On obstructions to the existence of a simple arc, connecting the multidimensional Morse-Smale diffeomorphisms Текст научной статьи по специальности «Математика»

Динамические системы, 2017, том 7(35), №2, 103-112 MSC 2010: 37C05, 37D15

On obstructions to the existence of a simple arc, connecting the multidimensional Morse-Smale diffeomorphisms1

A. Dolgonosova, E. Nozdrinova, O. Pochinka

Higher School of Economics

Nizhny Novgorod. E-mail: [email protected], [email protected], [email protected]

Abstract. In this paper we consider Morse-Smale diffeomorphisms defined on a multidimensional nonsimply connected closed manifold Mn, n > 3. For such systems, the concept of trivial (nontrivial) connectedness of their periodic orbits is introduced. It is established that isotopic trivial and nontrivial diffeomorphisms can not be joined by an arc with codimension one bifurcations. Examples of such pair of Morse-Smale cascades on the manifold Sn_1 x S1 are constructed. Keywords: Morse-Smale diffeomorphisms, bifurcation, smooth arc

1. Introduction and a formulation of results

The present paper has deal with a solution of the Palis-Pugh problem [10] on the existence of an arc with a finite or countable set of bifurcations connecting two Morse-Smale systems on a smooth closed manifold Mn. S. Newhouse and M. Peixoto [8] proved that any Morse-Smale vector fields can be connected by a simple arc. Simplicity means that the arc consists of the Morse-Smale systems with the exception in a finite set of points in which the vector field deviates by at least way (in a certain sense) from the Morse-Smale system. Below we give a definition of the simple arc for discrete Morse-Smale systems.

Let Diff (Mn) be the space of diffeomorphisms on a closed manifold Mn with Cl-topology and MS(Mn) be the subset of Morse-Smale diffeomorphisms. Smooth arc in Diff(Mn) is a smooth map

£: Mn x [0,1] ^ Mn,

that is a smoothly depending on (x,t) E Mn x [0,1] family of diffeomorphisms

{£t E Diff(Mn),t E [0,1]}.

The arc £ is called simple if £t E MS(Mn) for every t E ([0,1] \ B), where B is a finite set and for t E B diffeomorphisms undergo bifurcations of the following types: saddle-node, doubling period, heteroclinic tangency (see section 3 for details).

1rThe construction of simple arc was financial supported by RSF (Grant No. 17-11-01041), a study of the simple connectivity components was supported by the fundamental research program of the HSE (project 90) in 2017.

© A. DOLGONOSOVA, E. NOZDRINOVA, O. POCHINKA

As follows from the papers by Sh. Matsumoto [6] and P. Blanchar [1], any oriented closed surface admits isotopic Morse-Smale diffeomorphisms, who can not be connected by a simple arc. In the paper by V. Grines and O. Pochinka [3] necessary and sufficient conditions were found for the fact that the Morse-Smale diffeomorphism without heteroclinic intersections on the 3-sphere is connected by a simple arc with the "source-sink" diffeomorphism. They also constructed examples of Morse-Smale diffeomorphisms on the 3-sphere that are not joined by a simple arc due to the wild embeddings of all saddle separatrices for one of them.

In the present paper we consider a f G MS(Mn) which defined on a multidimensional not simply connected manifold Mn for n > 3. Denote by MS0(Mn) the class of homotopic to identity Morse-Smale diffeomorphisms. Let f G MS0(Mn). Through Ox we denote the orbit of the point x G Mn under the diffeomorphism f. Let 7 G Hi(Mn).

Following to [6], we say that a periodic orbit Op is homologically y-related to a periodic orbit Oq if there is a curve c С Mn such that dc = {q} — {p} and for some integer N such that fN (p) = p and fN (q) = q, [fN (c) — c] = Ny. The definition independents on the choice c,N,p G Op, q G Oq. We say that f is trivial if all periodic orbits of the diffeomorphism f are 0-related otherwise f is nontrivial.

In section 2 isotopic diffeomorphisms fo,fi G MS0(Sn-1 x S1),n > 3 will constructed, one of which is trivial, the other is nontrivial. The main result of this paper is the following theorem.

