On periodic translations on n-torus Текст научной статьи по специальности «Математика»

Динамические системы, 2017, том 7(35), №2, 113-118

MSC 2010: 37B05

On periodic translations on n-torus

E. D. Kurenkov, K. A. Ryazanova

Higher School of Economics

Nizhny Novgorod. E-mail: ekurenkov@hse.ru, ksu_ryaz@mail.ru

Abstract. We consider periodic translations on n-torus and investigate the set of all conjugating homeomorphisms for topologically conjugated translations. It was shown by J. Nielsen [Dansk Videnskaternes Selskab. Math.-fys. Meddelerer, 1937, Vol.15, 1-77] that two periodic homeomorphisms of two-torus such that all points have the same period are topologically conjugate if and only if they have the same period. We consider the problem when two periodic translations on n-torus are topologically conjugate by means of toral automorphism. The main result is that two periodic translations on n-torus of the same period are topologically conjugate by means of countable family of toral automorphisms. Moreover, we show that for two periodic translations that are topologically conjugate each homotopy class in the set of all conjugating homeomorphisms contains continuum of homeomorphisms.

Keywords: topological conjugacy, toral automorphism, homotopy.

1. Introduction and statement of results

Periodic translations of the n-dimensional (n ^ 2) torus Tn = Rn/Zn are studied in this paper. The transformation f: Tn ^ Tn is called periodic of period k, if fk = id, and for any k' < k the inequality fk = id holds.

Periodic translations of two-dimensional surfaces were considered in detail by J. Nielsen [4]. Generally speaking, periodic translations of period k may have points of a period less than k. However, orientation-preserving periodic homeomorphisms have only a limited number of points of a period less than k, while all other points have the same period k.

In this article we study periodic n-dimensional translations such that all points have the same period, that are described by a shift to an n-dimensional vector with rational coordinates. The transformations f: Tn ^ Tn have the following form:

It is known (see, for example, [3]) that depending on the value of y%, the transformation f can have the following type:

xThe authors thank V. Z. Grines for posing the problem and for useful discussions. The first author also thanks the Russian Science Foundation (project 17-11-01041) for financial support in the proof of Theorems 1 and 2. The study of periodic surface mappings was carried out within the framework of the fundamental research program of the HSE in 2017 (T-90 project).

(1.1)

w

© E. D. KURENKOV, K. A. RYAZANOVA

1. If all Yi are rational, then all the points of f have a period equal to the lowest common denominator of all Yi;

2. If Yi are linearly independent over the field Q, then f is topologically transitive and the trajectory of each point x E Tn is dense in Tn;

3. If Yi are linearly dependent over Q, but not all Yi are rational, then the closure of the trajectory of any points of x E Tn is a finite union of fc-dimensional tori Tk, where 1 ^ k ^ n — 1.

We denote by Gn the set of all periodic homeomorphisms of the n-dimensional torus of the form (1.1).

Nielsen [4] had shown that two periodic homeomorphisms of the two-dimensional torus T2 such that all points have the same period are topologically conjugate if and only if their periods coincide. However, the problem whether two homeomorphisms of a torus are conjugate by means of a group automorphism of the torus was not considered. In this article we study the problem when two transformations from the class Gn are topologically conjugate by means of the group automorphism of the torus Tn. The main result of the article is presented in the following theorems.

Theorem 1. If two periodic homeomorphisms of the n-torus f: Tn ^ Tn and f': Tn ^ Tn, f, f' E Gn have the same period, then there exists a countable family {hi}, i E N of group automorphisms of the n-torus hi: Tn ^ Tn conjugating the maps f and f'.

From the results of D. Z. Arov [1] it follows that if two transitive translations of the n-torus are topologically conjugate, then the conjugating homeomorphism must be linear (a composition of a group automorphism and a shift). The opposite result holds for periodic shifts.

Theorem 2. If two periodic homeomorphisms of the torus f: Tn ^ Tn u f': Tn ^ Tn, f,f' E Gn are topologically conjugate by means of homeomorphism h: Tn ^ Tn , then there is a continuum set of homeomorphisms of an n-dimensional torus {hpE A, h^: Tn ^ Tn homotopic to h such that hp o f = f' o hp.

2. Proof of the results

We denote the greatest common divisor of the set of integers a\,a2,... ,an by (ai, a2,..., an).

Lemma 1. For any pair of n-dimensional vectors

AM fpi/q\

Vi

0

V0/

и V2

P2/q

\Pn/qJ

q E N, Pi E Z,

such that (pl,p2,... ,pn,q) = 1 there exists an unimodular matrix A and integers ml, m2,..., mn, such that an equality Avl = v2 + (ml,m2,..., mn)T holds.

Proof. Suppose that the matrix A has the form

^an ... a\„\

A = . ••. .

yara1 • • • annJ

Let's set the elements of the matrix A and numbers m1,m2, • • •, mn, in such a way that they satisfy the conclusion of the lemma. Let elements ail7 i = 1, n be of the form ai1 = pi + qmi. We denote the complement minor of the element ai1 by Mi. Let us show that the numbers m1, • • •, mn can be chosen in such a way that (p1 + qm1, • • • ,pn + qmn) = 1.

