On a Classification of Periodic Maps on the 2-Torus Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 1, pp. 91-110. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220702

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 37E30

On a Classification of Periodic Maps on the 2-Torus

D. A. Baranov, V. Z. Grines, O. V. Pochinka, E. E. Chilina

In this paper, following J. Nielsen, we introduce a complete characteristic of orientation-preserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of the classes of orientation-preserving periodic homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic automorphism. Moreover, it is shown that the number of such classes is finite. According to V. Z. Grines and A. Bezdenezhnykh, any gradient-like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. Thus, the results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.

Keywords: gradient-like flows and diffeomorphisms on surfaces, periodic homeomorphisms, torus

1. Introduction

According to the Nielsen-Thurston classification (see, for example, [1]), 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 homeomorphism; T2) reducible nonperiodic homeomorphism of algebraically finite order; T3) a reducible homeomorphism that is not a homeomorphism of algebraically finite

Received April 10, 2022 Accepted June 10, 2022

The publication was prepared within the framework of the Academic Fund Program at the HSE University in 2021-2022 (grant 21-04-004).

Denis A. Baranov denbaranov0066@gmail.com Vyacheslav Z. Grines vgrines@yandex.ru Olga V. Pochinka olga-pochinka@yandex.ru Ekaterina E. Chilina k.chilina@yandex.ru

National Research University Higher School of Economics

ul. Bolshaya Pecherskaya 25/12, Nizhny Novgorod, 603155 Russia

order; T4) pseudo-Anosov homeomorphism. It is known (see [2, p. 369, Theorem 13.1]) that the homotopic types of homeomorphisms of a torus are T1, T2, T4 only.

Let S be a connected compact (possibly with boundary) orientable surface. Let us recall that homeomorphisms f, f: S — S are called topologically conjugate if there is a homeomorphism h: S — S such that f' = h o f o h-1.

A homeomorphism f is called periodic of period n if fn = id and fm = id for each natural m < n.

In class T1 there is a classification of structurally stable gradient-like orientation-preserving diffeomorphisms of an orientable surface in [3] and [4]. It was proved that any such diffeomor-phism is topologically conjugate to a diffeomorphism which is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism.

A homeomorphism h: S ^ S is called reducible by a system C of disjoint simple closed curves Cj, i = 1, ..., I, nonhomotopic to zero and pairwise nonhomotopic to each other if the system of curves C is invariant under h. A reducible nonperiodic homeomorphism h: S — S is called a homeomorphism of algebraically finite order if there exists an h-invariant neighborhood C of curves of the set C consisting of the union of two-dimensional annuli and such that for each connected component S-, j = 1, ...,??. of the set S \ C there is a number nij e N such that hmj \S : Sj — Sj is a periodic homeomorphism.

In class T2 there is a classification of structurally stable Morse-Smale diffeomorphisms with orientable heteroclinic intersection in [5-8]. The paper [9] contains a review of the results obtained so far in the classification of Morse-Smale diffeomorphisms in general.

All representatives of class T4 have chaotic dynamics. For the case of the 2-torus, structurally stable representatives of class T4 are Anosov diffeomorphisms. Each Anosov 2-diffeomorphism is topologically conjugate to an algebraic automorphism (see [10, Theorem 5.2]). For the case of orientable surfaces of genus greater than one there is a classification of structurally stable orientation-preserving diffeomorphisms containing a one-dimensional spaciously situated perfect attractor in [11, Corollary 1.11 to Theorem 1.10].

In [12], J. Nielsen found necessary and sufficient conditions for topological conjugacy of periodic transformations of closed orientable surfaces. Describing all topological classes for periodic maps is a difficult and boundless task. However, this problem was completely solved in the case of the two-dimensional sphere by Kerkyarto in [14] and was partially solved in the case of the two-dimensional torus by Brauer in [15]. In addition, in [16] there is an algorithm to find the number of classes of topological conjugacy for periodic maps and there is a construction of the canonical representative for each homotopy class of the topological conjugacy. In the present paper we describe all classes of topological conjugacy for periodic maps on the two-dimensional torus by means of orientation-preserving homeomorphisms. Moreover, the work contains a realization of nonhomotopy of periodic maps to the identity on the two-dimensional torus by algebraic automorphisms. In addition, it is shown that the number of such classes is finite. Notice that there is an infinite set of topological conjugacy classes for periodic maps of the two-dimensional torus that are homotopic to the identity. The realization of such classes is presented in [17].

From the results of J. Nielsen [12, Sections 1-4] the following statements are true for any orientation-preserving periodic map f on a closed orientable surface S, whose period is n:

1. For each / we denote by Bf C S a set of periodic points of the homeomorphism / with a period strictly less than n. This set is either empty or consists of a finite number of orbits O1, ..., Ok, k ^ 1. Denote by ni the period of an orbit Oi, i g{1, ..., k}. Then ni is a divisor of n. Set Xi = then for any orbit Oi C Bf there exists a unique num-

ber 5i G {1, ..., \ — 1} such that it is coprime to A^ and there is some neighborhood Dw of a point xi G Oi such that the restriction \D_ is topologically conjugate to the rotation

by the angle -j—of the complex plane about the origin:

z —> e

T^1

z.

