Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 3, pp. 321-334. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd210306

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 37D05

Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams

M. K. Barinova, E. Y. Gogulina, O. V. Pochinka

The present paper gives a partial answer to Smale's question which diagrams can correspond to (A, B)-diffeomorphisms. Model diffeomorphisms of the two-dimensional torus derived by "Smale surgery" are considered, and necessary and sufficient conditions for their topological conjugacy are found. Also, a class G of (A,B)-diffeomorphisms on surfaces which are the connected sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class G realize any connected Hasse diagrams (abstract Smale graph). Examples of diffeomorphisms from G with isomorphic labeled Smale diagrams which are not ambiently Q-conjugated are constructed. Moreover, a subset G* C G of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient Q-conjugacy is singled out.

Keywords: Smale diagram, (A,B)-diffeomorphism, Q-conjugacy

1. Introduction and formulation of the results

Let f be a diffeomorphism of a connected closed n-manifold Mn. In 1967 S. Smale [1] introduced a concept of A-diffeomorphism, i.e., a diffeomorphism whose nonwandering set is hyperbolic and whose periodic points are dense in it. He proved that the nonwandering set NWf of an A-diffeomorphism f is a union of pairwise disjoint subsets A1, A2, ..., Ar each of which is compact, invariant, topologically transitive and is called basic set. Moreover,

Received July 14, 2021 Accepted September 7, 2021

This work was supported by the Russian Science Foundation (project 21-11-00010).

Marina K. Barinova mkbarinova@yandex.ru Ekaterina Y. Gogulina ekaterinagogulina@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

r

r

Mn = U = U wl •

P=1 P=1

Basic sets Aj, Aj are said to be in Smale relation — (Aj — Aj) if

W{. n W'U. = 0.

An A-diffeomorphism f satisfies axiom B (is an (A, B)-diffeomorphism) if from the condition Ai — Aj it follows that there exist periodic points p G Aj, q G Aj such that the manifolds Wf and Wu have a transverse intersection point. For (A, B)-diffeomorphisms the Smale relation — is a partial order relation.

A sequence of pairwise distinct basic sets Ai = Ajo, A^, ..., Ai = Aj (m ^ 1) such that Aio — Aii — ... — Aj is called a chain with the length m G N connecting the basic sets Aj and Aj. Such a chain is called maximal if no new basic set can be added to it.

The Smale diagram Aj of an (A, B)-diffeomorphism f: Mn ^ Mn is a graph whose vertices correspond to the basic sets and whose directed edges sequentially connect vertices of maximal chains. In fact, the Smale diagram is a special case of a Hasse diagram. Let us recall that a Hasse diagram of a partially ordered set (X, —) is a graph whose vertices are elements of the set X, and a pair (x, y) forms an edge if x — y and : x — z, z — y.1

Lemma 1. A Smale diagram of any (A, B)-diffeomorphism is a connected Hasse diagram.

In [1] the following question is formulated as a problem (Problem 6.6a): which diagrams can correspond to (A, B)-diffeomorphisms?

This paper provides a partial answer to this question. Namely, let us define a model diffeo-morphism FCos 0u : T2 ^ T2 derived from an algebraic Anosov diffeomorphism C with a hyperbolic matrix C by "Smale surgery" along invariant manifolds of a finite (possibly empty) set of pairwise disjoint periodic orbits: along the stable manifolds of periodic orbits Os = [Of, ..., Osk} and along the unstable manifolds of periodic orbits Ou = [Of, ..., Of}.

Theorem 1. Model diffeomorphisms Fc 0s 0u, Fc, 0,s 0,u : T2 ^ T2 are topologically conjugate iff there exists a matrix H G GL(2, Z) such that HC = C'H and H(Os) = O's, H(Ou) = = O'u, where H is the induced automorphism of the 2-torus.

In Section 5, the class G of (A, B)-diffeomorphisms of closed orientable surfaces that are the connected sum of model diffeomorphisms is introduced. In Section 6 we prove the following result.

Theorem 2. Any connected Hasse diagram can be realized by some diffeomorphism from the class G.2

A labeled Smale diagram is the Smale diagram in which the topological conjugacy class of the restriction of the diffeomorphism to the corresponding basic set is additionally specified near each vertex. Two labeled diagrams are called isomorphic if there is an isomorphism of the corresponding graphs that preserves the incidence, the orientation of the edges, and the topolog-ical conjugacy classes of the vertices. Thus, the isomorphism of the labeled diagrams Aj, Ag of (A, B)-diffeomorphisms f, g is a criterion for an Q-conjugacy, that is, for the existence of a home-omorphism h: NWj ^ NWg such that hf\NW = gh\NWf. But in general this homeomorphism

g

does not extend to the ambient manifold.

