Determination of the Homotopy Type of a Morse – Smale Diffeomorphism on a 2-torus by Heteroclinic Intersection Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 4, pp. 465-473. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd210408

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 37D05

Determination of the Homotopy Type of a Morse-Smale Diffeomorphism on a 2-torus by Heteroclinic Intersection

A. I. Morozov

According to the Nielsen-Thurston classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types: T1) periodic home-omorphism; T2) reducible non-periodic homeomorphism of algebraically finite order; T3) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; T4) pseudo-Anosov homeomorphism. It is known that the homotopic types of homeomorphisms of torus are T1, T2, T4 only. Moreover, all representatives of the class T4 have chaotic dynamics, while in each homotopy class of types T1 and T2 there are regular diffeomorphisms, in particular, Morse-Smale diffeomorphisms with a finite number of heteroclinic orbits. The author has found a criterion that allows one to uniquely determine the homotopy type of a Morse-Smale diffeomorphism with a finite number of heteroclinic orbits on a two-dimensional torus. For this, all heteroclinic domains of such a diffeomorphism are divided into trivial (contained in the disk) and non-trivial. It is proved that if the heteroclinic points of a Morse-Smale diffeomorphism are contained only in the trivial domains then such diffeomorphism has the homotopic type T1. The orbit space of non-trivial heteroclinic domains consists of a finite number of two-dimensional tori, where the saddle separatrices participating in heteroclinic intersections are projected as transver-sally intersecting knots. That whether the Morse-Smale diffeomorphisms belong to types T1 or T2 is uniquely determined by the total intersection index of such knots.

Keywords: Morse-Smale diffeomorphisms, Nielsen-Thurston theory, heteroclinic intersections, homotopy class of a map

1. Introduction and formulation of the results

Let X be a topological space. Two continuous maps f0, f1: X ^ X are called homotopic (denoted by f0 ~ f1), if there is a continuous map H: X x [0, 1] ^ X such that H(x, 0) =

Received December 04, 2021 Accepted December 13, 2021

This work was supported by a grant from the Russian Science Foundation, contract 21-11-00010.

Andrei I. Morozov aimorozov@hse.ru

National Research University Higher School of Economics ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150 Russia

= f0(x) and H(x, 1) = f1(x). A map H is called a homotopy between f0 and f1. For every fixed t G [0, 1] we set H(x, t) = Ht(x). Denote by [f] the homotopy class of the map f, that is f' G [f] ^ f - f.

The set of homotopy classes of homeomorphisms of the space X forms the group r(X) with the group operation [f1 ] o [f2] = [f1 o f2], which is called mapping class group. Thus,

r(X): <=/ Homeo(X)/-,

where Homeo(X) is the set of homeomorphisms of the space X onto itself.

A homotopy Ht: X — X, t G [0, 1] is called an isotopy if, for any t the map Ht is a home-omorphism of the space X.

According to the Nielsen-Thurston classification [1] there are four types of homotopy classes of surface homeomorphisms. Each subset contains only homotopy classes of homeomorphisms of one of the following types:

T1) periodic homeomorphism;

T2) reducible non-periodic homeomorphism of algebraically finite order;

T3) reducible homeomorphism that is not a homeomorphism of algebraically finite order;

T4) pseudo-Anosov homeomorphism.

Let M2 be a connected compact (possibly with boundary) orientable surface of genus g. Recall that a homeomorphism h: M2 — M2 is called a periodic homeomorphism, if there exists m G N, such that hm = id, where id is the identity transformation. The smallest such number m is called the period of the periodic homeomorphism.

A homeomorphism h: M2 — M2 is called a reducible by a system C of disjoint simple closed curves Cj, i = 1, ..., l, non-homotopic to zero and pairwise non-homotopic to each other if the system of curves C is invariant under h.

A reducible non-periodic homeomorphism h: M2 — M2 is called a homeomorphism of algebraically finite order, if there exists an h-invariant neighbourhood C of curves of the set C, consisting of the union two-dimensional annuli and such that for each connected component M2, j = 1, ..., n of the set M2 \ int C there is a number mj G N such that hmj \M2 : Mj — Mj is a periodic homeomorphism.