(1.1)

Fig. 1. The action of the homeomorphism fUi in some neighborhood of the point xi

Figure 1 illustrates the action of the homeomorphism fni in the neighborhood of the point xi. This point has period n^ with respect to the homeomorphism / and is a fixed point with respect to the homeomorphism f ni. The action of the homeomorphism fn is topolog-ically conjugate to the rotation by the angle ^^ about the point in the counterclockwise

i

direction.

2. Denote by di the inverse for Si number in ZA . Due to conjugacy with the map (1.1) there exists a curve which is homeomorpic to the circle and such that it is invariant under the homeomorphism fnt and bounds a disk containing a point xi G Bf. Then the number d^ has the following property: the open arc of the invariant curve joining the points x and fnidi(x) (in the counterclockwise direction) does not contain points of the orbit of point x. A pair of numbers (ni, di) is called the valency of the orbit Oi.

Figure 1 illustrates the closed curve. This is invariant with respect to the homeomorphism fni and contains each of points of the orbit of the point x under fnt. The open arc of the curve connecting points x and (x) (in the counterclockwise direction) does not contain points of this orbit.

3. For each f denote by G = {id, f, ..., fn-1} the group which is isomorphic to Zn = = {0, ..., n — 1}. The orbit space £ = S/G is called the modular surface. Also, this is a closed surface, and the natural projection tt : S —> £ is an ??,-fold covering map everywhere except at the points of the set Bf. Put xi = 7r(Oj).

4. Denote by Di (i = 1, k) an open disk on the modular surface £ such that it contains the only point xi G Bf. Let ci denote the boundary of Di and let n-1 (Di) denote the full preimage

k . k of Di on the surface S. Put S = S \ (J n-1 (Di), £ = £ \ (J Di. Choose a point x on £

i=1 i=1

and draw a closed path c(t) (t G [0, 1]) through x on £ such that x = c(0) = c(1). Then, by the monodromy theorem (see, for example, [18, Statement 10.27]), for a point x G S such that. 7r(x) = x there is a unique path c(t) on S which is a lift of c(t) and such that. c(0) = x. Then there is m G Zn such that. c(l) = fm(x) (see Fig. 2).

Fig. 2. Path c(t) on surface E and its lift c(t) on surface S

Denote by n the map from the fundamental group of the surface ^(£) to the group Zn by the rule n(ic]) = m, where [c] is the class of loops homotopic to c. Then n: ^i(^Ö) ^ Zn is a group homomorphism.

Let p be the genus of the surface S and g be the genus of the modular surface £. It follows from the Riemann - Hurwitz formula (see, for example, [25]) that

2p + ni - 2 = n(2g + k - 2).

(1.2)

i=1

For each periodic transformation f of a closed orientable surface S we define the collection of the numbers

K = (n, p,ni,..., nk, di, ..., dk),

which we call the complete characteristic of the periodic transformation f. Let us choose a point O on £ and draw the canonical system of loops

a^ ß^ ..^ ag, ßg, ...,Yk

on £ starting and finishing at the point O such that it defines the family of generators of the fundamental group ^(£) of the surface £ (see Fig. 3). The only defining relation of the fundamental group ^(£) is the equality

Ki '... ' K„ ■ Yi ■ ... ■ Yk = 1

(1.3)

where Kj = aj ßj a- i ß- i.

Let us choose a point O on the surface S such that tt(O) = O G E. For any m G Zn through the point O G S it is possible to draw a path l(t) (t G [0, 1]) on the surface S such that 1(0) = O, 1(1) = fm(0) G S. Its projection l(t) = ir(I) is a closed path on the surface E such that l(0) = l(1) = O. Therefore, the map n: ^(E) ^ Zn defined in 4. is an epimorphism. Therefore, the following equality holds:

gcd(n([ai]), niPi], ••• n[ag], niPg], n[7i], nhk]> n) = 1> (14)

where gcd(a1, •••, am) is the greatest common divisor of integers a1, •••, am.

By definition of the valence of the ith orbit jy^J = n^ (i = 1, k), and f}[K \ = = 'ijlajpjaj1/]'1} = 'rilcxj] + rjlpj] + '^[o;"1] + 'liPj1} = 0 (mod n) (j = 1, q) since rj[c] + ,r][c~1} = = n[cc_1 ] = n[1] = 0 (mod n) for any path c G E. From this, taking Eq. (1 • 3) into account, we get

k

'<^dini = 0 (mod ^^ (1 • 5)

i=1

For the case where the modular surface is a sphere (g = 0) it follows from Eq. (1 • 4) that

gcd(n1 d1, • • •, nkdk, n) = 1 • (1 • 6)

Statement 1 ([12, p. 56]).

There is a periodic homeomorphism whose complete characteristic is k = (n, p, • • • , nk, d1, • • •, dk) and the genus of the modular surface is g if and only if the following conditions are satisfied: k

!) E dn = 0 (mod n);

i=1

k

2) 2p +£ n - 2 = n(2g + k - 2);

i=1

3) if g = 0: gcd(n1 d1, • • •, nkdk, n) = 1.

Statement 2 ([12, p. 53]). Two periodic maps f and f1 on the orientable surface S are topologically conjugate by an orientation-preserving homeomorphism if and only if the complete characteristic of f coincides with the complete characteristic of f'.

If Bf = 0 the periodic transformation / is completely described by the set of the numbers (n, p). In this case the natural projection n: S ^ £ is an n-fold covering map of the modular surface £ of genus g by the surface S of genus p. This implies the following statement.

