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6Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 3, pp. 335-345. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd210307
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 37D15
Cantor Type Basic Sets of Surface A-endomorphisms
V. Z. Grines, E. V. Zhuzhoma
The paper is devoted to an investigation of the genus of an orientable closed surface M2 which admits A-endomorphisms whose nonwandering set contains a one-dimensional strictly invariant contracting repeller Ar with a uniquely defined unstable bundle and with an admissible boundary of finite type. First, we prove that, if M2 is a torus or a sphere, then M2 admits such an endomorphism. We also show that, if Q is a basic set with a uniquely defined unstable bundle of the endomorphism f: M2 — M2 of a closed orientable surface M2 and f is not a diffeomorphism, then Q cannot be a Cantor type expanding attractor. At last, we prove that, if f: M2 — M2 is an A-endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type Qr with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of Qr is regular, then M2 is a two-dimensional torus T2 or a two-dimensional sphere S2.
Keywords: A-endomorphism, regular lamination, attractor, repeller, strictly invariant set
1. Introduction and main results
By an endomorphism we mean a C^smooth surjective map of a manifold to itself. If an endomorphism f is a one-to-one map and the inverse mapping f-1 is C^smooth, then f becomes a diffeomorphism. We see that the notion of endomorphism generalizes the notion of diffeomorphism.
We denote by End(M) the space of endomorphisms of a closed manifold M endowed with the standard C 1-topology. Let g G End(M). The orbit or g-orbit of the point x0 G M is a countable or finite set = O(x0) such that g(xi) = xi+1 for any i G Z. The set
{Xi}i=o = O+(x0) C O(x0) is called the positive semi-orbit of the point x0. A positive semi-orbit
Received July 30, 2021 Accepted August 25, 2021
This work is supported by the Russian Science Foundation under grant 17-11-01041.
Vyacheslav Z. Grines vgrines@yandex.ru Evgeny V. Zhuzhoma zhuzhoma@mail.ru
National Research University Higher School of Economics ul. B. Pecherskaya 25/12, Nizhny Novgorod, 603150 Russia
is uniquely defined by the original point x0, while the set of orbits passing through x0 (which we denote Q(x0)) is generally a set of cardinality continuum. To a fixed element a G Q(x0) there is uniquely appropriated the orbit [xf = Oa(x0), where g(xf) = xf+1 for any i G Z and xa = = x0. The set [xf}°_J = Oa-(x0) is called the negative semi-orbit of the orbit Oa(x0). Below for simplicity we will omit index a when we consider an orbit of the point x0. A point x G M of the endomorphism f: M — M is called nonwandering if for any neighborhood U of a point x there is i > 0 such that f i(U) n U = 0. The set of nonwandering points forms the nonwandering set of the endomorphism f and is denoted by NW(f). It is known that a nonwandering set is always forward-invariant, that is, f (NW(f)) C NW(f).
An orbit O(x0) is called hyperbolic if there is a continuous splitting of the tangent space
To{x0)M = U Tx.M = Es © Eu = U EX. © EX.
i=-J i=-J
which is invariant with respect to Dg and such that
0 < \\Dgm(v)\\ < cjm\\Dgm(w)\\ ^ c-1 j-m\\w\\ where v G Es \ [0}, w G Eu, Vm G N
for some constants c > 0, 0 < j < 1 and a Riemannian metric on TM.
Note that the unstable subbundle Eu(x0) depends, in general, on the negative semi-orbit [xi}0=-^, so we will write Eu(x0, [xi}0=-00). It may be Eu(x0, [x^-J = Eu(y0, &}=-«>) for x0 = y0 with [xi}-J = O-(x0) = O-(y0) = [yl}0-J. Such an effect is impossible for a stable subbundle Es(x0) that depends only on the point x0.