Since the Nielsen-Thurston theory was originally described for surfaces of genus g ^ 2, it is worth mentioning the book [2], in which this theory is also generalized to a torus and a fact is presented that the homotopic types of homeomorphisms of torus are T1, T2, T4 only (Theorem 13.1, p. 369).

Recall that the diffeomorphism f: Mn — Mn, given on a smooth, closed, connected n-ma-nifold (n ^ 1) Mn is called Morse-Smale diffeomorphism, if

1) the non-wandering set Q/ consists of a finite number of hyperbolic orbits;

2) the manifolds Ws, W^ intersect transversally for any non-wandering points p, q.

In the set of periodic orbits of such a diffeomorphism, one can introduce a complete order relation, which is an extension of S. Smale's partial order [3]. Namely, let Oi, Oj be the periodic orbits of the Morse-Smale diffeomorphism f. One can say that the orbits Oi, Oj are in the

relation -< (Oi -< O,), if WS n WU = 0. A sequence of different periodic orbits Oi = Oi ,

i j 0

O, , ..., O, = O. (k ^ 1), such that O, — O, — ... — O, is called a chain of the length k,

n lh j l0 n lh

connecting periodic orbits Oi and Oj. The chain connecting the periodic orbits of saddle points will be called saddle chain. Due to the finiteness of the non-wandering set, for any diffeomor-phism f G MS(Mn) there is a well-defined number equal to the length of the maximal saddle chain, which is denoted by beh(f).

Denote by MS1(T2) the set of orientation-preserving Morse-Smale diffeomorphisms f defined on the two-dimensional torus T2 and having beh(f) ^ 1. According to [4, 5] the Morse -Smale diffeomorphism with beh(f) ^ 1 exists in both the first T1 and the second T2 Nielsen -Thurston type. We state F = fd: T2 ^ T2, where d G N is the smallest number such that Vx G Qf: f d(x) = x and the map f \Wu preserves orientation. The dynamics of any diffeomorphism F G MS1(T2) can be represented as follows. The set QF of periodic orbits of the map F can be splatted into subsets QiF, i g{uj, s, u, a}:

• Q'F — the set of all sink orbits;

• QSF — the set of saddle orbits, whose unstable sets do not contain heteroclinic points;

• Q'F — the set of the remaining saddle orbits of the system;

• QF — the set of source orbits.

Let

Af = QF U W' , Rf = QaF U WSu , VF = T2 \ (Af URF).

By virtue of the paper [6], the sets AF and RF are an attractor and a repeller of the diffeomorphism F, respectively. All heteroclinic points of the diffeomorphism F belong to the set VF, which consists of a finite number of connected components Vi, i = 1, ..., m. Each component Vi is homeomorphic to an open two-dimensional annuli (see, for example [7]) and has a fundamental domain Ki = [0, 1] x S1, bounded by the circles bi ^ {0} x S1, F(bi) = {1} x S1, so that

Vi = U Fn(Ki).

n£Z

The component Vi of the set VF will be called trivial, if the curve bi is homotopic to zero on the torus, and non-trivial otherwise. The component Vi of the set VF will be called a heteroclinic domain, if the component Vi contains at least one heteroclinic orbit.

The following lemma is proved in section 2.

Lemma 1. If all heteroclinic domains of a diffeomorphism F are trivial, then the diffeomorphism F is isotopic to the identity map and, therefore, the homotopy class [f ] is of the type T1.

In what follows, assume that the diffeomorphism F has non-trivial heteroclinic domains. Let us renumber the connected components of the set VF: Vi, i = 1, ..., m in such a way that the first m0 components are non-trivial heteroclinic domains, and the rest are trivial.

Consider the covering of the torus q: R2 ^ T2, representing the torus as a quotient group of the plane R2 by the group of integer shifts. Let

a = q(Ox), b = q(Oy).

Then the curves a, b are generators on the torus with a positive traversal direction inherited from the coordinate axes, and their homotopy classes are [a] = (1, 0), [b] = (0, 1). Then the homotopy class [c] of any simple closed curve c G T2 is determined by two numbers [c] = (^, v), H G Z, v G N, where n, v denote the number of revolutions around the generators a, b. Moreover, the numbers n, v are coprime, that is, (^, v) = 1.