Statement 3. Two periodic transformations /, /' of the surface S such that Bf = 0, Bft = 0 are topologically conjugate by an orientation-preserving homeomorphism if and only if f and f1 have the same periods.

An orientation-preserving homeomorphism f: T2 T2 induces a group automorphism f of n1(T2) = Z © Z given by matrix Af £ SL(2, Z) (see, for example, in [22, Section D]). Homeomorphism f is isotopic to the identity if and only if Af = I. Recall that a matrix A is called periodic if there exists m £ N such that Am = I. Since the action of the induced automorphism fn is defined by the matrix An, An = E for homeomorphism f of period n. Therefore, matrix Af is periodic for any periodic homeomorphism f. However, the period of matrix Af may not coincide with period n of homeomorphism f in the general case, but does not exceed n.

The main results of this work are the following theorems.

Theorem 1. There is an orientation-preserving periodic homeomorphism f: T2 ^ T2 such that Bf / 0 if and only if the complete characteristic of f coincides exactly with one of seven complete characteristics:

1) «1: n = 2, n1 = n2 = 3 00 n4 = 1, = d2 = d 00 = d4 = 1

2) K2: n = 3, n1 = n2 = n3 = 1, d1 = d2 = d3 = 1;

3) K3: n = 3, n1 = n2 = n3 = 1, d1 = d2 = d3 = 2;

4) K4: n = 6, n1 = 3, n2 = 2, n3 = 1, d1 = d2 = d3 = 1;

5) K5: n = 6, n1 = 3, n2 = 2, n3 = 1, d1 = 1, d2 = 2, d3 = 5

6) K6: n = 4, n1 = 2, n2 = n3 = 1, = d2 = d3 = 1;

7) Kj: n = 4, n1 = 2, n2 = n3 = 1, d1 = 1, d2 = d3 = 3.

Theorem 2. Let f: T2 ^ T2 be an orientation-preserving periodic homeomorphism of period n £ N. Then the following conditions are equivalent:

1) f is homotopic to the identity;

2) Bf = 03) g = 1;

4) f is topologically conjugate to the shift on the torus ^n (ei2xn, ei2yn) = (ei2n(x+1/n), ei2yn).

Corollary 1. Periodic homeomorphisms of the two-dimensional torus satisfying complete characteristics Kj (j = 1, 7) and only they are orientation-preserving periodic homeomorphisms of the two-dimensional torus that are nonhomotopic to the identity.

Let A = (Cd) e SL(2, Z),thatis, let A be a second-order integer square matrix with det A = = 1. Then A induces an orientation-preserving algebraic automorphism of the two-dimensional torus f a : T2 ^ T2 given by the formula

{x = ax + by (mod 1), y = ex + dy (mod 1).

Theorem 3. Any orientation-preserving periodic homeomorphism f: T2 ^ T2 nonhomo-topic to the identity is conjugate by an orientation-preserving homeomorphism with exactly one

map fA induced by the matrix A- (j = 1, 7): j J

A1 = ("o1 -1} a2 = ("i1 -1} A3 = (-1 -1} a4 = (°

A= = ("1O} A6 =(! A7 = ("13 *

The complete characteristic of fA coincides with a complete characteristic Kj .

j J

1.1. Structure of the paper

• Section 2 contains lemmas necessary for the proof of Theorem 1.

• Section 3 is the proof of Theorem 1.

• Section 4 is the proof of Theorem 2.

• Section 4 contains a classification of algebraic automorphisms of the two-dimensional torus and the proof of Theorem 3.

2. Auxiliary inequalities for finding complete characteristics of periodic transformations

In this section we prove some useful consequences from Eq. (1.2):

k

2p + J2 ni - 2 = n(2g + k - 2). i=l

Lemma 1. For any orientation-preserving periodic homeomorphism f: S ^ S such that Bj = 0 the following equality holds:

p = n(g - 1) + 1. (2.1)

_ k

Proof. Due to Bf = 0, the number k = 0. Using Eq. (1.2) 2p + ni ~ 2 = n(2g + k — 2),

i=1

we get 2p - 2 = n(2g - 2). Therefore, p = n(g - 1) + 1. □

Lemma 2. For any orientation-preserving periodic homeomorphism f: S ^ S such that Bf / 0 the following equality holds:

p > n(g - 1) + 1. (2.2)

Proof. By definition n^ is a divisor of n not exceeding n for all i = 1, k, therefore, n^ ^ § and the following relation holds:

nk ,

0 < k ^ 2-^n.i < — < nk. (2.3)

i=1

Let us consider all possible cases:

k

1. p = 0. Then Eq. (1.2) is equivalent to the equality ^ ni — 2 = n(2g + k — 2). Transforming

i=1

k

this, we get ^ ni — 2 = n(2g — 2)+nk. It follows from Eq. (2.3) that nk — 2 > n(2g — 2)+nk.

i=1

Hence, n(2g — 2) < —2 and n(1 — g) > 1. Then 1 — g > 0. Considering that g £ N, we get g = 0.

k

2. p = 1. Then Eq. (1.2) is equivalent to the equality ^ ni = n(2g + k — 2). Using Eq. (2.3),

i=1

we obtain n(2g + k — 2) < nk. Therefore, g — 1 < 0 and g < 1. Considering g £ N, we get g = 0.