Theorem. There is no simple arc connecting a trivial diffeomorphism with a nontrivial diffeomorphism from the class MS0(Mn).

2. The construction of a trivial-nontrivial pair of isotopic diffeomorphisms

Let

Sn = {(xi,x2,...,xn+i) G Rn+1 : xi2 + X22 + ••• + Xn+12 = 1}. For the sphere S1 also consider its complex form

S1 = {ег2пвG [0; 1]}. Define a diffeomorphism ф: [0; 1] ^ [0; 1] by the formula:

Ф(в)=в+в (в—1)(Р—2) •

Dynamics of a diffeomorphism of the circle sending a point ег2жв to the point ег2п^(в) is shown in Figure 1. Notice that the diffeomorphism ф is isotopic to the identity since there is an isotopy ф1: [0; 1] ^ [0; 1] given by the formula:

ф*(в) = в + te (в — 1)(p — 1) ,

W

a

Fig. 1. Source-sink on the circle

for which ф0 = id and ф1 = g.

For n> 2 let us define a diffeomorphism ф : Sra_1 ^ Sra_1 by the formula

ф(хг,х2,... ,Xn)

f 4xi

\5 - 3:

4x2

5 3xn 5 3xn

5xn 3 5 3xn

.

Figure 2 depicts the dynamics of such a diffeomorphism for n = 3. The diffeomorphism

ОС

CO

Fig. 2. Source-sink on the sphere

0 is also isotopic to the identity since there exists an isotopy 0t: Sn_1 ^ Sn_1 given by the formula:

( ( xi(1 + 3t) X2(1 + 3t) t(4 - 3) + xn \

...,Xn)=\^ ^(4 - 3xn) + 1' t(4 - 3xn) + 1 ,---,t(4 - 3xn) + 1) '

for which 0o = id and 01 = 0.

Fig. 3. Dynamic of the trivial diffeomorphism f0: S2 x S1 ^ S2 x S1

Let us consider the Cartesian product of our spheres Sn 1 x S1 and define a diffeomorphism f0: Sn-1 x S1 ^ Sn-1 x S1 by the formula:

fo{xi,...,Xn,ei2nß) = (^(xi,...,xn),e

).

By construction, the diffeomorphism f0 is a Morse-Smale diffeomorphism and its non-wandering set consists of one sink, one source, and two saddle points whose invariant manifolds do not intersect. Figure 3 shows a phase portrait for the case n = 3. Since there is an isotopy

fo,t(xi ,...,Xn, ei2nß) = (^t(xi,..., Xn), e

г2жфгф)

)

such that f0>0 = id and f0t1 = f0, hence the diffeomorphism f0 is isotopic to the identity. In addition, it is easy to see that all its fixed points are trivially related.

On the sphere Sn-1 let us consider a subset of points (x1,... ,xn), for which xn E [0, 3] (see Figure 4 for the case n = 3). It is diffeomorphic to n-dimensional annulus, denote it by L. In the cartesian product Sn_1 x S1 we obtain a subset K = L x S1, xn E [0, 3]. We define a diffeomorphism p: Sn_1 x S1 ^ Sn_1 x S1 which is the identity outside K and on K is given by the formula:

p(x1,...,xn,e'l2nl3) = (x1,...,xn,ei2n(l3+3 x")).

We show that the diffeomorphism p is isotopic to the identity. To do this, we construct the isotopy pt as follows:

1) pt = id on the set K" = {x1,... ,xn,ei2n?) E Sn-1 x S1 : xn < 0};

2) pt(x1,... ,xn,ei2n'6)(x-]_,... ,xn,ei2n(l3+3Xntt)) on the set K;

Fig. 4. The parts of S2 x S1

Fig. 5. Dynamic of the nontrivial diffeomorphism f0 : S2 x S1 ^ S2 x S1

3) фг (x1,...,xn,ei2nß ) = (x1,... ,xn,ei2n(ß+t)) on the set K+ = {(x1,... ,xn,ei2nß ) G Sn_1 x S1 : xn G [§,1)}.