Without loss of generality (see, for example, [8]), we assume that all curves bi, i g{1, ..., m0} are of the form {y} x b, and the generator a intersects the annuli Ki along the segment ai, connecting the point Yi = yi x {0} with the point F(Yi). We state Vi = Vi/F = Ki/F is the quotient

space by the action of the diffeomorphism F. Denote by pi: Vi — V the natural projection, which

is also a covering map for the space Vi. Then (see, for example, [7]) Vi is homeomorphic to the

two-dimensional torus and the curves ai = pi(ai), bi = pi(bi) are generators on V such that [ai] =

= (1, 0) and [bi] = (0, 1) (see, Fig. 1), while the orientations on ai and bi are induced by the orientations on a and b.

Fig. 1. The projection q defines generators a, b on the torus T2 as images of the axes Ox and Oy. The

projection pi defines the generators ai, bi on the torus VVi as the images of the segments ai, bi. The

connection between the basis of the torus T2 and of the torus V is established in a natural way using the map pi for all i g 1, ..., m0

Denote by Lf (Lf) the union of the stable (the unstable) separatrices belonging to the domain Vi. Let Lf = pi(Lf) (Lf = pi(Lf)). Denote by rf and rf the number of curves in the sets Lf and Lf respectively. Since the curves of the set Lf (Lf) are pairwise disjoint and F = fd, then any curve if C Vf (if C Lf) is of homotopy type (5i, df) ((Si, df)), where 5i = ±1 and d is a multiple of df (df). Since the separatrices are oriented in a natural way from the repeller to the attractor, the number 5i always matches for if and if. For any two curves if C Lf, if C Lf at each point x G if nif we define v'X (vf) as a tangent vector to if (if) at the point x, directed according to the orientation if (Vf). Then we define an index £x of the point x equal to 1 (—1), if the orientation that an ordered pair of vectors (vf, vsx.) defines on the manifold V/ri, coincides (does not coincide) with the orientation of the manifold Vi. Then we will call the number

xeifntv

by the index of heteroclinic intersection of curves If and if. The index of the heteroclinic intersection of curves coincides for any two curves if and if from the sets Vf and Lf respectively. We state

m0 i=1

and will call the number £f by the heteroclinic intersection index of the diffeomorphism F G G MS1 (T2). The following result shows that the heteroclinic intersection index uniquely determines the type of the homotopy class [f ] of the diffeomorphism f G MS1 (T2).

Theorem 1. Let f G MS1(T2) and F = fd. The homotopy class [f ]

is of type T1, if and

only if £f = 0. The homotopy class [f ] is of type T2, if and only if £f = 0.

2. Proof of the Lemma 1

Let f e MS1(T2), F = fd: T2 ^ T2 and all heteroclinic domains Vi, i e{1, ..., m'} of the diffeomorphism F are trivial. Then the curves bi, i e{1, ..., m'} bound the disks Di C T2 such that F(Di) C int(Di) or F-1(Di) C int(Di). Since the domain Vi contains heteroclinic points, then Di contains at least one saddle point. By virtue of [9, Lemma 6.3, p. 1115]1 there is an isotopy Ht: T2 ^ T2, t e [0, 1], such that H0 = F, H1 = F1, where F1: T2 ^ T2 Morse-Smale diffeomorphism coinciding with F on the set T2 \ (D1 U ... U Dm,) and such that each disk Di, i e{1, ..., m'} contains a single fixed sink or source of the diffeomorphism F1 (see Fig. 2). By construction, the diffeomorphism F1 is a gradient-like diffeomorphism, all non-wandering points of which are fixed and have a positive orientation type. According to [10, Theorem 4.2], the diffeomorphism F is isotopic to the identity map. Since F = fd, then fd = id. Thus, the matrix fx is periodic and, therefore, the diffeomorphism [f] has the homotopy type T1.