3. p > 1. Using Eqs. (1.2) and (2.3), we find that n(2g + k — 2) < 2p + nk — 2. Hence, n(g — — 1) < p — 1 and p > n(g — 1) + 1.

Thus, in all cases the following inequality holds: p > n(g — 1) + 1. □

Lemma 3. For any orientation-preserving periodic homeomorphism f: T2 ^ T2 such that Bf / 0 the following equality holds:

2n

2 <-- < k < 4. (2.4)

n— 1

Proof. In the case of the two-dimensional torus p = 1. Then it follows from Lemma 2 that the genus of the modular surface is equal to 0. Therefore, Eq. (1.2) is equivalent to the equality

k

n

i=1

nk = ni + 2n. (2.5)

Using Eq. (2.3), we see that. 2n + k ^ nk ^ 2n + From the left, side of this inequality we find that, k ^ > = 2. From the right, side we obtain ^ ^ 2n. Therefore, n(4 - k) ^ 0. Considering that k, n £ N, we have k ^ 4. □

3. Complete characteristics of periodic transformations of the two-dimensional torus

It. follows from Statement 3 that, if the set Bf = 0 for the homeomorphism /, then the complete characteristic of f is determined by the genus p of the surface and the period n of the transformation. Thus, complete characteristics of periodic homeomorphisms on the two-dimensional torus corresponding to the empty set Bf have the form (n, 1), where n is the period of f.

In this section we find all complete characteristics of an orientation-preserving periodic homeomorphism of the two-dimensional torus corresponding to the nonempty set Bf and prove that there are only 7 such complete characteristics. Using transformations of a modular surface, we realize all such classes by the periodic homeomorphism.

Proof of Theorem 1. Let us prove the necessity. Let f be an orientation-preserving periodic homeomorphism of the two-dimensional torus such that Bf ^ 0.

Let us find all admissible complete characteristics of the transformation f. Due to Lemma 3, we have that the number k of orbits of the set Bf should only be equal to 3 or 4. Consider these cases separately.

4

Case 1: k = 4. Substituting 4 for k in Eq. (2.5), we get 2n = J2ni- Due to n^ 7 §

i=1

Eq. (2.5) holds only if n^ = § for all i = 1, ..., 4. Substituting | for n^ in Eq. (1.6), we get gcd (§, ??.) = 1. Therefore, n = 2. Since n% < n, we have n% = 1 for each i = 1, 4.

Hence, \ = j = 2. Since d^ < Ai? we have = 1 for each i = 1, 4. We obtain the complete characteristic of (2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1), which we denote by k1.

3

Case 2: k = 3. Substituting 3 for k in Eq. (2.5), we get n = ^ ni. From this it is obvious

i=1

that at least one n% is not less than For definiteness put ??1 ^ Consider 2 admissible cases: a) = § and b) n^ = §.

a) = Then n2 + n3 = Therefore, at least one of n2, n3 is not less than For definiteness put n2 ^ Then a1) n2 = § and n3 = or a2) n2 = § and n3 =

a,1) Substituting ^ for n^ in Eq. (1.6), we get gcd n^j = 1. Hence, n = 3.

3

Then, substituting the data n1 = n2 = n3 = 1 for ni in Eq. (1.5), we get ^ di = 0 (mod n).

i=1

Due to di G {1, 2} there are 2 admissible number collections: d1 = d2 = d3 = 1 (the complete characteristic (3, 1, 3, 1, 1, 1, 1, 1, 1), which we denote by k2), or d1 = d2 = d3 = 2 (the complete characteristic (3, 1, 3, 1, 1, 1, 2, 2, 2), which we denote by k3).

a2) Taking Eq. (1.6) into account, we get n = 6. Using Eq. (1.5) and the fact that di G G {1, ..., 5}, we obtain 2 admissible number collections: d1 = d2 = d3 = 1 (the complete characteristic (6, 1, 3, 3, 2, 1, 1, 1, 1), which we denote by k4) and d1 = 1, d2 = 2, d3 = 5 (the complete characteristic (3, 1, 3, 1, 1, 1, 1, 2, 5), which we denote by k5).

b) = Then n2 + n3 = §, whence § > n2 ^ j. Therefore, 61) n2 = | and n3 = or b2) n2 = f and n3 = f.

b1) The complete characteristics for this number collection is found in (a2).

b2) For these number collections, taking Eq. (1.6) into account, we have n = 4. Using Eq. (1.5) and the fact that di G {1, ..., 5}, we obtain 2 admissible number collections: d1 = = d2 = d3 = 1 (the complete characteristic (4, 1, 3, 2, 1, 1, 1, 1, 1), which we denote by k6)

and d1 = 1, d2 = d3 = 3 (the complete characteristic (4, 1, 3, 2, 1, 1, 1, 3, 3), which we denote by K7).

Let us prove the sufficiency. Let us show that for each complete characteristic Ki (i = = 1, 7) there is an orientation-preserving periodic homeomorphism of the two-dimensional torus. Using Eq. (1.2), we find that for such complete characteristics the genus of the modular surface is equal to 0. Therefore, a modular surface is the sphere.