From the construction of isotopy фг it's clear that ф0 = id^1 = ф. We define a diffeomorphism f1 : Sn_1 x S1 ^ Sn_1 x S1 formula

/1 = Ф/о.

Then the diffeomorphism f1 is isotopic to the identity by means of the isotopy f1tt = ф/о,г. By construction, the diffeomorphism f1 is a Morse-Smale diffeomorphism, its

non-wandering set consists of one sink, one source, and two saddle points whose two-dimensional manifolds intersect along a countable set of compact heteroclinic curves. In addition, the saddle points are not trivially related. In Figure 5 we show the Morse-Smale diffeomorphism f\ for the case n = 3.

3. Simple arcs

Let us consider a smooth map £ : Mn x [0,1] ^ Mn — a smooth arc such that G MS(Mn) for every t G ([0,1] \ B), where B is a finite set. For a generic set of such arcs, the diffeomorphism b G B has the finite non-wandering set, has no cycles and under the direction of motion along the arc, undergoes bifurcations of the following types: saddle-node, doubling period, heteroclinic tangency, for exact details see, for example, [7]. Below we give an information about these bifurcations, for exact details see, for example, [5].

Let p be a fixed point of a diffeomorphism f : Mn ^ Mn. Differential Dfp induces a decomposition of the tangent space TpMn into a direct sum of invariant subspaces

Tp Mn = Eu © Ec © Es.

Linear maps Dfp\Eu, Dfp\Ec, Dfp\Es have eigenvalues, respectively, outside, on the boundary, inside the unit disc. There exists a unique smooth invariant submanifold W'pU (WpS) of the manifold Mn tangent to Eu (Es) at the point p and possessing the property

WU = {y G Mn : lim fk(y) = p} (WpS = {y G Mn : lim fk(y) = p}).

It is called by unstable (stable) manifold of the point p. In particular, if dimEc = 0, the point p is hyperbolic. Otherwise, there exists a smooth invariant submanifold W£ of the manifold Mn tangent to Ec at the point p. It is called the central manifold of a nonhyperbolic fixed point. A central manifold is not unique but the maps f \W£ and f \wc are topologically conjugated for any central manifolds W£ and W^.

The central, stable and unstable manifolds of a periodic point of period k is defined as the corresponding manifolds of this point as a fixed point of the diffeomorphism fk.

In the explanatory drawings, double arrows schematically show the directions of motion with exponential contraction and expansion, and single directions indicate the directions of motion along the central manifold of the nonhyperbolic point.

1. All periodic orbits of the diffeomorphism are hyperbolic with an exception in a one orbit Op of the period k for which all eigenvalues of (Dfk)p different from 1 by absolute values except one A =1. The stable and the unstable manifolds of different periodic orbits of the diffeomorphism intersect transversely and WpS fl Wu = {p}. The transition through is accompanied by a confluence and further disappearance of hyperbolic periodic points of the same period. This bifurcation is called saddle-node (see Figure 6).

V

V

V

V

->

Fig. 6. Saddle-node bifurcation

2. All periodic orbits of the diffeomorphism are hyperbolic with an exception in a one orbit Op of the period k for which all eigenvalues of (Dfk)p different from 1 by the absolute value except one A = —1. The stable and the unstable manifolds of different periodic orbits of the diffeomorphism intersect transversely and Wp H Wu = {p}. By passing through along the central manifold an attractor becomes a repeller and a periodic hyperbolic orbit of the period 2k is generated. Such a bifurcation is called a doubling period (see Figure 7).

\ f V

v \ V t \r

i < ;

Fig. 7. Doubling period bifurcation

3. All periodic orbits of the diffeomorphism are hyperbolic, their stable and unstable manifolds have transversal intersections everywhere except for one trajectory along which the intersection is quasi-transversal. Such a bifurcation is called a bifurcation of a heteroclinic tangency (see Figure 8).

4. Proof of the main result

In this section we prove Theorem. Namely, we consider two homotopic to identity Morse-Smale diffeomorphisms f0, fi given on a not simply connected n-manifold Mn such that fo is a trivial and the diffeomorphism fi is a nontrivial. Let us prove that there is no simple arc joining the diffeomorphisms f0 and fi.