1For the first time this kind of visualization was systematically described by Birkhoff [2] in 1940; he named it in honor of Helmut Hasse, who used similar diagrams, however, such drawings were found in earlier works, for example, in the textbook of the French mathematician Henri Vogt [3], published in 1895.

2The idea of such a realization is developed in [13], but we present it here for completeness.

Recall that diffeomorphisms f: Mn ^ Mn, g: M n ^ M are called ambiently Q-conjugate if there exists a homeomorphism h: Mn ^ M/n such that hf \NW = gh\NWf.

In Section 7 examples of diffeomorphisms from G with isomorphic labeled Smale diagrams which are not ambiently Q-conjugated are constructed. Let us single out a subclass G* C G of diffeomorphisms in which any two model diffeomorphisms are connected by at most one orbit. For such diffeomorphisms, the isomorphic class of the labeled Smale diagram is a complete invariant of the ambient Q-conjugacy.

Theorem 3. Diffeomorphisms f, f £ G* are ambiently Q-conjugate iff their labeled diagrams are isomorphic.

2. Every Smale diagram is a connected Hasse diagram

The connectivity of a Smale diagram will be proved in this section.

Proof of Lemma 1.

Let f be an (A, B)-diffeomorphism given on a connected closed manifold Mn and let NWf be the nonwandering set of f. Let — be a Smale partial order relation given on the set NWf. Suppose the contrary: the Hasse diagram (NWf, —) = r is not connected, that is, there is a connected component r of r, different from r. Let Li = {A^, ..., Ai } be the basic sets corresponding to the vertices of T^. Since r is a connected component of the graph r, it follows

mm mm

that (J Wl. = IJ Wl. .Let Mn = (J W^. = (J . We show that Mn is an open subset

j=i j j=i j j=i j j=i j of Mn.

To this end it is sufficient to show that every point x £ Ml1 possesses an open neighborhood Ux C Mn such that each point y £ Ux belongs to an intersection of Wl with Wl

for some m1(y), m2(y) £{1, ..., m}. Indeed, if we suppose the contrary, then there is a basic set A* which does not belong to the set L and whose closure of the invariant manifolds contains the invariant manifold of some basic set from Li. By virtue of the (A, B)-axiom we find that A* is connected by the order relation — with a basic set from Li and, hence, belongs to Li, which contradicts the assumption.

In the same way the complement of M n in Mn is also open. Hence, the manifold M n is open and closed simultaneously and, therefore, coincides with the ambient manifold Mn. Thus, the connected component r coincides with r, which contradicts the assumption. □

3. Model diffeomorphisms on the torus

In this section model diffeomorphisms on the two-dimensional torus whose connected sums realize an arbitrary Smale diagram will described.

3.1. Smale surgery

Let C £ SL(2, Z) be a hyperbolic matrix with eigenvalues Àl5 X2 such that A = |A1| > 1 and |A2| = 1/A. Since the matrix C has a determinant equal to 1, it induces a hyperbolic automorphism C : T2 ^ T2 with a fixed point O. According to [4], this diffeomorphism is an Anosov diffeomorphism and has two transverse invariant foliations (stable and unstable), each of which is dense on the torus. Moreover, the set of periodic points of the diffeomorphism C is also dense on T2.

S. Smale [1] proposed a so-called "surgery" to obtain a diffeomorphism with a one-dimensional attractor and a fixed source point. Let us provide one of possible versions of such a surgery (for details, see [5]).

Let (x, y) be local coordinates in some neighborhood U(O) of the point O G T2 such that the diffeomorphism C: T2 ^ T2 in this coordinates can be represented as C(x, y) = (x/X, Xy). Then {y = const} and {x = const} are stable and unstable foliations of C. Let j: R ^ [0, 1] be a C^-smooth function defined by the formula

0,

j(x) = S j(x), 1,

x < X"3, X"3 < x < 1, x > 1,

where j(x): (X 3, 1) ^ (0, 1) strongly monotonically increases (see Fig. 1a).

