THE TOPOLOGICAL CLASSIfiCATION OF DIffEOMORPHISMS OF THE TWO-DIMENSIONAL TORUS WITH AN ORIENTABLE ATTRACTOR Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 4, pp. 595-606. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200405

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 37D20

The Topological Classification of Diffeomorphisms of the Two-Dimensional Torus with an Orientable Attractor

V. Z. Grines, E. V. Kruglov, O. V. Pochinka

This paper is devoted to the topological classification of structurally stable diffeomorphisms of the two-dimensional torus whose nonwandering set consists of an orientable one-dimensional attractor and finitely many isolated source and saddle periodic points, under the assumption that the closure of the union of the stable manifolds of isolated periodic points consists of simple pairwise nonintersecting arcs. The classification of one-dimensional basis sets on surfaces has been exhaustively obtained in papers by V.Grines. He also obtained a classification of some classes of structurally stable diffeomorphisms of surfaces using combined algebra-geometric invariants. In this paper, we distinguish a class of diffeomorphisms that admit purely algebraic differentiating invariants.

Keywords: A-diffeomorphisms of a torus, topological classification, orientable attractor

Received November 30, 2020 Accepted December 14, 2020

This work was performed with support of the Laboratory of Dynamical Systems and Applications NRU HSE of the Ministry of science and higher education of the RF grant no 075-15-2019-1931.

Vyacheslav Z. Grines vgrines@yandex.ru Olga V. Pochinka olga-pochinka@yandex.ru

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

Evgeniy V. Kruglov kruglov19@mail.ru

Lobachevsky State University of Nizhny Novgorod prosp. Gagarina 23, Nizhny Novgorod, 603950 Russia

1. Introduction and main results

In this paper we study structurally stable diffeomorphisms f defined on the two-dimensional torus T2 and containing a one-dimensional orientable attractor A in the nonwandering set NW(f). It follows from the results of V. Grines [4, 6] (see also the monograph [5]) that in this case

• NW(f) does not contain orientable basic sets other than A,

• the basic set A has only bunches of degree two that divide the boundary points into associated pairs pi, Qi, i e {1,..., k},

• the induced isomorphism f*: ^(T2) — n1(T2) is uniquely defined by the hyperbolic matrix Af e GL(2, Z).

We will denote by A: T2 — T2 the diffeomorphism given by the formula

A(x, y) = (ax + ffy, yx + 5y) (mod 1)

for any matrix A =

G GL(2, Z).

According to J.Franks [1], there is a unique, isotopic to the identity, continuous map hf : T2 — T2 that semiconjugates the diffeomorphism f with the diffeomorphism Af. The image hf (A) of the set A is the whole torus T2, and the set Bf = {x G T2 : h(x) consisting of more than one point} is the union of finitely many periodic points Pf = {g1 ,q2,...,qu } of the diffeomorphisms Af and their unstable manifolds. In this case, h-1 (Qi) n A, i G {1,2,... ,k}, consists of a pair of associated boundary points pi, qi of the basic set A.

Denote by G the class of structurally stable diffeomorphisms f : T2 — T2 such that NW(f ) consists of an orientable one-dimensional attractor A and a finite number of isolated periodic points whose stable manifolds closure belongs to simple arcs Lpiqi bounded by pairs of boundary points pi, qi (see Fig. 1). Then hf (Lpiqi) = Qi. Denote by nSi the number of sources on the

arc LPiqi.

The following theorem is the main result of this paper.

Theorem 1. The diffeomorphisms f,f' G G are topologically conjugate if and only if there exists a matrix H G GL(2, Z) such that HAf = Aft H, H (Pf ) = Pf/ and nSi = n^ ^ i = 1,...,k.

Fig. 1. Isolated periodic points located on a simple arc Lpq.

2. Necessary definitions and facts

2.1. A-diffeomorphisms with an expanding attractor of codimension one

Let Mn be a closed smooth orientable manifold of dimension n > 1, f: Mn — Mn be a diffeomorphism and NW(f) be its nonwandering set.

The diffeomorphism f is said to satisfy an axiom A (to be an A-diffeomorphism) if the set NW(f) is hyperbolic and the periodic points are everywhere dense in NW(f).

The following statement is called Smale's spectral decomposition theorem.