The set A c M is called hyperbolic if f (A) = A, any orbit lying in A is hyperbolic and the constants c > 0, 0 < j < 1 in the above estimates do not depend on the choice of the orbit (therefore, one sometimes talks about uniform hyperbolicity). Note that the equality f (A) = A guarantees that for any point x0 G A there is at least one negative semi-orbit lying in A.
By analogy with a diffeomorphism satisfying the Smale axiom A, an endomorphism f: M — — M is called an A-endomorphism if its nonwandering set NW(f) is hyperbolic and periodic points are dense in NW(f). Recall that a map N — N is called transitive if there exists a point x G whose positive semi-orbit is dense in N.
In [19], the Spectral Theorem for an A-endomorphisms is proved, which is a generalization of the Spectral Theorem by Smale [21] for A-diffeomorphisms. According to [19], the nonwandering set NW(f) of the A-endomorphism f: M — M is represented in a unique way up to numbering as a union of closed and pairwise disjoint sets
NW(f ) = Q1 U ... U Ql, Qi n Qj = 0 for i = j,
such that f (Qj) = Q and f : Q — Q is transitive for all 1 ^ j ^ l. The sets ..., Ql are called basic sets.
To date, there are a number of complete classification results for one-dimensional basic sets of A-diffeomorphisms on closed surfaces M2 [2-5, 11] and manifolds of higher dimension [6, 7, 12, 13]. It is proved that one-dimensional basic sets on M2 are Cantor laminations, that is, each basic set Q is the union of pairwise disjoint curves locally arranged as a direct product of a segment and the Cantor set. Moreover, each leaf of such a lamination is homeomorphic in the intrinsic topology to the line R, and the admissible boundary from the interior of any connected component of the set M2 \ Q consists of a finite number of periodic leaves.
For A-endomorphisms, which are not A-diffeomorphisms, there are few classes of systems for which it is possible to describe the structure of basic sets and obtain complete classification
results. Such classes include interval and circle endomorphisms, endomorphisms arising in complex dynamics on the Riemann sphere, and expanding endomorphisms of manifolds of higher dimension [20]. It should be noted that the structure of the basic sets of A-endomorphisms has not yet been studied so exhaustively even for endomorphisms of surfaces.
In this paper, we consider one-dimensional strictly hyperbolic basic sets of A-endomorphisms of closed surfaces. From the point of view of applications, an important role is played by basic sets which are attractors and repellers. We investigate the topology of orientable close surfaces which admits A-endomorphisms with such basic sets.
Let us start presenting the main results. A hyperbolic set A is called uniquely hyperbolic, or a set with a uniquely defined unstable bundle, if the unstable subbundle Eu(x0, {%i}°i=-oo) = Eu(x0) does not depend on the negative semi-orbit {xi}0=_^ for any point x0 G A.
A basic set fia is called an attractor if there exists a neighborhood U of the set fia such that
f (clos U) C U, p| f\U) = fia
i^0
An attractor fia is called expanding if its topological dimension is equal to the dimension of the unstable subbundle Eu(x0, {xi) for any point x0 G fia and any negative semiorbit {xi}0=_^ C fia.
A basic set fir is called a repeller if there exists a neighborhood U of the set fir such that
clos U C f (U), f| f -i(U) = fir.
A repeller fir is called contracting if its topological dimension is equal to the dimension of the stable subbundle Es(x0) for any point x0 G fir.
A basic set fi is called strictly invariant if f-1(fi) = fi = f (fi). It follows directly from the definition of hyperbolicity that, in a certain neighborhood of the basic set, an endomorphism is a local diffeomorphism. Below, saying that an endomorphism is not a diffeomorphism, we will assume for simplicity that an endomorphism is a finitely multiple cover of a multiplicity at least two, with the exception of a finite number of branching points.
Let K be an open set of a topological space N, and dK be the boundary of the set K. A subset of S(K) C dK is called the admissible boundary from K (sometimes one says, admissible boundary from interior K) if for each point z G dK there exists an arc d with the end point z such that d \ {z} C K.