Fig. 1. (a) The graph of the function j(x); (b) The graph of the function v(x)

Define the function v: R ^ R by the formula v(x) = X"^1"^^^x (see Fig. 1b). Let D2 = {(x, y) G R2 | x2 + y2 ^ 4}. A diffeomorphism BC Qs: D2 ^ D2 defined by the formula BC Os (x, y) = (v"1(x), y) has the form BC Qs (x, y) = (X2x, y) if x2 + y2 ^ X"6 and is identical on <9D2. _

Let BC Os: T2 ^ T2 be a diffeomorphism which coincides with BC Os in U(0) and is identical outside it. Then, according to [6], the diffeomorphism FC Os = BC Os oC is a ^A-diffeomorphism whose nonwandering set consists of a one-dimensional attractor AC Os with the so-called bunch of degree 2 formed by two different boundary fixed points pOS and qOs, and a fixed source aOS (see Fig. 2).

The construction described above is called the Smale surgery along the stable manifold of a fixed point. Since in the neighborhood of a source point in the local coordinates x, y the diffeomorphism FC Os is a linear extension, it follows that in the basin WUOs there exists a pair of transversal FC Os-invariant foliations having the form {y = const} and {x = const} in the local coordinates x, y. Thus, the diffeomorphism FC Os has a pair of global transversal invariant

T1l L0s Po°

a0s

.................................... ...................... T s

1o°

Fig. 2. DA-diffeomorphism FC

C,Os

foliations LsC Os, L'C Os containing the manifolds W£, x G Ac, Os, W, x G AC Os, respectively, as leaves (see Fig. 2).

3.2. Generalization of the surgery

The surgery described above is performed in a neighborhood of any periodic orbit of the Anosov diffeomorphism C. It can also be generalized to a surgery along the unstable manifold of a periodic orbit.

A model diffeomorphism of the torus is a diffeomorphism FC Qs Ou: T2 ^ T2 obtained from

the algebraic Anosov diffeomorphism C by Smale surgery along invariant manifolds of a finite (possibly empty) set of pairwise distinct periodic orbits: along the stable manifolds of orbits OS = = {OS, ..., O'SS} of periods m1, ..., mk, respectively, (s-orbits) and along the unstable manifolds of periodic orbits Ou = {Of, ..., O'} of periods n1, ..., nl, accordingly, (u-orbits).

By construction, the nonwandering set NWp^^^ ^ contains a unique nontrivial basic set ACC)s 0u and the diffeomorphism FCos 0u has a pair of transversal invariant foliations L'C Os Ou, L'C Os Ou containing, as leaves, stable and unstable manifolds, respectively, of points from A

c, os, ou

The set aQB = [ac

a0s} is a set of source periodic orbits and the set u =

= K

..., wOu} is a set of sink periodic orbits of the diffeomorphism FC Qa Ou. Sets pOs =

= {Pos, ..., Pof}, s = s, ..., Qof}, Pou = {Pou, ..., Pou }, ^u = {QQu, ..., u} consist i k i k i i i i

of boundary periodic points, such that for each i £ {1, ..., k} there exist stable separatrices lp f and l® f of the orbits ^s and ^s which belong to the basin of the same source periodic

orbit ^s and for each j £ {1, ..., l} there exist unstable separatrices l® u and 10

% Oj

orbits p0u and qQu which belong to a basin of the same sink periodic orbit u.

C os ou has one of the following possible structures:

and l® of the

qou

J

yu ^^^ HQu

JJ

In this case, the basic set A

has the topological dimension 2 and is an attractor and a repeller simultaneously if OS = = 0u = 0;

has the topological dimension 1 (see Fig. 3) and is an attractor (repeller) if Os = 0,

Ou = 0 (Os = 0, Ou = 0);

has the topological dimension 0 (see Fig. 4) and is a saddle basic set if Os = 0 and Ou = 0.

Fig. 3. The diffeomorphism FC {Os Osy with s-orbits with periods 2, 3 and the attractor AC {Os Osj; the diffeomorphism FC {O„ Ouy with w-orbits with periods 1, 2 and the repeller AC {O„ Ouy

Fig. 4. Diffeomorphism FC {Osy {O„y with s-orbit of period 4, w-orbit of period 2, and saddle basic

set AC,{Osy , {Ouy

4. Topological conjugacy of the model diffeomorphisms

In this section, we prove Theorem 1. Firstly, notice that necessary and sufficient conditions for model Anosov diffeomorphisms follow from [7-9]. For the case where exactly one of the sets Os, Ou is empty, Theorem 1 follows from [7].