Proposition 1. Let f: Mn — Mn be an A-diffeomorphism. Then:

1) NW(f) is uniquely represented as a finite union NW(f) = Ai U.. .U Am of pairwise disjoint subsets of Ai, each of which is compact invariant and topologically transitive;

m m

2) Mn = U Ws(Ai) = U Wu(Ai), where Ws(Ai) = {y e Mn: fk(y) — Ai,k — and

i=1 i=1

Wu(Ai) = {y e Mn: f-k(y) — Ai, k —

The set Ai is called a basic set. Using dimAi, denote its topological dimension.

Let A be a basic set of an A-diffeomorphism f. The set A is called an attractor (repeller)

if it has a closed neighborhood UA C Mn such that f(UA) C intUA, fl fk(UA) = A

ken

(f-1(Ua) C intUA, n f-k(Ua) = A). In this case, A = (J Wu(x) (A = (J Ws(x)).

ken xeA xeA

If dim A = dim Wu(x) (dim A = dim Ws(x)), then the attractor (repeller) A is called expanding (contracting).

Due to [8, Theorem 3], any basic set A of codimension one of an A-diffeomorphism f: Mra — Mra is either an attractor or a repeller. In this case A is called orientable if for any point x e A and any fixed numbers a > 0, /3 > 0 the index of intersection of local manifolds W S (x) H WjU'(x) is the same at all intersection points (+1 or —1). Otherwise, the basic set A is called nonorientable [2].

Everywhere below A is an orientable expanding attractor of codimension one of an A-diffeomorphism f: Mra — Mra.

Diffeomorphisms f,g: Mn — Mn are called topologically conjugate if there is a homeo-morphism h: Mn — Mn such that h o f = g o h. If the last equality holds for a continuous map h: Mn — Mn (which is not a homeomorphism), then the diffeomorphisms f, g are called semiconjugate.

A diffeomorphism f is called structurally stable if there exists a neighborhood in the space of diffeomorphisms Mn — Mn such that any diffeomorphism from this neighborhood is topolog-ically conjugate to the diffeomorphism f. By virtue of the results of R. Mane [7] and C. Robinson [9], diffeomorphism f is structurally stable if and only if it is an A-diffeomorphism and satisfies the strong transversality condition. The latter means that Vx,y e NW(f) the stable manifold Ws(x) of the point x and the unstable manifold Wu(y) of the point y have only transversal intersections, that is, the sum of the tangent spaces to these invariant manifolds coincides with the tangent space to the ambient manifold at the intersection points.

For each point x e A, the set Ws(x)\x consists of two connected components, and by virtue of [3], at least one of them has a nonempty intersection with the set A. Point x e A is called boundary if one of the connected components of the set Ws(x)\x does not intersect A. Let us denote this component by Ws0(x).

The set of boundary points of the basic set is finite. The union of the unstable manifolds Wu(p1),.. .,Wu(prb) boundary points p1,...,prb of the attractor A whose components Ws0(p1),..., Ws0(prb) belong to the same path-connected component of the set Ws(A) \ A is called a bunch b of the attractor A. The number rb is called the degree of the bunch b. According to [4] and [6], if the diffeomorphism f is given on the torus Tn, then the attractor A admits only 2-bunches and the pair of boundary points included in the bunch is called associated (see Fig. 2).

In [6] it is proved that, if f : Mn — Mn (n ^ 3) is a structurally stable diffeomorphism whose nonwandering set NW(f ) contains an expanding orientable attractor A of codimension one, then:

• the manifold Mn is homotopically equivalent to the torus Tn, and for n = 4, Mn is homeo-morphic to Tn ([6, Theorem 5.1]);

• the set NW(f ) \ A consists of a finite number of isolated sources and saddles, and the union of the stable manifolds of isolated points closure and components Ws0(p), Ws0(q) splits into a finite number of simple arcs, each of which contains a finite nonzero number of sources, a finite (possibly zero) number of saddle points, and exactly two associated boundary points of the attractor A ([6, Corollary 5.2], see Fig. 3).

Note that the attractor orientability requirement can be omitted in the case n = 3 by [10].

For the case n = 2, any one-dimensional basic set of an A-diffeomorphism is either an expanding attractor or a contracting repeller. However, the statements formulated above are not true in dimension two. Namely, the ambient surface M2 of an A-diffeomorphism f whose nonwandering set NW(f) contains an orientable attractor A is not necessarily a torus, such diffeomorphisms admit any orientable surfaces other than 2-sphere. In the case where M2 = T2, according to [4], the set NW(f ) \ A consists of a finite number of isolated periodic points, which can be located on simple arcs (see Fig. 1) or have other disposition (see Fig. 4).

