A is locally flat at every point x G X. Otherwise the embedding A is said to be wild and the

manifold X is said to be wildly embedded. A point X(x) which is not locally flat is said to be the point of wildness.

Statement 1 ([4, Theorem 4]). Let T be a two-dimensional torus tamely embedded in the manifold S2 x S1 in such a way that T(n1 (T)) = 0. Then T bounds in S2 x S1 a solid torus.

Statement 2 ([5, Proposition 0.1]). Let M3 be a closed connected 3-manifold and let X: S2 — M3 be a topological embedding of the 2-sphere which is smooth everywhere except one point. Let £ = X (S2). Then any neighborhood V of the sphere £ contains a neighborhood K diffeomorphic to S2 x [0, 1].1

Let us assume that the empty set and only the empty set has a dimension —1 (dim 0 = — 1). The separable metric space X has a dimension ^ n (dim X ^ n) if any neighborhood Vp of a point p G X contains a neighborhood Up such that dim(dUp) ^ n — 1. The space X has a dimension n (dim X = n) if the statement dim X ^ n is true and the statement dim X ^ n — 1 is false.

It is said that a subset D of a connected space X divides it if the space X\D is disconnected.

Statement 3 ([1, Corollary 1, p. 48]). Any connected n-manifold cannot be divided by a subset of dimension ^ n — 2.

1.2. Morse —Smale diffeomorphisms

Here and below, we assume that Mn is a closed connected 3-manifold with a metric d and a map f: Mn — Mn is a diffeomorphism.

The trajectory or the orbit of a point x G Mn is the set Ox = {f m(x), m G Z}.

A set A c Mn is said to be f -invariant if f (A) = A, i.e. the trajectory of any point x G A belongs to A.

A compact f-invariant set A c Mn is called an attractor of the diffeomorphism f if it has a compact neighborhood UA such that f (UA) c int UA and A = f] fk(UA). The neighborhood UA in this case is said to be trapping. The basin of the attractor A is the set

Mn : lim d(fk (x), A) =4. k^+œ V /J

WS = j x G

A repeller and its basin are defined as an attractor and its basin for f-1.

A point x G Mn is said to be a wandering point for the diffeomorphism f if there is an open neighborhood Ux of x such that fk(Ux) n Ux = 0 for all k G N. Otherwise, the point x is said to be a non-wandering point. The set of all non-wandering points is called the non-wandering set and it will be denoted by Qj. The non-wandering set Qj is f-invariant, and if Qj is finite, then it consists only of periodic points, i. e., points p G Mn such that there exists a natural number m for which fm(p) = p. If this equality is not satisfied for any natural number k < m, then m is called the period of a point p, which we denote by mp.

For a periodic point p, let us define the sets

Wp = jx G Mn : fclini df kmp(x), p) = 0}

1 This fact was proven in [5] for an orientable manifold M3, but the proof does not use the orientability anywhere. So we can use this result in our case too.

and

Wu = " p

u- ]x G Mn : lim d(f kmp (x),p) =0I,

V /J

which are called stable and unstable manifolds of the point p respectively. These sets are also known as invariant manifolds of the point p.

A periodic point p with period mp is said to be hyperbolic if the absolute value of each

eigenvalue of the Jacobi matrix (9gxP

is not equal to 1. If the absolute values of all the

p

eigenvalues are less than 1, then p is called an attracting point, a sink point or a sink; if the absolute values of all the eigenvalues are greater than 1, then p is called a repelling point, a source point or a sink. Attracting and repelling fixed points are called nodes. A hyperbolic periodic point which is not a node is called a saddle point or a saddle.

The hyperbolic structure of the periodic point p and the finiteness of the non-wandering set implies that its stable and unstable manifolds are smooth submanifolds of Mn which are diffeomorphic to Rqp and Rn-qp respectively, where qp is a Morse index of p, i.e. the number of the eigenvalues of the Jacobi matrix whose absolute value is greater than 1.