We introduce the notation ri = (pi,q), R = max ri. Since pi/ri and q/ri are relatively

i

prime, it follows from the Dirichlet theorem (see, for example, [2]) that any arithmetic progression with coprime the first term and the difference contains infinitely many prime terms. That is why, the sequence p + qmi, where p- and q are fixed, and mi are different integer values, contains infinitely many simple terms. We choose mi in such a way that p- + qmi will be different prime numbers exceeding in absolute value R. If we chose mi in such a way, then an equality (a11, • • •, an1) = 1 holds, since (r1, • • • ,rn) = 1.

Expanding det(A) by the first column, we note that det(A) = YHn=1(—1)l+iai1Mi. We will prove the lemma if we can choose integer elements aij, i = 1,n, j = 2,n in such a way, that the corresponding values of Mi, i

n

J2(-1)1+ianMi = 1.

i=1

1,n, will be such that

n,

For this, we show that we can choose the corresponding integer values aj, i = 1, j = 2,n for any fixed set of integer values M* of complement minors Mi, i = 1, n. Let matrix A be of the form (2.1), i.e. a matrix whose a^

ij

(

A

a11 a12 a13 a24 • • • a1n-1 a 1n

a21 a22 a23 a24 • • • a2n-1 a2n

a31 0 a33 a34 • • • a3n-1 a3n

a41 0 0 a44 • • • a4n-1 a4n

an_ и 0 0 0

0, г - j > 1, j = 1. \

an— 1n— 1 an— 1n

0

an

(2.1)

/

y an1 0 0 0

We prove this statement, using mathematical induction on n. The statement is obvious for n = 2, since the minors M1, M2 coincide with the elements a12, a22. Suppose that it is true for n = k and show that it is also true for n = k + 1. Let

( a11 a12 a13 a24 • • • a1fc a1fc+1 N a24 a34

a44

A

a2i

a3i a41

ai2 a22 0 0

a13 a23 a33 0

ak1 0 \afc+11 0

0 0

0 0

a1k a2k a3k a4k

a2k+1 a3k+1 a4k+1

akk akk+1 0 ak+1k+1 J

(2.2)

By Mi we denote the minors obtained from A by deleting the z-st and (k + 1)-th rows and the 1-th and (k + 1)-th column. Note that Mi = ak+lk+lMi, i = 1, k, expanding Mi by the last line. We set ak+lk+l = (Mf, ..., Mf). By the induction hypothesis the numbers aij, Z — j ^ 0, Z = 2, k, j = 2, k can be chosen in such a way

that M

M*

1,k.

(M*,M2*,...,M*) '

Consider the minor Mk+1. We notice that Mk+1

E(-l)1+kMiüik+i, expanding

i=1

Mk+1 by the last column. We chose the elements in such a way that (M\, M2,..., Mk) = 1. For this reason we can choose the elements a1k+1,..., akk+1 in such a way that the

expression Y1 (—1)l+kMiaik+1 will take the value M}

i=1

*

k+1.

□

Let's an elementary lemma before proceeding to the proof of theorem 1.

Lemma 2. Let A and B be unZmodular n x n matrices, and f: Tn ^ Tn and g: Tn ^ Tn are the group automorphisms of the n-dZmensZonal torus induced by them,. Then the group automorphism of the n-dZmensZonal torus induced by the product AB coincides with the composition of the maps f o g and the group automorphism induced by the matrix A-1 coincides f

1

Proof. Let x G Tn = Rn/Zn, that is, x = (xi,..., Xn)T, 0 ^ Xi < 1. We set Bx = y + v, 0 ^ yi < 1, vi G Z. Therefore we have f o g(x) = A(Bx mod 1) mod 1 = A(y + v mod 1) mod 1 = A(Bx — v) mod 1 = ABx — Av mod 1 = ABx mod 1 by the definition of f and g, and taking into account the integer property of the elements of the matrix B. Let h: Tn ^ Tn be a group toral automorphism induced by the matrix A-1, then, we have f o h = h o f = id by virtue of the just proved, considering the product of matrices AA-1 = A-1 A = E. □

Lemma 3. If two periodic homeomorphZsms of n-torus f: Tn ^ Tn u f: Tn ^ Tn, f, f' G Gn have the same periods, then they are topologZcally conjugate by means of the group automorphism of the n-torus h: Tn ^ Tn2.

Proof. Let q be the period of the maps f and f'. Then f and f' can be represented in the form f (x) = x + u mod 1, f (x) = x + u' mod 1, where

(pi/q\

P2/q

u

u

\Pn/q)

and (pi,p2,... ,Pn,q) = 1, (pi,p2,. ■ ■ ,Pn,q)

2 Two dynamical systems f: X ^ X и f': X ^ X , defined on a topological space X topologically conjugate if there exists a homeomorphism h: X ^ X, что f о h = h о f'.

fp'i/q\ pp2/q

\p'n/qj -1.