Fig. 4. Isolated periodic points outside the associated arc.

2.2. Orientable expanding attractors on a 2-torus

It is known that any diffeomorphism f on a two-dimensional torus induces an automorphism of the fundamental group f *: ^1(T2) ^ n1 (T2) , where the group n1(T2) is isomorphic to the Abelian group Z2. Then the automorphism f* is uniquely determined by the matrix Af belonging to the set GL(2, Z) of unimodular integer matrices. An automorphism f* is called hyperbolic if the matrix Af is hyperbolic, i.e., it has no eigenvalues with modulus one.

An algebraic automorphism A: T2 ^ T2 of the torus T2 = R2/Z2 is a diffeomorphism

( a f\

defined by the matrix A = e GL(2, Z), i.e.,

s J

A(x, y) = (ax + ffy, yx + Sy) (mod 1).

If the matrix A is hyperbolic, then A is an Anosov diffeomorphism, that is, the hole ambient manifold M2 is its hyperbolic set.

Proposition 2 ([1]). Let f: T2 ^ T2 be a diffeomorphism and let the induced isomorphism f*: n1(T2) ^ n1(T2) be hyperbolic. Then among the homotopic to the identity continuous maps of the torus T2 there is a unique map hf that semiconjugates the diffeomorphism f with the algebraic automorphism Af (see the diagram in Fig. 5). In this case, if f is an Anosov diffeomorphism, then hf is a homeomorphism.

V -U V

4,hf \,hf

V V

Fig. 5. Semi-conjugation.

Proposition 3 ([4, 5]). If an A-diffeomorphism f: T2 — T2 has an orientable basic set A, then the induced automorphism f*: n1 (T2) — n1(T2) is hyperbolic.

Proposition 4 ([4, 5]). If A is an orientable basic set of an A-diffeomorphism f: T2 — T2, then f has no orientable basic sets other than A.

Let A be a one-dimensional orientable attractor of the diffeomorphism f : T2 — T2 and hf: T2 — T2 be the semiconjugating map to the algebraic automorphism Af. Let Bf = = {x e T2: h-1(x) consist of more than one point}.

Proposition 5 ([4, 5]). The image hf (A) of the set A is the whole torus T2. The set Bf is the union of a finitely many periodic points Pf = {q1, Q2,..., Qk} of the algebraic automorphism Af and their unstable manifolds. The set h-1 (qi) H A, i e {1, 2,... ,k}, consists of two boundary points p,, qi of the set A.

Proposition 6 ([4, 5]). Let f,f': T2 — T2 be A-diffeomorphisms and A, A1 be their orientable attractors, respectively. Then there is a homeomorphism ^: T2 — T2 for which p(A) = A', f '|a' = ^f^~1\a/ if and only if there is a matrix H e GL(2, Z) such that HAf = = Af/ H and H(Pf) = Pf,.

3. Proof of conjugation criteria (proof of Theorem 1)

Consider a structurally stable diffeomorphism f: T2 — T2 from class G. We describe the properties of the diffeomorphism f, which will be used substantially in the proof of the theorem.

The nonwandering set of the diffeomorphism f e G contains a unique nontrivial basic set A which is a one-dimensional orientable expanding attractor. For different points a,b e Ws(x), x e A we denote by [a, b]s a compact segment of the manifold Ws(x) bounded by the points a, b. Let (a,b)s = [a,b]s \ (a U b).

The set Ta of all boundary points of the set A is not empty and consists of a finite number of periodic points that are split into associated pairs (pi,qi), i = 1,...,kf of points of the same period so that the bunch Bpiqi = Wu(pi) U Wu(qi) is accessible from within the boundary 1 of some connected component Gpiqi of the set T2 \ A. Denote by Ba the set of all bunches of the basic set A.