Thus, everywhere below we suppose that both sets Os, Ou are not empty and, hence, AC os ou is a zero-dimensional basic set. Let us provide some necessary information and prove auxiliary lemmas using ideas of the paper [10] and the book [11].

We say a point x G Ac qs qu is s-dense (w-dense) if both connected components of the set W%. \ x (WXU \ x) contain sets which are dense in a periodic component of the AC Qs Qu containing the point x. A point x G AC qs Ou is said to be an s-boundary point (w-boundary point) if one of the connected components of the set W£ \ x (WU \ x) is disjoint from AC Qs Qu.

Any model diffeomorphism FC Qs Qu corresponds to a hyperbolic matrix C and by construction this diffeomorphism has the form FC Qs Qu = BC Qs Qu ◦ C, where BC Qs Qu : T2 — T2 is isotopic to the identical diffeomorphism. Thus, the diffeomorphism FC Qs Qu is isotopic to the diffeomorphism C.

Proposition 1 ([11, Lemma 9.8]). Let f: T2 — T2 be a diffeomorphism such that the induced isomorphism f *: n1 (T2) —> n^(T2) is hyperbolic and let C be the matrix that defines the isomorphism f*. Then there is a unique continuous map h of the torus T2 which is ho-motopic to the identity and such that it semiconjugates the diffeomorphism f to the algebraic automorphism C.

Let 7r: R2 —> T2 be a universal covering 7r_1(Ac,C)s qu) = Ac os ou. If i e T2, let x G R2 denote the point in the preimage n~l(x). Denote by w5 G {s, t/,} the curve on R2 containing x such that 7r(«4) = Wj, x G ACQS Qu. For points y, ~z G w^ (y / z) let [y, [y, ~z)s, (y, (y, z)s denote the connected arcs on the manifold w§ with the boundary points y, z.

By Proposition 1, there exists a unique continuous map h: T2 — T2 which is homotopic to the identity and such that hFcQs Qu = Ch. Let h: R2 —> R2 be a lift of the map h. For every x G T2 let W£ (Wu) denote the stable (unstable) manifold of the point x with respect to the automorphism C and let wf- denote the connected component of the preimage of the manifold W^ of the automorphism C which passes through the point x.

Lemma 2. Let x, y G Acos ou, w§ = Wg (w§ = Wg) and (x,y)s n Ac,os,ou / 0 {{x, y)u n Ac,o\o™ / 0)- Th'en Hx) / h{y).

Proof.

We consider the case «>§■ = Wg (the case w§ = Wg is similar). By [11, Lemma 9.10] there is r > 0 such that d(x1, x2) < r for any two points xlt x2 G AC Os OU for which /i(ar1) = h(x2), where d is the Euclidean metric on R2. Let p G AC Qs Qu be an s-dense periodic point and let m be the natural number for which FmOs ou(p) = P. Then there are points x*, y* G (x, y)s such that x*, y* e Wp+, where is one of the connected components of W^\p. Let p G Ac,os,ou be a point in the set n_1(p) and let 0, 0- be the lifts of the diffeomorphisms FmOs Ou, Cm such that (f)(p) = p, h(f) = (f)~h. Denote by yx* the preimages of the points y*, x* belonging to the curves Wg , respectively. On the curve Wg there are points x', y' congruent to

the points x, y, respectively, with respect to integer plane shifts. Since the arc (x', y')s C Wg contains the arc (a7*, C wig , the points x', y' are separated by the curve w^ on R2. By the A-lemma (see, for example, [12, A-lemma]), there is N > 0 such that d((f)~N(x'), (f)~N(y')) > r. Suppose that, contrary to the assumption, h(x) = h(y). Then h(x') = h(y'), h((f)~N(x')) =

- 0qN(Hx')) = (pQN(h(y>)) = h(rN(y')) and, therefore, d(rN(x'), rN(y — -

impossible. □

Lemma 3. If for a point x G Ac, 0s 0u its manifold \ x contains no 5-boundary points, then h(w|r) = Wj.

Proof.

Let us provide the proof for the case 5 = u (the proof for 5 = s is similar). First we show that h(w§) C w^y Let y (y ^x) be an arbitrary point on the curve The definition of an

unstable manifold implies lim p(Fpj0s 0u{x), F'AQs Qu(y)) = 0 for x = p(x), y = p(y) where

n^ — tt ' ' ' '

p is the metric on the torus. Continuity of h implies lim p(h(Fn0* ou (x)), h(FCn 0* ou(y))) =

n^—œ

u\

is

= p(Cn(h(x)), Cn(h(y))) = 0, therefore h(y) £ W'H{x), that is, h(W^) C W'H{x). Since h i

a lift, of h, we have h(u& C The curve h(u& is a connected set on the curve and

h(x) J- h(x)

it. contains the point. h(x).