A connected component (ip) of the set Wpu \ p (Wp> \ p) is called an unstable (stable) separatrix of the periodic point p. For p let vp be +1 (—1) if fmp\Wu preserves (reverses) orientation and let ¡ip be +1 (—1) if fmp\Ws preserves (reverses) orientation.

A diffeomorphism f : M3 ^ M3 is called a Morse-Smale diffeomorphism f G MS (M3)) if

1) the non-wandering set Qj is finite and hyperbolic;

2) for every two distinct periodic points p, q the manifolds Wp>, WU intersect transversally.

Note that all the facts below are proved in the case when Mn is orientable, but a direct check allows us to verify the correctness of these results for non-orientable manifolds as well.

Statement 4 ([2, Theorem 2.1]). Let f G MS (M3). Then

1) Mn = Wp

u p,

pen.

2) W'U is a smooth submanifold of the manifold Mn diffeomorphic to R?p for every periodic point p G Qf,

3) cl [¿U) \ {¿U U p) = U WU for every unstable separatrix £U of a periodic point p G

renf : iuC\Ws=0

G Qf.

If g 1, g2 are distinct saddle points of a diffeomorphism f G MS (Mn), then the intersection Wn W'U2 = 0 is called a heteroclinic intersection. If dim (w^ n W'U^j > 0, then a connected component of the intersection W^ n W%2 is called a heteroclinic manifold, and

if dim (w^ n WU^j = 1, then it is called a heteroclinic curve. If dim (w^ n W'U^j = 0, then the intersection WS n W%2 is countable, each point of this set is called a heteroclinic point and the orbit of a heteroclinic point is called a heteroclinic orbit.

Statement 5 ([2, Proposition 2.3]). Let f G MS (Mn) and a be a saddle point of f such that the unstable separatrix i'%. has no heteroclinic intersections. Then

cl(C) \ (C U a) = M,

where u is a sink point. If qa = 1 then cl (¿U) is an arc topologically embedded into Mn, and if q& ^ 2 then cl (¿U) is the sphere Sqa topologically embedded into Mn.

A diffeomorphism f £ MS (Mn) is called a " source-sink" diffeomorphism if its non-wandering set consists of a unique sink and a unique source.

Statement 6 ([2, Theorem 2.5]). If a diffeomorphism f £ MS (Mn), n > 1, has no saddle points, then f is a "source-sink" diffeomorphism and the manifold Mn is homeomorphic to the n-sphere Sn.2

Statement 7 ([7, Theorem 1]). Let f £ MS (Mn) and ÜA be a subset of üf such that the set

A = ^o U WUA

is closed and f -invariant. Then

1) the set A is an attractor of diffeomorphism f;

2) WA = U Wp;

pe(Annf)

3) dim A = max {qp}.

pe(Annf) p

For an orbit Op of a point p, let m0 = mp, q0 = qp, = vp, = ßp, WS = U W^,

wu = u WU.

Following the classical paper by S.Smale [14], we introduce on the set of periodic orbits of f £ MS (Mn) a partial order

O, -< Oi ^ WS n WU = 0-

J i j

According to Szpilrajn's theorem [16], any partial order (including the Smale order) can be extended to a total order. Let us consider a special kind of such total order on the set of all periodic orbits.

We say that the numbering of the periodic orbits O1, ..., Ok^ of the diffeomorphism f £ £ MS (Mn) is said to be dynamical numbering if it satisfies the following conditions:

1) Oi - Oj i < j;

2) q0.<qo. i< j.

* j

Statement 8 ([2, Proposition 2.6]). For any diffeomorphism f £ MS (Mn) there is a dynamical numbering of its periodic orbits.

2The second part of this statement can be known as a special case of Reeb's theorem [12].

1.3. Orbit spaces

In this section, we present concepts and facts, a detailed proof of which can be found in the monograph [2].