Let T(f) = NW(f) \ A and Tpiqi = T(f) H Gpiqi. Closures of the stable manifolds of isolated periodic points from the set Tpiqi belong to simple arcs Lpiqi bounded by pairs of boundary

1 Let V C M be an open set with boundary dV (dV = cl(V)\int(V)). A subset SV C dV is called

accessible from within the domain V if for any point x e SV there is an open arc that lies completely in V and such that x is one of its endpoints.

points pi, Qi. For any point x e Wu(pi)\pi, there is a unique point y e (Wu(qi) n Ws(x)) such that the arc (x, y)s does not intersect A. Define a map

£ptqt: Bpiqi\{pi, qi} ^ BPiqi\{pi,Qi}

assuming £pm (x) = y and £pm (y) = x. Then £pm (W u(pi)\pi) = Wu(qi)\qi and

ipiQi(Wu(qi)\qi) = Wu(pi)\pi, i.e., the map £piqi maps punctured unstable 2-bunch manifolds to each other and is an involution (£^iqi(x) = id). By virtue of the theorem of continuous dependence of invariant manifolds on compact sets, the map £piqi is a homeomorphism.

Denote by mpi the period of the point pi and by mi the period of the component Gpiqi. Then the restriction fmpi\Wu(pi) has exactly one hyperbolic repelling fixed point pi, so there is a smooth closed segment Dpi c Wu(pi) such that pi e Dpi c int(fmpi(Dpi)). Then the set Cpiqi = U [x,£piqi(x)]s consists of two segments and the points £piqi(dDpi) bounded

x£dDPi

in Wu(qi) segment Dqi such that qi e Dqi c int(fmi (Dqi)). The set Spiqi = Dpi U Cpiqi U Dqi is homeomorphic to a circle. We will call Spiqi the characteristic circle corresponding to the bunch Bpiqi. Each set Spiqi bounds a two-dimensional disk Qpiqi such that the set Lpiqi containing periodic points from the set Tpiqi is a subset of Qpiqi.

The induced isomorphism f*: n1(T2) ^ n1(T2) is uniquely defined by the hyperbolic matrix Af e GL(2, Z), and among the homotopic to identity continuous maps of the torus T2 there is a unique map hf that semiconjugates the diffeomorphism f with the diffeomorphism Af. The map hf sends the set A to the whole torus T2, and the set Bf = {x e T2: h-1(x) consisting of more than one point} consists of a finite number of periodic points Pf = {q1, q2,..., Qk} of the diffeomorphism Af and their unstable manifolds. In this case, h-1(Qi) n A, i e{1,2,..., k}, consists of a pair of boundary associated points pi,qi of the basic set A and hf (Lpiqi) = Qi. Denote by nei the number of sources on the arc Lpiqi.

In addition to the diffeomorphism f, we consider the diffeomorphism f e G, whose basic set contains a one-dimensional orientable expanding attractor A'; we provide strokes for all other objects considered in connection with the mapping f'.

We prove that the diffeomorphisms f, f' e G are topologically conjugate if and only if there exists a matrix H e GL(2, Z) such that HAf = Af/H, H(Pf) = Pf/ and nei = nq^), i = 1,...,k.

3.1. Necessity

If the diffeomorphisms f, f' e G are topologically conjugate, then there is a homeomorphism h: T2 ^ T2 such that hf = f 'h. The induced isomorphism of h*: n1(T2) ^ n1(T2) is uniquely defined by the hyperbolic matrix H e GL(2, Z), which by conjugation satisfies the condition HAf = Af/ H and consequently the condition HAf = Af/ H.

Since h is a conjugating homeomorphism, it converts the boundary points pi of the diffeomorphism f to the boundary points of the diffeomorphism f'. Let us say pi = h(pi) and

qi = h(qi).

It follows from the semiconjugations that hf f = Af hf and hf/f = Af/hf/. Since hf (pi) = Qi, it follows that hf (f (pi)) = Af (Qi) and hence Af (Pf) = Pf. Similarly, hf/(pi) = Qi, hf/(f '(pi)) = = Af/(Qi) and hence Af/(Pf/) = Pf/. So H(Pf) = Pf/ and H(Qi) = Qi.

Since h is a conjugating homeomorphism, the number nei of sources on the arc Lpiqi coincides with the number ng/ of sources on the arc Lpiq/, where nei = nq ^ ), i = 1,...,k.

3.2. Sufficiency

Let f,f' G G and suppose there exists a matrix H G GL(2, Z) such that HAf = Af/H, H(Pf) = Pf/ and nei = uqi = l,...,k. Let us construct a homeomorphism p: T2 ^ T2 conjugating the diffeomorphisms f and f'. We will construct a conjugation step by step.