Let fi be a strictly invariant one-dimensional basic set with a uniquely defined unstable bundle of an A-endomorphism f: M2 — M2. It follows from the closeness of the set fi and Proposition 2.7 [15] that, if fi is an expanding attractor, then unstable manifolds of points from fi form the lamination L(fi) = |J Wu(x). Similarly, if fi is a contracting repeller, then its
xen
stable manifolds also form a lamination, which we will denote by the same symbol L.
We will say that L(fi) is a Cantor lamination with an admissible boundary of finite type if the following conditions hold:
1) all leaves of lamination L(fi) are the images of injective immersion of the line R, and the union of leaves of L(fi) is locally homeomorphic to the product of a segment and Cantor set;
2) for any component K of the set M2 \ L(fi) the boundary S(K) admissible from K consists of a finite number of invariant manifolds (unstable manifolds if fi is an expanding attractor, and stable manifolds if fi is a contracting repeller).
We will say that Q is a basic set of Cantor type with an admissible boundary of finite type if L(Q) is a Cantor lamination with an admissible boundary of finite type.
Denote by T2 the two-dimensional torus, and by S2, the two-dimensional sphere.
Theorem 1. There is an endomorphism f: M2 — M2, where M2 = T2 or M2 = S2, such that the nonwandering set of f contains a one-dimensional strictly invariant contracting repeller Qr with uniquely defined unstable bundle and with an admissible boundary of finite type.
It is known that on any closed surface M2 there is an A-diffeomorphism f : M2 — M2 with a one-dimensional expanding attractor and a one-dimensional contracting repeller of Cantor type (in both cases, the basic sets have an admissible boundary of finite type). The following statement contrasts with the situation where an endomorphism is not a diffeomorphism, even if the attractor of the endomorphism is strictly invariant and has a uniquely defined unstable bundle (for a diffeomorphism, this is done automatically).
Theorem 2. Let Q be a basic set with a uniquely defined unstable bundle of endomor-phism f : M2 — M2 of a closed orientable surface M2, and suppose f is not a diffeomorphism. Then Q cannot be a Cantor expanding attractor with an admissible boundary of finite type.
The A-endomorphisms satisfying Theorem 1 on the two-dimensional torus T2 and the two-dimensional sphere S2 have nonwandering sets consisting of a finite number of isolated sinks and a one-dimensional strictly invariant contracting repeller of Cantor type. In addition, the laminations corresponding to one-dimensional repellers continue on S2 to foliations with a finite number of singularities of a nonzero index, and on T2 to foliations without singularities. It is natural to consider the existence of A-endomorphisms with similar properties on other closed surfaces. Note that the index of a singularity of foliation in the component C of the set M2\L(Ar) was related to the number of stable manifolds belonging to the admissible boundary of C as follows.
Let v be the number of leaves (stable manifolds) in the boundary of the component C admissible from C. Then in C the foliation has the singularity of the index 1 — f. On §2, four components of the set S2 \ Q have exactly one admissible leaf. Therefore, in each such component the singularity has the index On T2, according to the construction, v = 2 for all components of the set T2 \ Q. Therefore, there are no singularities in each component.
We call the lamination L regular if it continues to a foliation F with the following property. Let C be a component of the set M2 \L, and 5(C) be its boundary admissible from C, consisting of v leaves. Then T has exactly one singularity of the index 1 — | in C provided v / 2. If v = 2, then there are no singularities in C. The following statement holds.
Theorem 3. Let f: M2 — M2 be an A-endomorphism on a closed orientable surface M2, such that the nonwandering set NW(f) consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type Ar with a uniquely defined unstable bundle. Suppose that the lamination L(Ar) is regular. Then M2 is the two-dimensional torus T2 or the two-dimensional sphere S2.