Let f: Mn — Mn be a diffeomorphism and let X c Mn be an f-invariant set. It can be checked directly that the relation x ~ y ^^ 3k G Z: y = fk(x) is an equivalence relation on X. The quotient set X/f induced by this relation is called an orbits space of the action of f on X. Let us denote by PX/f: X — X/f the natural projection. A fundamental domain of the

action of f on X is a closed set DX c X such that there is a set DX satisfying:

1) cl(Dx) = Dx ;

2) fk(DX) n DX = 0 for each k G Z \ {0};

3) U fk (Dx ) = X.

kez

If the projection PX/f is a cover and the orbits space X/f is connected then, due to the monodromy theorem (see, for example, [2, p. 60]), for a loop c c X/f, closed at a point x, there exists its lift c c X, which is a path joining points x G PX/f (x) and fk(x). In this case, the map nX/f: n1(X/f) — Z, defined by the formula nX/f (C]) = k, is a homomorphism which is called induced by the cover Px/f.

Statement 9. Let f and f' be diffeomorphisms defined on an f - and f1 -invariant set X. If h: X/f — X/f' is a homeomorphism for which Vx/f = nx/f'h, then there is a homeomor-

phism h: X — X which is a lift of h (Px/f'h = hpx/f) and such that hf = f 'h.

Statement 10 ([2, Theorem 2.1.3]). Let f G MS (Mn), A be an attractor of f, £sA = = WA \ A, IA = lA/f and D£s be a fundamental domain of the action of f on IA. Then the projection is a cover and the orbits space lA is a smooth closed n-manifold homeomorphic

to Pfe (D£a) . In particular, if the attractor A coincides with a sink orbit, then the manifold l'A is homeomorphic to the following manifolds:

• S1 for n = 1;

• Sn_1x S1 for n> 1, vA = -1;

• Sn-1 x S1 for n > 1, vA = +1.

2. Topology of 3-manifolds admitting G-diffeomorphisms

Recall that G is a class of Morse-Smale diffeomorphisms f: M3 — M3 defined on a closed connected 3-manifold M3 (not necessarily orientable), with a non-wandering set Qf whose all saddle points have the same dimension of their unstable manifolds.

This section is focused on the proof of the main result of this paper.

Theorem 1. Any closed connected 3-manifold M3, admitting a diffeomorphism f gG, is homeomorphic to the 3-sphere.

To prove the main result, let us state some auxiliary facts.

Remark 1. Further, without loss of generality, up to the power of the diffeomorphism, one may assume that Qf consists of fixed points only and for all p e Qf the numbers vp and ^p equal to +1. Moreover, for definiteness, we suppose that the set is empty.

Lemma 1. For any diffeomorphism f eG, the set Q0 consists of a unique sink.

Proof. Let

R = ^2 U Q3.

By virtue of Statement 7, the set R is a repeller of the diffeomorphism f and dim R = 1. It follows from Statement 3 that M3 \ R is connected. On the other hand, according to Statement 4, M3 \ R = WSo. From the above we conclude that the set Q0 consists of a unique sink. □

Let us denote by w the unique sink of the diffeomorphism f eG.

Lemma 2. In the non-wandering set of any diffeomorphism f eG there exists a saddle a such that £'U C .

Proof.

By Lemma 1, the fixed points of the diffeomorphism f admit the following dynamical numbering:

w - ai - ■ ■ ■ - ak - -< ■ ■ ■ -< a„,

1 k 1 s> (2.1)

where Q2 = {ai, ..., ak}, Q3 = {ai, ..., as}. Assume that a = a1. Then, it follows from order (2.1) that

yp e Qf \ w C n Ws = 0.

In other words, £%. can only intersect with W^- By Statement 4 (1), any point x G £%. has to be in the stable manifold of some fixed point. Hence, C £s^. □

Further, let the saddle a G satisfy the conclusion of Lemma 2, and let = cl(£U). It follows from Statements 5 and 4 (2) that = U {w} U {a} is an embedding of the two-dimensional sphere (see Fig. 1). This embedding is smooth everywhere except perhaps the point w. Let Ma = M3 \ .