3.2.1. Conjugation on nontrivial basic sets

Semiconjugation hf maps the attractor A to the hole torus T2, with hf (Lpiqi) = A similar action is performed by semiconjugating hf/ with the attractor A'. Moreover, the maps qf = hf \a\ba : A \ BA ^ T2 \ Wu(Pf), qf/ = hf/\A/\ba, : A' \ BA/ ^ T2 \ Wu(Pf/) are homeomor-phisms. By condition H(Pf) = Pf/, this means that the map

p = q— Hqf: A \ Ba ^ A' \ Ba/

is a homeomorphism conjugating f and f' by [4] (see also [5, pp. 224-225]).

We show how to continue p to the set Ba. For the boundary point p G Ta (p' G rA/) denote by Wsr(p) (Wsr(p')) the stable separatrix of the point p (p') that has a nonempty intersection with A (A'). From the fact that p is a conjugating homeomorphism, it follows that p(Wsr(rA)) = Wsr(rA/). Thus, the homeomorphism p is uniquely extended to the set rA. Since hf (Ws'r(pi) U Wsr(qi)) = Ws(gi) \ Qi for paired points pi, qi and H(gi) = gi, it follows that pi = pp), qi = p(q-i) are paired points such that gi = hf/ (Lp/q/).

Since for any point x G Wu(rA) there is such a sequence of points xn G Wsr(x) n (A \ Ba) for which x is a limit point, then the homeomorphism p can be continued to the set Wu{Ta) by the formula x' = p(x), where x' is the limit point of the sequence of points p(xn).

An exact proof of the fact that the map p is a homeomorphism conjugating the basic sets A and A' of f and f', respectively, at all points of the above-mentioned sets, is given in [4] (see also [5, pp. 224-225]). Thus, to prove Theorem 1, it is sufficient to continue the map p to the set T2 \ A.

3.2.2. Conjugation on trivial basic sets

For each associated pair of boundary points (pi,qi) there is a natural number uei such that the set of isolated periodic points Tpiqi consists of exactly uei periodic sources a\,..., aing and uei — 1 periodic saddle points r1 ,...,rn _i, alternating with sources on a simple arc

nei-1 nBi

Lp qi = Ws0(pj) U U Ws(rj) U U aj U Ws0(qi) from the point pi to the point qi (see Fig. 1). j=i j j=i j Similarly, by primes we denote the periodic points of the

equality pi(aij) = aj, pi(rj) = rji defines a homeomorphism pi: Tpiqi ^ Tp/qq/. Denote by pT : T(f) ^ T(f') a homeomorphism composed by p1 ,...,pk.

We show that pT conjugates the diffeomorphisms f \T(f) and f '\T(f/). Since the diffeomorphism H conjugates the diffeomorphisms Af, Af/ and H(gi) = gi, the periods of the points gi and gi are mi, where mi is the period of the connectivity component Gpiqi (Gp/.q/) of the diffeomorphism f (f'). Since hf (Lpiqi) = gi and hf/(Lp/q/) = gi, the arcs Lpiqi and Lp/q/ also have a period mi. In this case, the diffeomorphism fmi\LpiV preserves (changes) the orientation if the eigenvalue A of the matrix Af is positive (negative). Similarly, for the diffeomorphism f 'mi\L / /. Thus, if A > 0, then all periodic points of the diffeomorphism fmi\L

PM

are fixed; otherwise, there is exactly one fixed point, and all the others have a period of two. By construction, pT(Lpiqi П T(f )) = П T(f ') and the homeomorphism p conjugates f |Гл

and f '|гл/, hence the homeomorphism pT conjugates the diffeomorphisms f lTf ) and f 'lTf /).

The homeomorphism pT naturally continues to unstable manifolds of isolated saddle points. Exactly, let z G Wu(rj) \ rj. Then there is a unique point xz G Wu(pi) \ pi such that 2 = [xz,(piqi(xz)]s П Wu(rj). The same is true for the diffeomorphism f '. Let

Pt(z) = z' = [xZ(xZ)]s П W

pqi

2mi j _ f/2m;

and pT fmi (z) = f'mipT (z) if Л > 0; pT f2mi (z) = f'2mipT (z) if Л < 0.