We now show that. h(w§) = w^^y Suppose the contrary, h(w§) / Then, by Lemma 2,

the image of one of the connected components of the set. w§ \ x by the map h is a bounded set. on the line But., by Lemma [11, Lemma 9.9], the map h is proper, i.e., the preimage of

every compact set is compact, and this is a contradiction. □

Lemma 4. Let points p, q £ Aco* 0u are 5-boundary periodic points from the same bunch. Then

1) the curves w^, w^, where 5 = s if 5 = u and 5 = u if 5 = s, bound on R2 a domain Qpg disjoint from the set AC Os qu and it(Q^ ^) is an injective immersion of the open

2) h(p) = h(q)',

3)

Proof.

Let. us provide the proof for the case 5 = s. The other case is analogous. Notice that, all the

curves where x G AC 0s 0u and x is u--dense point., have the same asymptotic directions3 [11,

Corollary 9.5]. Since the domain Q^ is disjoint, from the set. AC0s 0u and since there are

~ , Lpi q >

no congruent, points on the curves the domain contains no congruent, points either.

Therefore, tt(Qp^) is an immersion of the open disk into T2.

We now prove items 2 and 3. Let m be the period of the points p. Since q and p are from the same bunch and FC O* Ou is an orientation-preserving diffeomorphism, their periods are equal.

Let. (f): R2 —> R2 denote the lift, of the diffeomorphism F™QS Qu such that. (f)(p) = p and cf)(q) = = q. Since hFc 0s Qu = Ch, we have hF^Qa Qu = Cmh, therefore there is a lift. of the

' ' ' ' C

diffeomorphism Cm such that. h(f> = (pQh. Suppose now h(p) / h(q). Then the points h(p), h(q) are distinct fixed points of the diffeomorphism 0 -, which is impossible.

We now show that. h(Qpg) C h(wSuppose the contrary: there is a point, y G Qp-q such that. h{y) £ h{w'§). Lemma 3 implies that. h{y) £ .. Then lim d{(f)%n {h{y)), (J)%n{h{p))) =

1 h\P) ra—s-+oo C C

= lim d(h(<fi~n (y)), h(<fi~n(p))) = +oo but. d(<fi~n(y), 4>~n(p)) < r0 for all n > 0, where r0 is

n^+tt

some positive number since y belongs to the basin of some source. This contradiction completes the proof of the lemma. □

3Let x G Ac, Qs c„, t G R be a parameter on the curve «4 such that «4(0) = x. A curve «4+ («4 ) has the asymptotic direction 6^ (S^) t —> +oo (t —> — oo) and there is a finite limit = lim

G,0S,0"

+ + s

I 1 O^l V~l £ I { £ \ V I /-v^ ( "/■ V /-v^ A V~l r] t" Vl ni^n in A fn V~l if n 1-1 W^I t" /\ I - 1 T y ■

(t)

_L_L 1 Tirnoro

= ^ lim pr^j^ where xs(t), ys(t) are the Cartesian coordinates of the point w^(t) on the plane _ RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2021, 17(3), 321 334 .

Lemma 5. The image of the set Kc 0s 0u by h is the whole torus T2. The set EF^ ^ ^^ =

= {x G T2 : h-1 (x) consists of more than one point} is the union of unstable manifolds of points from Os and stable manifolds of points from Ou of the algebraic automorphism C. Moreover,

h-1 (WO. ) = wuos u wpuos u wuoa and h-1 (Wj ) = wsou u wpsou u wsou.

u> ~n. „. „.

j j j

Proof.

Let us prove that the image of the set ACC)s 0u by the map h is the whole torus T2. Let x be a point from Ac Qs Ou both components of whose unstable manifold are dense in Ac Qs Ou.

Lemma 3 implies h(wu) = w^). Since the automorphism C is hyperbolic, the manifold wU(x)

is dense in T2. This and the continuity of h imply that h(AC,Os ,ou) = T2.

Let x be an arbitrary point from Ac os ou. Two cases are possible: 1) w§ is disjoint from the preimage of any s-boundary and w§ is disjoint from the preimage of any «-boundary periodic point of the set Acos 2) w§ (w§) intersects the preimage of an s-boundary («-boundary) periodic point p of the set Ac Qs Qu at a point p.