W0J

Fig. 1. Sphere

Lemma 3. The manifold Ma is disconnected. Proof.

Since for any manifold the notions of connectedness and path-connectedness are equivalent, they will be used interchangeably hereafter.

Step 1. First of all, let us proof that the set La = t^ \ PU is disconnected. Suppose the contrary: any two distinct points x, y GCa can be connected by a path in Ca (see Fig. 2).

Consider the orbits space tst = tst/f of the sink w and let pt = : tst ^ tst, nt =

= : n1 (¿t^ ^ Z. By Statement 10, the map pt is a cover, tt is homeomorphic to S2 x S1

and t% is homeomorphic to the two-dimensional torus (see Fig. 3).

Since tU C tt, it follows that C tst. Moreover, = pt (t"a), which, by Statement 4 implies that tva is a smooth embedding of the 2-torus into tst (see Fig. 3). By Statement 10 homomorphism nt is nontrivial and it follows from its definition that ^ (n1 (t%)) =0.

Then, using Statement 1, one may conclude that bounds in tst a solid torus and hence it divides this orbits space into two connected components. Let us choose a point in each component and denote them by x and y. From their preimages, we take two points x G p-1(x) and y G p-1(y). Since we assumed that Ca is path-connected, there exists a path 7: [0, 1] ^ Ca: Y(0) = x, 7(1) = y. Then, by continuity of pt, the map 7 = pt7: [0, 1] ^ tst \ tU is a path between x and 7 in tst \ tva, that is a contradiction.

Thus, Ca is disconnected.

Step 2. Let us prove that Ma = M3 \ is disconnected as well. Suppose the contrary. Note that dimM.a = 3, because it is an open subset of the manifold M3. Then, by Statement 3, Ma \ R is connected. On the other hand, Ma \ R = (M3 \ £CT) \ R = WSS \ = Ca and it contradicts the conclusion of the previous step. So, Ma is disconnected. □

Let us introduce a diffeomorphism a: R3 —>• R3 given by the formula a(x, y, z) = (|, |). It has a unique non-wandering point, a sink 0(0, 0, 0). Let t = R3 \ O.

As before, let f G G, let a satisfy the conclusion of Lemma 2 and = cl(tU). By Statement 7, sphere is an attractor of the diffeomorphism f with its basin W£ = WS U W£.

Let = \ S0

Lemma 4. The manifold £- consists of two connected components £1, £2, and for each of the components £i there exists a diffeomorphism hi: £i — £ conjugating f with a|r

Proof.

By virtue of Statement 10, the orbits space £f^ = £— /f is a smooth closed 3-manifold. Let us prove that £- = S2 x S1 U S2 x S1.

Fig. 4. The neighborhood Ka = S2 x [0, 1] of the sphere Sa

By Statement 2, the attractor has a neighborhood Ka c WS diffeomorphic to S2 x [0, 1]

(see Fig. 4). Let us show that there exists a natural number N such that fN(x) G int Ka for any x G Ka. Since dKa c W£ , it follows that for all x G dK there exist a closed neighborhood Ux c dKa and a natural number vx such that for any v ^ vx it is true that fv(Ux) c int Ka. Due to the compactness of dKa, there exists a finite subcover of dKa in {Ux, x G dKa}. Thus, one may choose the desired number N as the maximum of numbers vx corresponding to the neighborhoods of Ux in the chosen subcover. Without loss of generality, we assume the number N to be 1, then f (K^c int Ka (see Fig. 5). It follows from Lemma 3 that the sphere separates in Ka the connected components of its boundary. Whence, according to [6, Theorem 3.3], Ka \ int f (Ka) = S2 x [0, 1] U S2 x [0, 1]. It follows from the construction that the manifold Ka \ int f (Ka) is a fundamental domain of the action of f on the space £ - . Then by

Statement 10, £- = S2 x S1 U S2 x S1.