3.2.3. Conjugation in source basins

Consider the source aj and put for convenience pi = r0, qi = r%n^. On the stable separatrices lj-1, lj of the saddles rj-1, rj lying in the basin of aj, we select points yj-1, yj and arcs vj-1, vj transversally intersecting the separatrices at these points. By A-lemma, we can draw these arcs so that each segment [x,(piqi, x G (Dpi \ pi) transversally intersects each of the arcs vj-1, vj at a unique point. Denote by Qj the closure of the connected component of the set Qpiqi \ (vj-1 U vj) containing aj. Let ф = fmi.

Let x1, x2 denote the boundary points of the segment Dpi. Let (piqi (x1) = x4 G Wu(qi) and £piqi(x2) = x3 G Wu(qi). So (x2,x3)s П Л = 0, (x1,x4)s П Л = 0, and both arcs intersect the manifold Wu(aj). Let A = (x1 ,x4)s П vj-1, B = (x2,x3)s П vj-1, C = (x2,x3)s П vj, D = (x1 ,x4) П vj. Then the closed curve Lj = ABCD is the boundary of the domain Qj. Let E = ф(А), F = ф(В), G = ф(С), H = ф^) (Figure 6 shows a case in which ф(р^ = pi, in the case of ф(р^ = Qi the reasoning is repeated verbatim). Then the closed curve ф(Lj) = EFGH is the boundary of the domain ф(Qj). Let Kj = cl (ф(Qj) \ Qj). It is obvious that the annulus Kj is a fundamental domain of the diffeomorphism ф restricted to Wu(aj) \ aj.

X2

Pi

4-1

Xi

В

У-1

ПЦ)

т%

a:-

G

D

K)

Fig. 6. Fundamental domain of the diffeomorphism f.

хз

qi

X4

Let us add a prime to similar objects for the diffeomorphism f', assuming pi = p(p-i),

qi = p(qi), xi = p(xi). By construction, for any point v G (AB \ Ws(rj-1)) there is a unique

point xv G (Wu(pi) \ pi) such that v — (xv ,^piqi (xv))s n AB. Define the homeomorphism pAB: AB — A'b' by the formula paB(v) = v' G A'b', where v' = (x'v,£p/q/ (x'v))s n a'b' and

pab(AB n Ws(rj-1)) = A'B' n Ws(r'i_ .1)) (see Fig. 7). Similarly, we construct the homeomorphism pcD: CD — C'D'. Finally, we compose a homeomorphism pLi : Lj — L'J. Let p^(Li) =

= $pLi : 4>(Lj) — $(L'i). In the same way, we construct homeomorphisms pae : AE — AE',

j j j

pBF: BF — B'f', pCG: CG — C'G' and pDH: DH — D'H', where all segments are chosen transversally to the foliation Ws(A). Denote by pd the union of the constructed homeomorphisms. Let x' = pd (x).

E

H'

Fig. 7. Conjugation on AB and CD.

Consider the quadrilaterals AEFB, BFGC, CGHD, DHEA obtained by dividing the annulus Kj into segments AE, BF, CG, DH. We foliate each of them by segments of stable manifolds Wxf,. The set of the segments thus obtained is denoted by W. Then we foliate each of the quadrilaterals by segments that are transversal to the segments from W, the set of which we denote by R (see Fig. 8).

F G

\ \ \ \ \ \ \ 1 / ' / / / / /

\ \ 1 t / / //

// / / /—/—/— -\-\-^-\ \

X / / / / / / \ \ \ \ \x-

Fig. 8. Foliation on the fundamental domain Kj.

Consider an annulus K'j that is similarly divided into four quadrilaterals. We foliate each quadrilateral of this annulus by segments as follows: if there is a segment from the set R with boundary points x and y, then there is a segment from the set R' with boundary points x' = po (x), y' = pa (y). Then on each quadrilateral of the annulus Kj homeomorphism pg induces a map pw ■ W — W', pR: R — R'. Then a homeomorphism pKj of the annulus Kj = ABCDEFGH is defined by the following rule: if z is the intersection point of segments w eW and r eR, then pKj(z) is the intersection point of segments pw(W) and pR(r). The

equality pWu(ai) = (pKj(z))), where z e Kj, k e Z, defines the required homeomorphism

( j) j

pwu(ai) ■ Wu(aj) - Wu(aj).

Conflict of Interest

The authors declare that they have no conflict of interest.

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