Let us construct the homeomorphism f1 satisfying the complete characteristic k1 thus found. We mark on the sphere (modular surface) 4 points: x^ X2, X3, X4 • each of which is the projection of orbits with a period less than the period of the homeomorphism f3. Construct an arc with the beginning at the point x1 and the end at the point x4 on the sphere such that it contains points x2 and x3 and the point x2 is located between the points x1 and x3 (see Fig. 4a). Cutting the sphere along the constructed arc, we obtain a disk with the boundary x1x2x3x4x3x2x1 (see Fig. 4b). Gluing two such disks along the boundary x3x4x3, we get a square (see Fig. 4c). Having identified the sides of the square as shown in Fig. 4c), we obtain the two-dimensional torus (see Fig. 4d).

Define the homeomorphism f1 by the following rule. Rotating disk 1 in Fig. 4c by the angle n in the counterclockwise direction, we map it into disk 2. Similarly, we map disk 2 into disk 1 by the angle n in the counterclockwise direction. The described map is an orientation-preserving periodic homeomorphism of the two-dimensional torus satisfying the complete characteristic k1.

Fig. 4. The construction of the homeomorphism with the complete characteristic k^

Let us construct homeomorphisms f2 and f3 satisfying the complete characteristics k2 and k3, respectively. We mark 3 points on the sphere: x1, x2, x3, each of which is the projection of orbits with a period less than the period of homeomorphisms f2 and f3. Construct an arc with the beginning at the point x1 and the end at the point x3 on the sphere such that it contains the point x2 (see Fig. 5a). Cutting the sphere along the constructed arc, we obtain a disk with the boundary x1 x2x3x2x1 (see Fig. 5b). Gluing in pairs three such disks along the boundary x1x2, we obtain a hexagon (see Fig. 5c). Having identified the sides of the hexagon as shown in Fig. 5c), we obtain the two-dimensional torus (see Fig. 5d).

Define the homeomorphism f2 by the following rule. Rotating disk 1 in Fig. 5c by the angle ^ in the counterclockwise direction, we map it into disk 2. Similarly, we map disk 2 into disk 3, and disk 3 into disk 1. Figure 5e1 illustrates the action of the map in the neighborhood

of a fixed point. The described map is an orientation-preserving periodic homeomorphism of the two-dimensional torus satisfying the complete characteristic k2 .

Define the homeomorphism f3 by the following rule. Rotating disk 1 in Fig. 5c by the angle in the counterclockwise direction, we map it into disk 3. Similarly, we map disk 3 into disk 2, and disk 2 into disk 1. Figure 5e1 illustrates the action of the map in the neighborhood of a fixed point. The described map is an orientation-preserving periodic homeomorphism of the two-dimensional torus satisfying the complete characteristic k3 .

Fig. 5. The construction of homeomorphisms with complete characteristics k2 and k3

Let us construct homeomorphisms f4 and f5 satisfying the complete characteristics k4 and k5, respectively. We mark 3 points on the sphere: x1, x2, x3, each of which is the projection of orbits with a period less than the period of homeomorphisms f4 and f5. Construct an arc with the beginning at the point x2 and the end at the point x3 on the sphere such that it contains the point x2 (see Fig. 6a). Cutting the sphere along the constructed arc, we obtain a disk with the boundary x3 x1 x2 x1 x3 (see Fig. 6b). Gluing in pairs six such disks along the boundary x1x3, we get a polygon (see Fig. 6c). Having identified the sides of the polygon as shown in Fig. 6c, we obtain the two-dimensional torus.

Define the homeomorphism f4 by the following rule. Rotating disk 1 in Fig. 6c by the angle f in the counterclockwise direction, we map disk 1 into disk 2. Similarly, we map disk 2 into disk 3, disk 3 into disk 4, disk 4 into disk 5, disk 5 into disk 6, and disk 6 into disk 1. Figure 6d1 illustrates the location of orbits with a period less than the period of the homeomorphism f4. The described map is an orientation-preserving periodic homeomorphism of the two-dimensional torus satisfying the complete characteristic k4.

Define the homeomorphism f5 by the following rule. Rotating disk 1 in Fig. 6c by the angle ^ in the counterclockwise direction, we map disk 1 into disk 6. Similarly, we map disk 6 into disk 5, disk 5 into disk 4, disk 4 into disk 3, disk 3 into disk 2, and disk 2 into disk 1. Figure 6d2 illustrates the location of orbits with a period less than the period of the homeomorphism f5.

The described map is an orientation-preserving periodic homeomorphism of the two-dimensional torus satisfying the complete characteristic k5.

Fig. 6. The construction of homeomorphisms with complete characteristics k4 and k5

Let us construct homeomorphisms f6 and f7 satisfying the complete characteristics k6 and k7, respectively. We mark 3 points on the sphere: x1, x2, x3, each of which is the projection of orbits with a period less than the period of homeomorphisms f6 and f7. Construct an arc with the beginning at the point x2 and the end at the point x3 on the sphere such that it contains the point x3 (see Fig. 7a). Cutting the sphere along the constructed arc, we obtain a disk with the boundary x3x1 x2x1x3 (see Fig. 7b). Gluing in pairs four such disks along the boundary x1 x3, we get a square (see Fig. 7c). Having identified the sides of the square as shown in Fig. 7c, we obtain the two-dimensional torus (see Fig. 7d).