We denote by £1, £2 the connected components of the set £ - and by ^ : ^1(£i) ^ Z, i = 1, 2

i

the restriction of the homomorphism to ^1(£i). Let us assume £i = p(£i). It follows from

the definition of the homomorphism ^ that it is an isomorphism, hence the set £i is connected.

Let £ = £/a. Since the point O is a sink of the three-dimensional map a, then by Statement 10, £ = S2 x S1 and the homomorphism ^1(£) Z is an isomorphism. Therefore, the manifolds £i

and £ are homeomorphic smooth 3-manifolds, hence there exists a diffeomorphism hi: £i — £i

(see [9]). Without loss of generality, we assume that nhi = ^ (otherwise, one may consider its

composition with a diffeomorphism d: S2 x S1 — S2 x S1, given by the formula d(s, r) = (s, —r)).

By Statement 9, there exists a lift hi: li — l of the diffeomorphism hi, smoothly conjugating f \e with a\r □

Now let Ma = R3 U Ma U R3, Ma = R3 U Ma U R3 and let pa:AJa ^ Ma be the natural

h1 h2

projection.

Lemma 5. The space Ma consists of two connected components Ml, M%, with both of them being a closed smooth 3-manifold such that

M3 = .

Moreover, the manifold Ml, i = 1, 2, admits a diffeomorhism fi: Ml — Ml belonging to the class G and having less saddle points than f.

Proof.

It follows from Lemma 4 that the manifold Ma is a disjoint union of two manifolds, and hence the space Ml has exactly the same number of connected components, which we denote as Mil and Ml. Since hi glues open subsets of 3-manifolds, then the projection pa induces the structure of a smooth 3-manifold on Ma. Since the glued manifolds have no boundary, the manifold Ma has no boundary as well. Due to the compactness of M3, the manifold Ma is closed. Moreover, it follows directly from the definition of the connected sum that M3 = Ml#M%.

According to [10, Theorem 18.3 (the pasting lemma)], the map fl: M1 — M1 , defined by the formula

,, , [a fp-V))), if x e Pi (M1),

fa(x) = < _

{ Pa(a(pa(x))), if xe pa (Apr\Ma),

is a diffeomorphism. Let fi = fl\Ma (see Fig. 6, 7). By the construction, the diffeomorphism fl is smoothly conjugated with / on pa(Ma) and with a on pa(Ma \ Ma) (Oi (i = 1,2) is a point

conjugated with the fixed sink 0(0,0,0) of a). Hence, fS G G and its non-wandering set have one saddle point less than the non-wandering set of the diffeomorphism f. □

Now let us prove the main result of this paper.

Theorem 1. Any closed connected 3-manifold M3, admitting a diffeomorphism f gG, is homeomorphic to the 3-sphere.

Proof.

Let f: M3 — M3 be in G. Moreover, we assume that f satisfies the Remark. We prove Theorem 1 by the induction on the number k of the saddle points of the diffeomorphism f. Base case. k = 0.

It follows from Statement 6 that M3 is homeomorphic to the 3-sphere. Step of induction. k > 0.

Inductive hypotheses. Any diffeomorphism from the class G in which the number of saddle points is less than some natural number k can be defined only on a manifold homeomorphic to the 3-sphere.

The diffeomorphism f: M3 — M3 lies in G and has exactly k saddle points. By Lemma 2, there exists a saddle a whose unstable manifold has no heteroclinic intersections. This saddle was chosen according to the order 2.1. By Lemma 5, M3 = MS#Mf and the manifold Mf, i = 1, 2, admits a diffeomorphism f: Mf — Mf from the class G which has less saddle points than f.

In this case, it follows from the inductive hypotheses that Mf = S3. Thus, M3 is a connected sum of 3-spheres and hence M3 = S3. □

Conflict of interest

The authors declare that they have no conflicts of interest.

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