Pi, pi e

are

Let g: Tn ^ Tn be a periodic toral homeomorphism of the following form g(x) = x + v mod 1, where v = (1/q, 0,..., 0)T

It follows from lemma 1 that there exist unimodular n x n matrices A and A' such that Av = u (mod 1), A v = u' (mod 1). Let us show that the group toral automorphisms h: Tn ^ Tn and h': Tn ^ Tn induced by the matrices A h A' are such that f o h = h o g and f' o h' = h' o g. Fix the point x E Tn, 0 ^ xi < 1 and set w = (x+v) — (x+v mod 1). We have hog(x) = A(x+v mod 1) mod 1 = A(x+v— w) mod 1 = Ax + Av + Aw mod 1 = Ax+u mod 1 = (Ax mod 1+u) mod 1 = foh(x). Since the point x was chosen arbitrary, we have an equality f o h = h o g. The second equality is proved similarly. We note that the group automorphism induced by the matrix AA'-1 coincides with h o h'-1 by the virtue of lemma 2. Now it is easy to see that the desired conjugation has the form f o (h o h'-1) = (h o h'-1) o f'. □

Lemma 4. Let a periodic homeomorphism of an n-dimensional torus g: Tn ^ Tn of

1/q 0

a period q has the form g(x) = x + v, where v = . . Then there exists a countable

0

set of unimodular n x n matrices ai such that the equality g o hi = hi o g holds for the group automorphisms of the n-dimensional torus hi: Tn ^ Tn induced by them,.

Proof. Let's denote a sent oif unimodular matrices such that the first column of the

1 0

by {Ai},i e N.

matrix Aj has the form

n0

Fix the point x E Tn and set w = (x + v) — (x + v mod 1). We have hi o g(x) = Ai(x + v mod 1) mod 1 = Ai(x + v — w) mod 1 = Aix + Aiv — Aw mod 1 = Aix + v mod 1 = (Aix mod 1 + v) mod 1 = g o hi(x) by the choice of Ai. The equality hi o g = g o hi is true since the point x was chosen arbitrary. □

2.1. Proof of the theorem 1

Proof. Let q be the period of the maps f and f', and let g: Tn ^ Tn be the periodic homeomorphism defined in lemma 4. It follows from lemma 3 that there are group automorphisms h: Tn ^ Tn and h': Tn ^ Tn induced by the matrices A and A, respectively, for which the equalities h o f = g o h, h o f = g o h hold. It follows from lemma 4 that there exists a countable family of unimodular matrices {A,i}, i E N such that the group automorphisms hi: Tn ^ Tn induced by them are pairwise non-homotopic, and the equalities g o hi = hi o g hold. It follows from lemma 2 that the group automorphism induced by the matrix A'-1AiA coincides with (h'-1 o hi o h) and the group automorphisms h -1 o hi o h are pairwise non-homotopic, since hi are pairwise non-homotopic. It is easy to see that the desired conjugation has the form (h'-1 o hi o h) o f = f' o (h'-1 o hi o h). □

2.2. Proof of the theorem 2

Proof. Let q be the period of the maps f and f', and let g : Tn ^ Tn be the homeomorphism defined in lemma 4. We have h o f = f o h by the condition of the theorem. Since f' and g have the same period, then there is a homeomorphism h' : Tn ^ Tn such that the equality h' o f = g o h' holds. We consider the set [hp} of all possible homeomorphisms hp : Tn ^ Tn of the form

hß (x) = x +

(uß (x2,

V

x3,

0

)

/

mod 1,

where up: Tn-1 ^ R is an arbitrary continuous function of n — 1 variables, periodic by each argument. Ntote that for any hp we have the equality hp o g = g o hp and each homeomorphism {hp} is homotopic to the identical mapping. We set ha = h'-1 ohp oh'o h. It is easy to see that for the homeomorphism h'-1 o hp o h' o h the equality (h'-1 o hp o h' o h) o f = f' o (h'-1 o hp o h' o h) holds. Moreover, the homeomorphism h'-1 o hp o h' o h is homotopic to h, since the homeomorphism h -1 o hp o h is homotopic to the identical mapping. Thus, the set {ha} is the desired family of conjugating homeomorphisms. □

References

1. Arov, D. Z. On the topological similarity of automorphisms and translations of compact commutative groups // Uspehi matematicheskih nauk. — 1963. — Vol.18, No.5. — P. 113— 138.

2. Gelfond, A. O, Linink, Y. V. Elementary methods in analytic number theory. — M: Fizmatgiz, 1962. — 272 pp.

3. Katok, A. B, Hasselblatt, B. Introduction to the Modern Theory of Dynamical Systems. — Cambridge: Cambridge University Press, 1999. — 768 pp.

4. Nielsen, J. Die Struktur periodischer Transformationen von Flachen // Dansk Videnskaternes Selskab. Math.-fys. Meddelerer. — 1937. — Vol.15. — P. 1-77.

Получена 10.06.2017