Note that the analog of Theorem 2 does not hold, in general, for endomorphisms of three-dimensional manifolds. Indeed, consider a 3-dimensional torus T3, which we represent as the product of T2 x S1 of the two-dimensional torus T2 and the circle S1. On the first multiplier T2, we define a DA-diffeomorphism f0 : T2 — T2 with an isolated source and a nontrivial one-dimensional expanding attractor A1 , and on the second multiplier S1 , we define the expanding en-domorphism E2 : S1 — S1 of the form x — 2x mod 1. The endomorphism (z, t) — (f0(z); E2(t)), z G T2, t G S1, has the expanding strictly invariant two-dimensional attractor A1 x S1 with a uniquely defined unstable Cantor type bundle, that is, A1 x S1 is locally homeomorphic to the
product of a Cantor set on a two-dimensional plane. At the same time, each connected component of the set T3 \ (A1 x S1) has a boundary admissible from T3 \ (A1 x S1) consisting of two lamination leaves L (A1 x S1), that is, an expanding attractor A1 x S1 has an admissible boundary of finite type.
Let us give some other examples of one-dimensional attractors and repellers of endomor-phisms of two-dimensional manifolds. Let h: S1 — S1 be a Morse-Smale diffeomorphism with one source sr and one sink sa. Then the nonwandering set of the endomorphism (h; E2): S1 x S1 = = T2 — T2 contains the strictly invariant one-dimensional expanding attractor {sa} x S1 with a uniquely defined unstable bundle and a strictly invariant one-dimensional repeller {sr} x S1 with a uniquely defined unstable bundle (both are homeomorphic to S1). Note that the repeller is not strictly contacting, since its stable bundle is zero-dimensional.
The nonwandering set of the endomorphism SH: S1 — S1 constructed in [20] consists of an isolated sink s1 and a nontrivial (zero-dimensional) repeller £r, homeomorphic to the Cantor set C. Then the nonwandering set of the endomorphism (SH; E2): S1 x S1 = T2 — T2 consists of a one-dimensional expanding attractor {s1} x S1, which is not strictly invariant, and a one-dimensional repeller R which is noncontracting and is homeomorphic to C x S1. Both basic sets have uniquely defined unstable bundles.
There are repellers that are not submanifolds. It is known, for example, that endomorphisms of surfaces arising in complex dynamics can have one-dimensional repellers having a fractal structure.
The structure of the article is as follows. Section 2 provides preliminary information and proves auxiliary results. The main theorems are proved in Section 3.
2. Auxiliary results
Let A be the hyperbolic set of a Cfc-endomorphism f: M — M (k ^ 1), and let p G A. It is known [15, 18, 19] that for a sufficiently small 5 > 0 set
Whoc(p) = {x G M: e (fi(p); fi(x)) <5 Vi > 0}
is a Cfc-embedded open disk BdimEs(p) with the tangent space Es(p) at the point p, and for any p°int x G Wlloc (p)
e (fi(p); f») —> 0 exponentially fast as i — <x>.
The set WSioc(p) is called a local stable manifold at the point p. A globally stable manifold (or, simply, a stable manifold) at the point p is called a set
Ws(p) = {x G M: e (f i(p); fi(x)) — 0 as i — x>} .
If A is a compact hyperbolic set, then the number 5 can be taken so that local stable manifolds WSloc(z) are defined for all points z G A. Then the global stable manifold Ws(p) can be represented as follows:
Ws(p) = U Compp f-i [WhoM)] , where pi = fi(p), where Comp^ K denotes the connected component of the set K containing z.
Unlike stable bundles Es, unstable bundles depend not only on the point, but also on the negative half-orbit of the point. Let O-(p) C A be the negative semi-orbit of the point p G A. According to [15, 18, 19], for a sufficiently small 5 > 0 set
W'uioc {O-(p)) = {x G M: 3O-(x), q (p-i;x-t) <5 Vi ^ 0, where p-i G O-(p), x-i G O-(x)}
is a Ck-embedded open disk BdimE"(O (p)) with a tangent space, which we will denote by Eu (O-(p)), at the point p. Moreover, for any point above x G Wgloc (O-(p))
Q (p-i; — 0 exponentially fast as i — <x>.