In the former case there is no point y (y ^x) from AC C)S Ou such that h(x) = h(y). Indeed, suppose the contrary. First, let y G But for all y G W™, (x, y)u n ACOs Qu / 0 so y / x by Lemma 2. So y fi w§. Consider the domain Q^-g on R2 bounded by the curves w: Pick a point J on the curve such that the curve «4 tends to infinity on R2 in both possible

и . -a

^ such that the curve «4 — тп>2

directions, then [11, Corollary 9.5] implies u4nwg ф 0. Let z' = w§nwg. By [11, Theorem 8.5],

Лс,

qs qu П Qxy Ф 01 therefore the open arc (z, z')u intersects Ac\os,ou and, by Lemma 2, h(z) Ф h(z'). On the other hand, z G l! G by [11, Corollary 9.5], h(w£) = h{uig) and

we get h(z) = h(z'), which is a contradiction.

Consider another case. Suppose that w§ intersects the preimage of an s-boundary periodic point p of the set ACOs Qu at a point p (another case is proved in a similar way). Let Qp^ be the domain on R2 satisfying item 1) of Lemma 4. If x = p, then by item 2) of Lemma 4 h(x) = h(y), where y = q. For all y G W™ (x, y)u П ACC)S Qu ф 0, so by Lemma 2 у is the unique point on R2 for which x = у. If x ф p, then by [11, Corollary 9.5] w^D иЩ ф 0. Let у = П w^, then, since h(w§) = we have h(x) = h(y). If simultaneously «>§■ intersects

the preimage of an u--boundary periodic point p' of the set Acos Qu at a point p\ then there is another point y' = w§ П ud,, where q' is from the same bunch as p', such that y' = x'. In this case there are 4 points on R2 with equal values of h. If «>§■ is disjoint from the preimage of any u--boundary periodic point, then у is the unique point such that h(y) = h(x) by Lemma 2.

Since h is a lift of h, it follows that Ep is the union of unstable manifolds of

rC,Os,Ou

points from Os and stable manifolds of points from Ou of the algebraic automorphism C

7 s U Ws и Wqs

and h-1 (WO, ) = Wu U Wu U W'u and h-1 {W^u ) = Ws U Ws U Ws . □

4 O-> s Ps-,s s 4 O • / sr\u Pf-,u u

j ' snu Pnu qnu

n. n. n.

Now let us prove Theorem 1: model diffeomorphisms FC Qa Qu, FC, O,s O,u : T2 ^ T2 are topologically conjugate if and only if there exists a matrix H G GL(2, Z) such that HC = C'H and H(Os) = O's, H(Ou) = O'u, where H is the induced automorphism of a 2-torus.

Proof of Theorem 1.

For simplicity of notation, let F = FC Qa Qu, F' = FC, 0,a 0,u, A = AC Qa Qu, A' =

_ A TU _ TU T 'u _ TU T s _ T s T 's _ T s

— AC' O's O'ui L — LC Os Ou i L — LC' O's O'ui L — LC Os Ou i L — LC' O's O'u'

Necessity. Let diffeomorphisms F, F': T2 ^ T2 be topologically conjugate by a homeomor-phism h: T2 ^ T2. The induced isomorphism h*: (T2) ^ n^(T2) is uniquely determined by

a matrix H G GL(2, Z) which, by virtue of conjugacy, satisfies the condition HC = C'H and, consequently, the condition HC = C'H holds.

Since h is a conjugating homeomorphism, it follows that h(Wf ) = Wf , i g{1, ..., k}

o'.s '

and h(Wf u) = Wf , j G {1, ..., l}. By Proposition 1 and Lemma 5, there exist maps hF

"oV Soh j

and hF, such that hFF = ChF, hF,F' = C'hF, and hF Ws ) = Of, hF (Wf^u ) = Os

hF, (Wf ) = O'U, hF, (WS ) = O'u. Due to the uniqueness of hF, and the equalities hF = F'h,

Oi Oj

HC = C'H we get hF, = HhFh-1. Then hF,h(W ,f s) = HhF(Wf s), hence hF,(Wf s) =

= H(Of). Thus, Of = H(Of). In a similar way it is possible to show that Oj = H(0s).

Sufficiency. Suppose there is a matrix H G GL(2, Z) such that HC = C'H and H(Os) = = O's, H(Of) = O'f. Let us construct a homeomorphism h: T2 T2 which conjugates the diffeomorphisms F and F'.