Proof. Assume the contrary: f0 and fi can be joined by a simple arc. Then on this arc there are two Morse-Smale diffeomorphisms g0 and gi such that:

1) g0 and g1 can be connected with a simple arc {gt}te[o,ij with only one bifurcation point, g 1 = h (see Figure 9);

2) g0 is a trivial diffeomorphism, g1 — nontrivial.

I

*

->■

Fig. 8. Bifurcation of the heteroclinic tangency

Fig. 9.

From the description of possible for a simple arc bifurcations (see the section 3) it follows that the non-wandering set Qh of the diffeomorphism h has a periodic orbit saddle-node Op that is not trivially related to some (and, consequently, with any) other periodic orbit; in a comparison with the Morse-Smale diffeomorphism gt, 0 < t < |,

two periodic orbits of the different indexes appear for the Morse-Smale diffeomorphism

gt, 2 <t< L

Let m,k E N be the dimensions of the unstable, stable manifolds W;of the point p. Then m + k = n + 1. By an analogy with the properties of Morse-Smale diffeomorphisms (see, for example, [2] Theorem 2.1), one can establish that

Mn = и wu = u w

x£Qh x£Qh

Since there are no cycles for the diffeomorphism h, there are hyperbolic points q,r E such that

w; n W* = 0, W* n WU = 0. (4.1)

These points are not node as in this case p will 0-related with q or r by means a curve c on W* or W;. From the transversality of the intersection of the stable and the unstable manifolds of non-wandering points of the diffeomorphism h it follows that the invariant manifolds W* and W; are arbitrarily close to the point p and, therefore, are close each to other, this means that W* n W; = 0.

It means that W * n W; = 0 for every diffeomorphism gt for t near 1. Moreover, if t < 2 then q and r are the nearest saddle points, that is the intersection W * n W; = 0 consists of a finite number connected components, denote it N. Then there is compact fundamental domains F* of f \w^\g containing exactly N connected components in the intersection with W;. For t = 2 the conditions (4.1) imply that there is a compact subset Cu of W; \ r such that the number of the connected components in the intersection F* n Cu is greater than N + 2. Due to transversality condition for gt, t sufficiently near 2 the number of the connected components in the intersection F* n Cu is preserving, that is contradicting the definition of N. □

References

1. Blanchard, P. R. Invariants of the NPT isotopy classes of Morse-Smale diffeomorphisms of surfaces, Duke Math. J. 47:1, 33-46 (1980).

2. Grines V., Medvedev T, Pochinka O. Dynamical Systems on 2- and 3-Manifolds. Springer International Publishing Switzerland, 2016.

3. Grines, V. Z., Pochinka, O. V. On the simple isotopy class of a source-sink diffeomorphism on a 3-sphere. Mathematical Notes 94, No.6, 828-845, (2013).

4. Grines V., Pochinka O. Morse-Smale cascades on 3-manifolds, Russian Mathematical Surveys 68, No.1, 117-173 (2013).

5. Hirsch, M. W., Pugh, C. C., Shub, M. Invariant Manifolds, Lecture Notes in Math., 583, Berlin: Springer-Verlag, 1977.

6. Matsumoto, S. There are two isotopic Morse-Smale diffeomorphism which can not be joined by simple arcs, Invent. Math. 51, No.1, 1-7 (1979).

7. S. Newhouse, J. Palis and F. Takens. Bifurcations and stability of families of diffeomorphisms, Publ. I.H.E.S. , Vol. 57, 5-71 (1983).

8. Newhouse, S., Peixoto, M. M. There is a simple arc joining any two Morse-Smale flows, Trois etudes en dynamique qualitative, Asterisque, 31, Soc. Math. France, Paris, 15-41 (1976).

9. Palis, J., de Melo, W. Geometric theory of dynamical systems: An introduction. New York: Springer, 1982.

10. Palis J., Pugh C. C. Fifty problems in dynamical systems, Dynamical Systems-Warwick. Lecture Notes in Math., 468, pp. 345-353, Berlin: Springer-Verlag, 1974.

Получена 15.06.2017