The set Wuioc (O (p)) is called a local unstable manifold with respect to the negative semiorbit O-(p) at the point p. A globally unstable manifold (or, simply, an unstable manifold) with respect to the negative half-orbit O-(p) at the point p is called a set
Wu (O-(p)) = [x G M: 3O-(x), q (p-i; x-^ — 0 as i — wherep-i G O-(p), x-i G O-(x)}, which can be represented in the form
Wu {O-(p)) = |J f (Wftoc {O-(p-i))), where p-i G O-(p).
Stable and unstable manifolds are called invariant manifolds.
We will need some auxiliary statements of a topological nature.
Lemma 1. Let X be a topological space and B its base of topology consisting of connected sets. Let N C X be a closed subset. Then each connected component of the complement X \ N is an open set.
Proof. Consider an arbitrary point x G X \ N. Since X \ N is an open set, there is a connected open set U gB such that x G U. Let K be a connected component X \ N containing the point x. Since K is the largest connected subset of X \ N containing the point x, and U is connected, the inclusion of U C K takes place. The proved statement now immediately follows from the arbitrariness of the choice of the point x. □
Corollary 1. Let M be a closed manifold, and N C M be its closed subset. Then each connected component of M \ N is an open set.
Proof. Let us take a topology base B for the manifold M as the union of open sets y(Br (p)), where y: Rra — M, a coordinate map of atlas of manifolds M and Br(p), is the open ball of radius r. Now the required statement follows from Lemma 1. □
Lemma 2. Let M be a simply connected topological space, and let the continuous map f: M — M be a cover (in particular, a local homeomorphism). Then f is a homeomorphism.
Proof.
Suppose that there are two different points x1, x2 G M, x1 = x2, such that f (x-J = f (x2) = = x0. Let G0: [0; 1] — M be a mapping of the segment [0; 1] such that G0(0) = x1, G0(1) = x2. Then the mapping F0 = f oG0 has the property F0(0) = F0(1). As M is simply connected, there is a homotopy Ft: [0; 1] — M, 0 ^ t ^ 1, F1([0; 1]) = x0. Since f is a covering, by virtue of the covering homotopy theorem, there exists a homotopy Gt: [0; 1] — M, 0 ^ t ^ 1, such that Ft = f o Gt for all 0 < t < 1. Therefore, F1 ([0; 1]) = f o G1 ([0; 1]) = f (x1) = f (x2). Hence, the path G1 ([0; 1]) connects in f-1(x0) various points G0(0) = x1, G0(1) = x2, which contradicts the fact that the set f -1(x0) consists of isolated points. The resulting contradiction proves the required statement, since f is a local homeomorphism. □
Let us recall the definition of lamination. A detailed explanation of the theory of laminations can be found in [1]. Cr'1, a smooth d-dimensional lamination on an n-dimensional manifold Mn (1 ^ d ^ n — 1, 0 ^ l ^ r ^ is a closed set L C Mn, which is the union of (J La pairwise disjoint images of La of some d-dimensional manifolds with respect to
a
some Cr-smooth injective immersion, and for any point x G L there is its neighborhood U(x) and C'-diffeomorphism ^ : U(x) — Rn such that any connected component of the intersection U (x) n La is mapped by a diffeomorphism ^ to an open subset of a d-dimensional subspace {(x1, ..., xn) G Rn | xd+1 = cd+1, ..., xn = cn}, while the restriction of ^ on any connected component of the intersection of U (x) n La is a Cr-diffeomorphism on the image.