Ht

Fig. 2. An example of the application of the lemma 1. On the left is the phase portrait of the diffeomorphism F on the torus T2 with marked curves bi; i e{1, ..., 4} and the disk D3 containing the repeller connected component R. On the right is a diffeomorphism F-± on the torus T2, which coincides with the diffeomorphism F everywhere except for the disk D3, on which, using the isotopy, we trivialize the connected component of the repeller to the only sink a

3. Proof of the Theorem 1

Let f e MS1(T2) and F = fd. Without loss of generality, we will assume that Vi = S1 x R, F\V. (s, r) = (s, r + 1) and Ki = S1 x [0, 1]. We state U+ = S1 x [0, +ro).

i

Consider the set EF of such diffeomorphisms G: T2 ^ T2 such that G coincides with F in

mo

some neighborhood U+ of the attractor AF, containing |J Ki, in some neighborhood U- of the

i=1

repeller RF and QF = QG. For any diffeomorphism G e EF we denote by LsG i and L— i the set of stable and unstable saddle separatrices of the diffeomorphism G, that belong to the set Vi. Then each connected component of the orbit spaces LsGi/G and L—i/G is homeomorphic to a circle.

Since Ki is a common fundamental domain of diffeomorphisms G\V and F\V, then LsG i/G is

_ i i '

1 Lemma 6.3 from [9] asserts that any gradient-like diffeomorphism f on the surface M2, with an attractor A belonging to the disk D and such that f (D) C D, A = p| fn(D), is connected by a stable

arc (diffeotopy admitting only flip and saddle knot bifurcations) with some gradient-like diffeomorphism g, coinciding with f on M2 \ D and having a single non-wandering point in the disk D, which is a fixed

homeomorphic to pi(LG i n Ki) and LG i/G is homeomorphic to pi(LG i n K). Since G\U + = = F\u +, then LG j n U+ = U Fn(LG j n Kj), LG i n U+ = U Fn(LG j n K) and, therefore,

Pj(LG, j n Kj) = Pj(LSG j j n U+), Pj(LG , j n Kj) = Pj(LG , j n U+). Let

LsG,j = Pj(LsG,j n U+), LG,j = Pj(LG,j n U+). Let us prove an auxiliary lemma.

Lemma 2. Let h: Vj — Vj is isotopic to the identity diffeomorphism. Then there exists a diffeotopy Ct C EF such that C0 = F, ^ = Fl and L'F j = h(LFj), LsF j = LsFj.

Proof. Take an open cover D = {D1, ..., Dq} of the manifold V by two-dimensional disks

so that P-1(Dj) n S1 x {rj, Vj + 1} = 0 for some v- e R. According to [11, Fragmentation lemma]

there exist diffeomorphisms w1, ..., wq: VVi — smoothly isotopic to the identity and with the following properties:

i) for each j e{1, . .., q} there is a smooth isotopy {Wj t}, identical outside Dj and joining the identity map and Wj;

ii) h = W1... Wq.

Let U+ j = Vj n U+, UJj = Vj n UJ and denote by r—, r+ real positive numbers such that S1 x (-m, -r—) C U-^ and S1 x (-r+, +rc>) C U+.

Take q disks D1, ..., Dq one in each of the sets p—l(D 1), ..., p—l(Dq) so that Dj C Kj, with Kj = S1 x (-rj - 1, -rj), r— < r1 and rj + 1 < rj+1 for j e{1, ..., q — 1}. Let Wj t: V — V be a diffeomorphism that coincides with (pj^) 1Wj tPj on the set Kj and coincides with the

identity map outside the set Kj. Let

Ct = W1,t . . . Wq, tF.

By the construction Ct C Ef and C0 = F. Let F1 = C1. Let us show that VFi j = h(VF j), Ls = Ls .

lf1 , j = lf, j.

Since the manifolds LsFi j are obtained from the local manifolds LsFi j n U+ j by iteration due to the map F—1 and LsFi, j n U+, j = LsFjj n U+, j, then LsFi, j n U+ = LsFjj n U+ and therefore LsFi, j =

= Ls .

We state UJ = S1 x (-<x>, -rq - 1) and Fj = Wj .. .wqF, j e {1, ..., q}. By the construction Fj e Ef. Then, to prove the lemma, it suffices to show that VF.i, = Wj(VF+1 j), where Fq+1 = F.