Let W = Wf U Wf U Ws U Wf , W' = Wf U Wf U U Wf and A = A \ W,

A' = A' \ W'. By Lemma 5, the maps hF = hF|x : A — T2 \ (W'fs U WOu), hF/ = hF/ |X/ : A' —

— T2 \ (Wf/s U Wf u) are bijections, moreover, since H is the algebraic automorphism of a torus

satisfying the conditions of the theorem, H(Wfs U Wfn) = Wf/s U Wfu. This means that the map

h = h— HhF: A — A

is a homeomorphism conjugating Fand F' . Furthermore, by the invariance of the stable and unstable foliations, the homeomorphism h can be extended to the whole A such that h(pOs) =

= pO/s and h(pOu) = pO/u for all i g{1, ..., k}, j g{1, ..., l}. Let us show how to extend h to

i j j

a homeomorphism h: T2 — T2, conjugating F and F'.

To do this, let us denote by Wf the union of the closures of stable separatrices l0 s and l® s

Ois Ois

lying in Wf s and by Wf the union of the closures of unstable separatrices l® u and l® u lying Oi Oi Oi in Wf . W/S, Wf are similar notations for the diffeomorphism F'. Let us continue the homeo-

"Ou j

morphism h

to the homeomorphism hf : Wf ^ W's, conjugating F\WS with F'\w;s and the

homeomorphism h

pos

I

u . TJT-U

to the homeomorphism hf: Wf — W'f conjugating FlW„ with F'lWu

pOu J J J j j

j

k k i i in an arbitrary way. Let Ws = (J Ws, W's = Q W's, Wu = U Wf, Wf = U Wj". Denote

i=l i=l j=1 j=l

by hs: Ws — W's a homeomorphism composed of hs and by hf: Wu — W'u a homeomorphism

composed of hf.

Let ri j be an arbitrary point belonging to the set Wfos n Wps^. Then (see Fig. 5) there is

Oi Oj

a unique path connected component of the set Wf s n Wf u, containing ri j in its closure, which

Oi Oj

we denote by Qrij. The closure of a similar component for the point r'j = h(rij) is denoted

by Q / . Let us show how to extend the homeomorphism h to the set Qr , which completes the

ri,j ri,j

proof of the theorem.

Fig. 5. Construction of a conjugating homeomorphism

Any point x E Qr is the intersection point of the leaves L'"x E Lu and Lsx E Ls. Define the homeomorphism h: Qr — Q , by the formula h(x) = hs(Lu) П hu(Lsx) П Q , , where hs: Lu — L'u, hu: Ls — L's is the map of leaves induced by the map hs, hu, respectively. □

5. Connected sum of the model diffeomorphisms

Consider model diffeomorphisms FCOs Ou, F'c, O,s ö,u : T2 — T2 such that Ou = 0, O's = 0 and periodic orbits Ou E Ou, O's E O's have the same period. Since Ou is a sink orbit and O's is a source orbit, there exists a diffeomorphism p: WOu \ Ou — Wq,b \ O's such that

Ф ◦ FC.O s .O u

\ou = Fc''O's'O'u 0 \ou •

Let us say

Q = (T2 \ Ou) U (T2 \ O's), Q = (T2 \ Ou) U^ (T2 \ O's)

and denote by p: Q ^ Q the natural projection.

A connected sum of model diffeomorphisms Fc os ou, F'c, 0,s 0,u along the orbits Ou, O's

is a diffeomorphism 0: Q ^ Q which coincides with the diffeomorphism pFC Qa Qup-1 ^ 2 on p(T2 \ Ou) and with the diffeomorphism pF'c, p(T2 \ O's).

An example of a connected sum of model diffeomorphisms is shown in Fig. 6. Here Q is an orientable surface of genus two (pretzel). The connected sum operation is naturally generalized to the case of several model diffeomorphisms and several periodic orbits.

Denote by G a class of (A, B)-diffeomorphisms which are a connected sum of model diffeo-morphisms on a torus.

6. Realization of a connected Hasse diagram by a connected sum of the model diffeomorphisms

In this section, following [13], we realize any connected Hasse diagram by some diffeomor-phism from class G. A scheme of the realization is provided below.

Proof of Theorem 2.