3. Proof of the main results
Proof of Theorem 1. The existence of an A-endomorphism f: T2 — T2 of torus T2 with a Cantor type one-dimensional contracting repeller satisfying Theorem 1 was proved in the work [8] of V. Z.Grines, E. V. Zhuzhoma and E. D. Kurenkov, see also [10]. The proof consists in constructing the so-called Smale surgery operation [21] for the algebraic Anosov endomorphism
T2 _y t2
For the sphere S2, the proof is based on the method of constructing the Plykin attractor on S2 using a two-leaf branched cover T2 — S2, see [16]. Consider the transformation group r of the plane R2 generated by the group Z2 of integer shifts x — x + m, y — y + k, m, k G Z, and the symmetry (x; y) — (—x; —y). Then the quotient space R2/r is the two-dimensional sphere S2, and the natural projection p: R2 — R2/r = S2 is a branched cover with four branching points a1, a2, a3, a4 G S2 of index two each. The set
E= (J {(m;fc)}U + +
(m; fc)€Z2
4
is the preimage of the set £0 = {a1, a2, a3, a4}, that is, p(S) = S0 = |J ai.
i=1
It is known that, if the transformation f : R2 — R2 satisfies the relations
,f(S) = S, f ◦ r = r o f, (3.1)
then f induces a transformation of f: S2 — S2 such that f o p = p o f.
Consider the linear transformation A: R2 — R2, defined by the matrix A = (^3 ). We denote the matrix and the transformation with one letter, which should not lead to confusion. It is not difficult to check that A satisfies the relations (3.1), and therefore A induces the transformation S2 — S2, which we denote by f0 and which has four fixed points a1, a2, a3, a4 G S2. The following construction applies to each of these points.
For simplcity, we will consider the point a1 = a = p(0, 0) (and omit the index). The point a has a single preimage a(0, 0) G R2, p(a) = a, such that A(a) = a. Since a is a branching point of index two, there are neighborhoods U, U of points a and a, respectively, such that p(U) = U and the restriction p\~: U — U is a natural projection of factor space U/s0 — U, where s0 is the symmetry (x; y) — (—x; —y). It is not difficult to verify that a is a hyperbolic saddle point of the linear transformation A with the eigenvectors eu, es corresponding to the eigenvalues \u > 1 and 0 < As < 1, respectively. We will consider the pair e^u, as the basis (unit) vectors of
2
the coordinate system (v1; v2) so that (1; 0) and (0; 1) are the coordinates eu, es, respectively, and 5(0; 0) = O.
For the set N C R2, we denote by Z2(N) the family of sets obtained by the action of the group Z2 on N (sets from Z2(N) are called congruent to the set N).
Let us take r0 > 0 so small that the \ur0-neighborhood U0 of the point O belongs to U and Aur0-neighborhoods of all points from the set p-1(a1 U a2 U a3 U a4) have no intersection. Let 5: [0; oo) —> [0; 1] be a C°°-function such that S(r) = 1 when 0 ^ r ^ ■y, and S(r) = 0 for r ^ r0. Consider a system of differential equations in the neighborhood U0 and in all congruent to U0 neighborhoods, assuming that the coordinate system in U0 is transferred to the congruent neighborhood by the corresponding integer shift.
= -v1 • In\u ■ 6 +
(3.2)
0.
Denote by ytr the t-time shift along the trajectories of the system (3.2). Let us fix some number t0 > 1, and put
fr = A o $. (3.3)
Since the system (3.2) is invariant under the symmetry of (v1; v2) — (—v1; -v2), the dif-feomorphism fr of the form (3.2) satisfies equation (3.1), and hence induces an endomorphism of fr: S2 — S2. Using a method completely analogous that of [8], one can show that fr is an A-endomorphism and the nonwandering set of fr has a strictly invariant one-dimensional and uniquely hyperbolic contracting repeller Ar = Q, locally homeomorphic to the product of the segment and the Cantor set. In this case, any connected component of the set S2 \ Ar has a boundary admissible from S2 \ Ar, which consists of exactly one stable manifolds of periodic points from Ar. □
Proof of Theorem 2. Let us show that the basic set A cannot be an expanding attractor. Let us assume the contrary. Then there is an A-endomorphism f : M2 — M2 with a Cantor type expanding attractor A such that any connected component of the set M2 \ Aa has a boundary admissible from M2\A consisting of a finite number of unstable manifolds of points from Aa. Consider an arbitrary point x G A. Let us show that the restriction f\WU(X): Wu(x) — f (Wu(x)) = = Wu(f (x)) of an endomorphism f on any unstable manifold Wu(x), x G A, is a homeomorphism in the intrinsic topology of the leaf Wu(x).