Define the homeomorphism f6 by the following rule. Rotating disk 1 in Fig. 7c by the angle j in the counterclockwise direction, we map disk 1 into disk 2. Similarly, we map disk 2 into disk 3, disk 3 into disk 4, and disk 4 into disk 1. Figure 7e1 illustrates the action of the map in the neighborhood of a fixed point. The described map is an orientation-preserving periodic homeomorphism of the two-dimensional torus satisfying the complete characteristic k6.

Define the homeomorphism f7 by the following rule. Rotating disk 1 in Fig. 7c by the angle in the counterclockwise direction, we map disk 1 into disk 4. Similarly, we map disk 4 into disk 3, disk 3 into disk 2, and disk 2 into disk 1. Figure 7e2 illustrates the action of the map in the neighborhood of a fixed point. The described map is an orientation-preserving periodic homeomorphism of the two-dimensional torus satisfying the complete characteristic k7.

Fig. 7. The construction of homeomorphisms with complete characteristics k6 and k7

Remark 1. By construction the homeomorphism /1 is mutually inverse to itself and each pair of homeomorphisms {/2, /3}, {/4, /5} and {/6, /7} is mutually inverse.

4. Classification of periodic homeomorphisms of the

two-dimensional torus that are homotopic to the identity

In this section we prove Theorem 2.

Let the vector field £(x, y) be defined and continuous at each of the points of the plane R2 except perhaps at some points. Denote by M0(x0, y0) the singular point of the vector field £(x, y).

Choose r > 0 such that, the disk dr = j(.r, y) G R2: \J(x — x0)'2 + (y — y0)'2 ^ r| does not.

contain singular points of the vector field different from the point M0(x0, y0). The number of rotations of the vector field (\9d in the counterclockwise direction in the neighborhood of the point M0 is called the index of the singular point M0 and is denoted by I(M0) (for detailed definitions, see [19, Section V, 10]).

Consider A and 5 such that A G N, 5 g{1, ..., A — 1} and (S, X) = 1. Then the index I(M0) of the fixed point. M0 of the plane rotation by the angle ^ in the counterclockwise direction is equal to 5 if 5 ^ | and is equal to —5 if 5 >

Let 2 G T2 be an isolated fixed point of a continuous map g: T2 ^ T2. Let p: U ^ R2 be a chart at z; put p(z) = w. The vector field /(x, y) = pgp-1 (x, y) — (x, y) is defined on a neighborhood of w in R2 and has w for an isolated zero. Define Le/z(g) = I(w). This is independent of (p, U) (see [20, p. 139, Ex. 10]). Put Le/(g) = ^Le/z(g), where z is a fixed

z

point of g.

For the surface S of genus p the map f: S — S homotopic to the identity, with a finite number of fixed points, we have the Lefschetz - Hopf formula [21]:

Lef (g) = 2 - 2p. (4.1)

Proof of Theorem 2. Let f: T2 — T2 be an orientation-preserving periodic homeomorphism of period n G N.

Let us prove that the following implications are true: 1 — 2, 2 — 3, 3 — 4, 4 — 1.

1 — 2. Assume the converse. Let the homeomorphism f be homotopic to the identity and Bf / 0. In this case, taking Theorem 1 into account, we have 7 admissible complete characteristics k • (j = 1, 7) for the homeomorphism /. On the one hand, the homeomorphism / is homotopic to the identity and by Eq. (4.1) we get Lef (f) = 0. On the other hand, by direct calculation we get a nonzero value of Lef (f) (see Table 4.2) for all admissible complete characteristics k • (j = 1,7). This contradiction proves the implication.

K «1 K2 00 h*4 K5 k6 tvj

Lef(f) 4 3 -6 1 -5 2 -6

(4.2)

2 —> 3. If Bf = 0, then it follows from Lemma 1 that p — 1 = n(g — 1). Substituting 1 for p in this equality, we obtain n(g — 1) = 0. Therefore, g = 1.

k

3 — 4. Substituting 1 for g and 1 for p in Eq. (1.2), we get ^ ni = nk. Using Eq. (2.3),

i=1

k _

we have ^ ni ^ if - Therefore, k = 0 and Bf = 0. It follows from Statement 3 that the periodic

i=1

homeomorphism f of period n is topologically conjugate to the diffeomorphism ^n (ei2xn, ei2yn) = = (ei2n(x+1/n), ei2yn^j, which has period n.

4 — 1. By construction, the diffeomorphism (ei2xn, ei2yn) = ('ei2n(x+1/n\ ei2yn) acts identically on the fundamental group and, therefore, it is isotopic to the identity map (see, for example, [22]).

Remark 2. Implication 2 — 3 is known to specialists since it is a consequence of some homotopy facts about covering maps and surface topology, but we have found a proof of this fact using the technique implication from Lemma 1.

The following statement is a corollary to Theorem 1 and Theorem 2.

Corollary 1. Periodic homeomorphisms of the two-dimensional torus satisfying the complete characteristics k,j (j = 1, 7) and only they are orientation-preserving periodic homeomor-phisms of the two-dimensional torus that are nonhomotopic to the identity.

In each topological conjugacy class of orientation-preserving periodic homeomorphisms of the two-dimensional torus that are homotopic to the identity there is the representative fa: T2 — T2, which is a translation of the two-dimensional torus by the vector a = (a1, a2) defined by the formula fa(x, y) = ((.t+ a1) (mod 1); (y + a2) (mod 1)), where a1, a2 G Q. Necessary and sufficient conditions for the conjugacy of two translations on the 2-torus can be found in [17].