Let r be a connected Hasse diagram with vertices v1, v2, ..., vr and a set of oriented edges by the form (v{, Vj), i = j. Let us number all edges in an arbitrary order, that is, for each

edge (vi, Vj) let k^^ be a corresponding number, so that each edge is defined by triple eij =

= (vi, vj, kij) (see Fig-

Let vi be a vertex with li incoming edges efj = (vi, vjn, kfj), j G {1, ..., li} and mi outgoing edges ej = (vi, v°ut, k°ut), j G {1, ..., m,} (li and mi can be zeros). Then consider a model diffeomorphism FVi = FC Qs Qu, where the sets Os, Ou consist of li periodic orbits of the periods kf\, ..., kin and mi periodic orbits of the periods kff, ..., kOUt., respectively.

The resulting diffeomorphim f G g realizing the graph r is the connected sum of Fv . Namely, if there is an edge e%n = (vi, v%n, kln) which coincides with an edge e^f = (vj, v°ut, k°f), then we connect FVi with Fv along the s- and u-orbits of the same period kln = kj^f. d

7. The labeled diagram is a complete invariant of the ambient ^-conjugacy

In this section we prove Theorem 3.

By virtue of Theorem 1, the labeled Smale diagram for a diffeomorphism f G G is a Hasse diagram corresponding to the order of the basic sets of the diffeomorphism f, where each vertex is equipped with a triple C, Os, Ou corresponding to the basic set ACos 0u topological conjugacy class.

For diffeomorphisms f, f' G G the isomorphism of their Hasse diagrams and the fulfilment of the conditions of Theorem 1 on isomorphic labeled vertices is a necessary and sufficient condition for their Q-conjugacy. However, in the general case the conjugating homeomorphism does not extend from the basic sets to the ambient surface. For example, Fig. 8 shows two diffeomorphisms obtained from the same set of model diffeomorphisms by different types of connected sums such that the diffeomorphisms are not ambiently Q-conjugate (see Fig. 8).

Fig. 8. Diffeomorphisms f and f' that are not ambiently ^-conjugate

Let us single out a subclass G* c G of diffeomorphisms in which any two model diffeomorphisms are connected along at most one orbit.

Let us prove that the diffeomorphisms f, f' G G* are ambiently Q-conjugate if and only if their labeled diagrams are isomorphic.

Proof of Theorem 3.

Necessity. Since any two model diffeomorphisms from class G* are a connected sums where any two model diffeomorphisms are connected along at most one orbit, it follows that the ambient Q-conjugacy of the diffeomorphisms f, f G G* implies the isomorphism of their labeled Smale diagrams.

Sufficiency. The isomorphism of the labeled diagrams of the diffeomorphisms f, f ' G G* and Theorem 1 imply the existence of homeomorphisms conjugating model diffeomorphisms corresponding to isomorphic vertices. The map composed of these homeomorphisms is defined everywhere on the ambient surface except on the annuli connecting the two models. Continuing the map to the annuli in an arbitrary way, we get the desired homeomorphism which executes the ambient Q-conjugacy of the diffeomorphisms f, f '. □

References

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[2] Birkhoff, G., Lattice Theory, 3rd ed., AMS Coll. Publ., vol.25, Providence, R.I.: AMS, 1967.

[3] Vogt, H. G., Leçons sur la résolution algébrique des équations, Paris: Nony, 1895.

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[6] Williams, R. F., The "DA" Maps of Smale and Structural Stability, in Global Analysis: Proc. Sympos. Pure Math. (Berkeley, Calif., 1968): Vol. 14, Providence, R.I.: AMS, 1970, pp. 329-334.

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[8] Sinai, Ya.G., Markov Partitions and C-Diffeomorphisms, Funct. Anal. Appl., 1968, vol.2, no. 1, pp. 61-82; see also: Funktsional. Anal. i Prilozhen., 1968, vol.2, no. 1, pp. 64-89.

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[10] Grines, V. Z., The Topological Conjugacy of Diffeomorphisms of a Two-Dimensional Manifold on One-Dimensional Orientable Basic Sets: 2, Tr. Mosk. Mat. Obs, 1977, vol. 34, pp. 243-252 (Russian).

[11] Grines, V., Medvedev, T., and Pochinka, O., Dynamical Systems on 2- and 3-Manifolds, Dev. Math., vol. 46, New York: Springer, 2016.

[12] Palis, J. Jr. and de Melo, W., Geometric Theory of Dynamical Systems: An Introduction, New York: Springer, 1982.

[13] Barinova, M., Gogulina, E., and Pochinka, O., Realization of the Acyclic Smale Diagram by an Q-Stable Surface Diffeomorphism, Ogarev-Online, 2020, no. 13, 10p. (Russian).