First, we note that due to the hyperbolicity of A the restriction f on the set A is a local diffeomorphism. Since A is an attractor, it follows by virtue of Theorem 1 from [9] that Wu(x) belongs to A. Therefore, Wu(x) does not contain branch points of the endomorphism f and, therefore, the restriction f\Wu<xX) is a cover without branch points. Now the required statement follows from the fact that Wu(x) is simply connected and from Lemma 2.
We now show that in the complement M2 \ A there is some periodic point O of the endomorphism f. Since A is an attractor, there is a closed neighborhood U D A such that f (U) C int(U),
<x
and P| f l(U) = A. Consider an arbitrary point y0 G M2 \ U and its arbitrary negative semi-
l=0
orbit O-(y0). It follows directly from the inclusion of f (U) C U that the intersection O-(y0)nU is empty. Since the a-limit set of any point x G M2 is contained in the nonwandering set NW(f), the closure clos(O-(y0)) contains a point w G NW(f) \ A. Then, by virtue of the spectral decomposition theorem, there is a basic set A1 c M2 \ U different from A containing the point w.
In the basic set A1, we choose an arbitrary periodic point O of some smallest period k. If k > 1, then further arguments can be carried out to map fk. Thus, below it is possible to assume that point O is a fixed point of the f (without loss of generality).
Let K be the connected component of the complement M2 \ A containing the fixed point O. Denote by S(K) the admissible boundary of the K from interior K and take a leaf l C A of the lamination A such that l belongs to S(K). By the condition, l is an unstable manifold of some point from A. We show that f r(l) = l for some r G N.
Indeed, recall that, as l belongs the admissible boundary from interior K, for any point 2 G l there exists an arc d with endpoints O and 2 such that d \ {z} C K. Since A is strictly invariant with respect to f, the point f (z) belongs to A, and f (d)\{f (z)} belongs to M2\A. Since f (O) = = O, it follows that f (d) \ {f (z)} belongs to K. Therefore, f (z) belongs to the leaf f (l) of lamination A such that l C S(K). This means that the set S(K) is invariant. Since S(K) consists of a finite number of leaves, fr(l) = l for some r G N. Taking the iterations fnr, if necessary, we will assume below that all leaves of the lamination A belonging to S(K) are invariant with respect to f.
Consider an arbitrary point p G l, where l G S(K). Since the restriction of f to A is a k-cover, k ^ 2, the complete preimage of f -1(p) contains some point p1 = p. Since f (A) = A = = f-1(A), p1 G A. Due to the injectivity of the restriction f\WU(p), we have that p1 G l. It is clear that p1 does not belong to the boundary S(K) admissible from K. Indeed, suppose p1 G l* where l* is the leaf of the lamination A admissible from K. The leaf l* is invariant respect to f, different from l and f (l*) = l, which is impossible.