5. Classification of periodic algebraic automorphisms of the two-dimensional torus

Let A = (Cd) £ GL(2, Z), that is, let A be a second-order integer square matrix with det A = ±1. Then A induces the map fA: T2 — T2 given by the formula

{x = ax + by (mod 1), y = ex + dy (mod 1),

which is an algebraic automorphism of the two-dimensional torus.

If the eigenvalues of A £ Gl(2, Z) are not equal in modulus to unity, then the algebraic automorphism of the two-dimensional torus induced by the matrix A is called a hyperbolic algebraic automorphism of the two-dimensional torus. Otherwise, the automorphism fA is called a nonhyperbolic algebraic automorphism of the two-dimensional torus.

Two algebraic automorphisms of the 2-torus f and g are called conjugate if there exists an automorphism h such that gh = hf.

The set Kf = {hfh-1 | h is an algebraic automorphism of the 2-torus} is called the conjugacy class of the automorphism f.

We denote by Z2x2 the set of integer matrices of order 2. The matrix B £ Z2x2 is called similar over Z to the matrix A £ Z2x2 if there exists a matrix S £ Gl(2, Z) such that B = = S-1 AS. If A £ Z2x2, then the set KA = {S-1 AS | S £ Gl(2, Z)} is called the similarity class of the matrix A.

Thus, the problem of finding the conjugacy classes of nonhyperbolic algebraic automorphisms of the two-dimensional torus is reduced to the problem of finding the similarity classes of second-order integer unimodular matrices, whose eigenvalues are equal in modulus to unity. The problem of finding similarity classes for second-order integer unimodular matrices whose eigenvalues are roots of unity was solved in [23] in the form of the following statement:

Statement 4 ([23, Lemma 3]). Let A £ Gl(2, Z) and suppose that both eigenvalues of A are roots of unity. Then A is similar over Z to exactly one of the following matrices:

(1 m\ (-1 m\ (1 0 N (1 1 N 1) , ^ 0 -1) , ^0 -1) , ^0 -1) '

(-1O) ■ (-1 -1) • • m £{012-}-

Lemma 4. The eigenvalues of second-order integer unimodular matrices are equal to the root of unity if and only if they are equal in modulus to unity.

Proof. Let A = (Cd) £ Gl(2, Z). Then A = ad - bc is a determinant of the matrix A (A = ±1) and f (X) = X2 - X(a + d) + ad - bc is a characteristic polynomial of the matrix A. Denote by X1 2 the eigenvalues of the matrix A.

Suppose that there exists n £ N such that X™ = 1 (i £ {1, 2}). Then |XJ = 1.

Let us prove the statement in the opposite direction. Let |XJ = 1 (i £ {1, 2}).

In [24], Kronecker proved that, if each of the roots of an integer polynomial with the leading coefficient 1 is equal in modulus to unity, then these roots are roots of unity. Since a, b, c, d £ Z and |X1|-|X21 = ^d - bC = 1, we have that all eigenvalues of the matrix A are roots of unity. □

The next corollaries follow from Statement 4 and Lemma 4.

Corollary 2. Each conjugacy class of nonhyperbolic algebraic automorphisms of the two-dimensional torus is given by exactly one of the following matrices:

/1 m\ /—1 m

M1 (m)= f 1 , M2(m) = o m1

M5 =

' 0 r -1 0,

M6 =

' 0 1 ^ 11

1 0

0 —1 0 -1

1 1

M4 =

1 1

0 —1

, m G {0, 1, 2, ...}.

Recall that a nonidentity matrix A is called periodic if there exists a number n G N such that An = I. The smallest of such n is called the period of the matrix A.

By direct calculation, we have that for m = 0 matrices M1(m) and M2(m) are not periodic. For m = 0, the matrix M1 (m) is identical and the matrix M2(m) is a matrix of period 2. Matrices M3 and M4 are also matrices of period 2. The periods of matrices M5, M6 and M7 are equal to 4, 3 and 6, respectively.

Corollary 3. There are 6 classes of periodic algebraic automorphisms of the two-dimensional torus, each of which is given by exactly one of the following matrices:

—1 0 \ /10' M2(0)= 0 -1 , M3 = 0-1

M5 =

' 0 1 -1 0,

M6 =

' 0 1 11

M4 =

M7 =

1 1 1 0 —1

1

A3 = M6,

A4 = M7,

Put A1 = M2 (0), A2 = M6 Remark 3. Consider the modular group PSL(2,

A = Mr

1

A6 = M5-1,

A7 = M5.

:= SL(2, Z)/{±I} which acts on the upper-half of the complex plane by fractional linear transformations: z —>• frijif (o, b, c, d G Z, ad — be = 1, z G C and Im(z) > 0) (see, for example, [13]). It is well known that the standard generators of this group are two matrices 5' = ( ^ J) and T = ( J }) so that PSL{2, Z) = (5', T \ S2 = I, (ST)3 = I).

Consider the projection p: SL(2, Z) —> PSL(2, Z). Let A'j = p(Aj) (j = 1, 7) be the image of the matrix A. by p. Then we have Ai = I, A'6 = A'7 = S, A3 = A'4 = (ST), and A2 = A'5 = (ST)2.