~ —k~~~~ k+ ~ Take natural numbers k—, k+ such that K— = F- j (K •) C UJ, K+ = F-j (K •) C U+. We

j j j j j i j j j i

state k = k-

+ k+. Since the manifolds LF , are obtained from the local manifolds LF , n U—

j 3 3 j _ Pj+l'j Fj

by iteration due to the map F- and LF j n U— = LF j n U—, then (see Fig. 3)

j j' j+i' k

ji = Pj 1(LUj+1j n K-), k

LUFj j = pjFjj (LFj+l'i n KJ).

ткУ7рМ*)

F-\y)

a)

b)

Fig. 3. The figure (a) shows an illustration of the action of the formula step by step. x -> y

F j

г—+ wj

F j J

phism Fx

z —> z'-> Fkj (z'). The figure (b) shows the form of separates LsF LF . for the diffeomor-

7/J - I- + 1 ' 1 '

Z— +

F J

Then

k+ k— ~ LI i = VlF+ WjFj++ № i П К-)

4+1'

: ptF k+ Wj F-k+ Fj ,(LUF г П K-) = ptF k+ Wj F-k+ ( pt\R+ )-1'р] ^ г П К-)

3

PiW

Ык)(Lw)= Wj (LI г)

г^] \

- 3+1

j+1 '

Let's return to the proof of the theorem. By assumption, the diffeomorphism F has nontrivial heteroclinic domains Vi, i G {1, ..., m0} and curves from the sets FsFi and L^ have

homotopy types (5i, df) and (5i, df) respectively on the torus Vi. Then, by virtue of [8, Chapter D, Exercise 7] the equality £i = Si(df — df) is holds.

We define the function v: [0, 1] ^ [0, 1] by the formula

f0,

* (t) =

1 + exp (—*——

\t2 (t-1)2

t = 0, 0 <t< 1,

t = 1.

1

1

Define on the annuli Ki the map Ri: Ki ^ Ki, by the formula

Ri (e*, r) = (ei(4,+2n*iv(r)),

The projection Ri = piRip-1 of the map Ri acts on the fundamental group of the torus 'Vi by the matrix Rit. Due to the consistency of the bases a, b and ai, bi} the equality = RiM holds. Then, the diffeomorphism Ri acts on the fundamental group of the torus T2 by the matrix R^ =

= ^0 1 j. Let F = R~mt0 ◦ ...◦ R-1 ◦ F. Then F e Ef and for any i ^{1, ..., m0} the equality holds

Since the global stable manifold of a saddle point is obtained from a local one by iteration due to the inverse map and in a neighborhood of the attractor AF the manifolds Lf and L~ , are

coincide, then Ls~ ,HKi = Lf HKi. Since the global unstable manifold of a saddle point is obtained

from the local by iterating by the direct map and in the neighborhood of the repeller Rf the manifolds Lf and Lu~ . are coincide, then Lu~ . n K■ = RALV n Ki).

i f i ' f i i i ^ i i'

Thus, the knots Ls~ ., Lu . have the same homotopy type on the torus Vi. By virtue of [8], there exists a diffeomorphism h: 'Vi ^ \/ri isotopic to the identity such that h(Lu ,) n Ly , = 0.

According to the lemma 2, the diffeomorphism F is diffeotopic to some diffeomorphism F1 e Ef such that L~ nLsy = 0. By construction, the diffeomorphism F1 belongs to the class MS(T2) and has only trivial heteroclinic domains (see Fig. 4).

Fig. 4. On the left is the phase portrait of the diffeomorphism F on the torus T2 with a non-trivial heteroclinic domain, F is reduced by some isotopy to the diffeomorphism F1; the phase portrait of which is shown on the right

By virtue of the lemma 1,the diffeomorphism F1 is isotopic to the identity diffeomorphism. Then the diffeomorphism F is isotopic to the diffeomorphism R1 o ... o Rm . Because

Ri* =

'1 £i 0 1

then

F=

'i £f 0 1

Then F belongs to the homotopy type T1 if and only if £F = 0. Since F = fd, then the matrix f^ is periodic if and only if £F = 0. Thus, the diffeomorphism [f ] has homotopy type T1 if and only if £f = 0.

Since the Morse-Smale diffeomorphisms are representatives of only homotopy classes from T1 and T2, then [f ] e T2 if and only if £F = 0.

Acknowledgments

The author thanks V. Z. Grines and O. V. Pochinka for formulation of the problem and fruitful discussions.

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