Consider the path jp connecting the points p and O such that int jp G K. Since p1 = p, there is a path y'p connecting the points p1 and O1, such that jp = f (y'p), f (O1) = O. Thus, p1 belongs to the admissible boundary of some connected component K1 of the set M2 \ A. Moreover, p1 belongs to a leaf, say l1, of the lamination A such that f (l1) = l. Since l1 G ^(K), K = K1. Similarly, it is shown that the complete preimage of f-1(p1) contains a point p2 G {p, p1} belonging to the component K2 of the set M2 \ A t such that K2 G {K, K1}, and there is a point O2 G K2 such that f (O2) = O1. Continuing this process, we get a negative semiorbit O-(p) = {p = p0, p1, p2, ...}, f (pj+1) = pj, and the sequence {K = K0, K1, K2, ...}, f (Kj+1) = Kj, with the following properties:
1) the point pi belongs to the boundary of the domain Ki admissible from Ki for all i G NU{0};
2) Ki are pairwise distinct connected components of the set M2 \ A;
3) there is a negative semi-orbit O-(O) = {O, O1, O2, ...}, f (Oj+1) = Oj such that Oi G Ki for all i G N U {0}.
Since O G M2 \ U and f (U) C U, there is an inclusion of O-(O) C M2 \ int(U). Hence, taking into account the compactness of the set M2 \ int(U), it follows that the sequence O-(O) has a limit point O* G M2 \ int(U). Let K* be the connected component of the set M2 \ A containing the point O*. According to Corollary 1, the connected component K* is an open set. Then there are two different numbers n1, n2 G N such that On G K* and On G K*, which contradicts the fact that Kn and Kn are different connected components of the set M2 \ A. □
Proof of Theorem 3. Due to the conditions, the stable manifolds of the repeller Ar form the lamination [15], which we denote by L(Ar). The set M2 \ L(Ar) is the union of a countable number of pairwise disjoint open disks that are basins of isolated sink periodic orbits. In addition, the lamination L(Ar) can be extended to some foliation F(Ar) on the surface of M2.
It is known that the sum of the indices of singularites of a foliation is equal to the Euler characteristic of the ambient surface (see, for example, [1, 14]). Suppose that M2 = T2, S2. Since in this case the Euler characteristic of M2 is negative, there is a component C of the set M2 \ £(A,r) containing the singularity of index ^ — Denote by v the number of leaves from 5(C). Since the index of the singularity from C satisfies the inequality 1 — | ^ — we see that v ^ 3. It follows from the strict invariance of Ar that f-1(5(C)) C Ar. Moreover, since the restriction f\A is a local diffeomorphism, the preimage f-1(5(C)) belongs to the admissible boundary of some component of M2 \ L(Ar). Since each component of the set M2 \ L(Ar) is homeomorphic to an open disk, it is simply connected. Thus, f-1(5(C)) is the union of the admissible boundaries of either one or several connected components of the set M2 \ L(Ar). In any case, since the lamination L(Ar) is regular, the preimage of f-1(C) contains a component of the set M2 \ L(Ar), different from C, with singularity of a negative index. Since f is not a diffeomorphism, the number of singularities of the negative index must be infinitely large. Since the surface M2 is compact, the foliation F(Ar) must have only a finite number of singularities of the nonzero index, by virtue of the Euler-Poincare formula (see, e.g., [1, 14]). The contradiction completes the proof.
Let us illustrate our arguments above for the simplest saddle-type singularities, when the set of singularities is invariant with respect to f. Let us assume for simplicity that the singularity s0 G C of the negative index has v separatrix. Then its index ind s0, according to [14], formula (3.11), equals 1 — |, where v ^ 3. If / has no branch points in C, then the foliation F(Ar) has in each component of /_1(C) a singularity of the index 1 — | < 0. If s0 is a branch point of degree k ^ 2, then each singularity of f-1(s0) has k ■ v separatrix and therefore its index equals 1 — < 0. In both cases, the index of a singularity from the preimage f~1(s0) is negative. □
Remark. The Euler characteristic of the surfaces T2 and S2 equals zero and two, respectively. Therefore, except for singularities of zero index, there are possible singularities of the index ^ for v = 1. However, their preimages for k = 2 have zero index. This shows that, for the surfaces T2 and S2, the proof of Theorem 3 does not hold.
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