Next we consider a subgroup H' = (S | S2 = I)x ((ST) | (ST)3 = I)

Z2 X Z3

Z6 of PSL(2,

A7 and

Thus, there is a connection between algebraic automorphisms, induced matrices Ai, actions of homeomorphisms on the upper-half of the complex plane. The images of these matrices under p generate a cyclic subgroup H' of the group of all such homeomorphisms. It is noteworthy that automorphism fA of period 3 and automorphism fA of period 6 correspond to the same homeomor-phism ^ —> — -q-j-, automorphism fA of period 3 and automorphism fA^ of period 6 correspond to homeomorphism: z —> — 1 — automorphism f A of period 4 and automorphism f A of period 4 correspond to homeomorphism: £ —> — ^ and automorphism fA of period 2 correspond to the identical map: z — z (z G C and Im(z) > 0).

Proof of Theorem 3. It follows from Statement 2 that two periodic surface transformations are conjugate by an orientation-preserving homeomorphism if and only if their complete characteristics coincide. It follows from Corollary 1 that any orientation-preserving periodic homeo-morphism of the two-dimensional torus that is nonhomotopic to the identity satisfies exactly one of the complete characteristics k,j (j = 1, 7). Let us show that the map fA has the complete

characteristic Kj and, therefore, any orientation-preserving periodic homeomorphism of the two-dimensional torus that is nonhomotopic to the identity is conjugate by an orientation-preserving homeomorphism with exactly one automorphism fA .

Let us find the complete characteristic of the periodic transformation fA induced by the

matrix A5 = (-11 o). The complete characteristics of automorphisms given by the remaining matrices are found in a similar way.

The period of A5 is 6. Consequently, it induces a periodic homeomorphism of the two-dimensional torus fa of period 6.

Let us find the set B^ with a period less than the period of the homeomorphism fA . As

mentioned before, the periods of such points are divisors of six, that is, they are fixed or have period 2 or 3.

We find fixed points from the system

{x = x + y (mod 1), y = -x (mod 1),

decomposed into a countable set of systems

{x + k = x + y, k £ Z, y + m = -x, m £ Z.

These systems are equivalent to

(x = -m - k,

k £ Z, m £ Z.

y = k,

Hence, by direct calculation we see that the map fA^ has a unique fixed point p(0, 0), where p: R2 — T2 is the natural projection.

Finding fixed points of the map fAg, induced by the matrix A5 = (— ^), we get points of period 2 of the map fA^.

By analogy with finding fixed points of the map fA , we consider a countable set of systems

{x + k = y, k £ Z,

y + m = -x - y, m £ Z.

These systems are equivalent to

x = 2m — k),

1 k £ Z, m £ Z,

V = g(k-m),

whence, by direct calculation we make sure that the map fAg has three fixed points p(0, 0), p(l, I), p (f, I). Therefore, the map fA has a unique orbit of period two 02 =

Finding fixed points of the map fAg induced by the matrix Aj = ( q1 -i ), we get points of period 3 of the map fA .

By analogy with finding fixed points of the map fA , we consider a countable set of systems

(x + k = -x, k £ Z, y + m = -y, m £ Z.

These systems are equivalent to

1

x = —k, 2 '

1

V =

k G Z, m G Z,

I V

2' 2/

whence, by direct calculation we have that, the map fA has 4 fixed points: p(0, 0), p p Q, 0), p (0, Therefore, the map fA has a unique orbit. 03 = {p , p 0) , p (0, }

of period 3.

Thus, the set. Bf of the map fA consists of three orbits: Ol = {p(0, 0)}, 0.2 =

= {p(h ¡)>p(i !)}X = Mi i)}.

^ i x = x + y,

Consider the map A1: < _ This map is covering with respect, to fA . The

\y = -y- 6

point 01(0, 0) is fixed under A1.

In some neighborhood of the point 01 we fix the point Q(1, 0). Under the action of A1 the orbit of the point 01 is the set

= {(1, 0), (0, 1), (-1, 1), (-1, 0), (0, -1), (1, -1)}.

Connect each of the points of the set Oq by means of a segment to the point 01. Consider a circle of center 01 such that it intersects each of the constructed segments at some point (see Fig. 8).

A(q)

Mq) / \ 0l \ i

\ \ 0

9

^(Q)l-1 A.iQ)

Fig. 8. The action of At in some neighborhood of the fixed point

Connect points of the set Oq in pairs by means of the segment following the counterclockwise order on the circle. We obtain a closed curve (see Fig. 8). It is an invariant curve of the map A1. The open arc (Q, A\ (Q)) does not contain points of the orbit of the point Q in the counterclockwise direction. Therefore, d3 = 5.

{X = y,

This

y = —x - y + 1. 5

point 02 is fixed under A2.

Analogously to finding d3, we construct a closed curve invariant under the map f\g and find the parameter d2 = 2.

( x = —x + 1,

Consider the map A3: _ ^ This map is covering with respect to fA . The

y = -y + 1-

point 03 Q, is fixed under A3.

The map A3 defines the rotation by the angle tt about the point 03 Q, of the plane. From this we find d1 = 1.

Thus, the complete characteristic of the map f A given by the matrix A5 is the following collection of the numbers: n = 6, n1 = 3, n2 = 2, n3 = 1, d1 = 1, d2 = 2, d3 = 5 (the complete characteristic k5).

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