On construction of axiom a 3-diffeomorphism with one-dimensional surface attractor-repeller dynamics Текст научной статьи по специальности «Математика»

Динамические системы, 2018, том 8(36), №4, 305-311 MSC 2010: 37D15

On construction of axiom A 3-diffeomorphism with one-dimensional surface attractor-repeller dynamics1

M. Barinova, V. Grines, O. Pochinka

Higher School of Economics

Nizhny Novgorod 603005. E-mail: mkbarinova@yandex.ru, vgrines@yandex.ru, olga-pochinka@yandex.ru

Abstracts. We suggest a method of a construction of axiom A 3-dieomorphisms whose non-wandering set consists of exactly one-dimensional surface attractor and one-dimensional surface repeller. Unlike from examples constructed by Ch. Bonatti, N. Guilman and Sh.Yi, our diffeomorphisms are not structurally stable, however suggested method gives rather simple construction of new types of 3-manifolds, admitting "hyperbolic sink-hyperbolic source" dynamics. Keywords: A-diffeomorphism, surface basic set

1. Introduction

Let M be a closed n-manifold and f : M ^ M be an axiom A diffeomorphism. By Smale's spectral theorem the non-wandering set of f consists of finite number f-invariant closed subsets, named basic sets. For a basic set Л a pair (a, b), where a = dim WU, b = dim WS is called type of the basic set.

A basic set A of the diffeomorphism f is called attractor if it has a trapping region,

that is a compact neighborhood UA С M such that f (UA) С int UA and P| fj(UA) = A.

jm

A basic set R is called a repeller if it is an attractor for f-1.

A hyperbolic attractor of diffeomorphism f is called surface if there exists a compact surface £ that A С £, f (£) С £. A surface repeller is defined as surface attractor for f-1.

There is a natural question: does some manifold admit an A-diffeomorphism with exactly two basic sets of the same dimension and attractor-repeller dynamics? Such a diffeomorphism is automatically П-stable. The simplest example of such a system is a Morse-Smale diffeomorphism on n-sphere (n > 1) whose non-wandering set consists of exactly one sink and one source. These examples are structurally stable and exhaust all possible diffeomorphisms with zero-dimensional attractor and repeller.

For an attractor and repeller with the dimension 2k (k E N) also not so difficult to realize a diffeomorphism on n-manifold (n > 2k) as a direct product of Anosov diffeomorphism on 2k-torus with the type (k, k) by sink-source diffeomorphism on

1rThis work was supported by the Russian Science Foundation (project 14-41-00044).

© M. BARINOVA, V. GRINES, O. POCHINKA

(n — 2fc)-sphere. Moreover, for 3-manifolds by V. Grines, Yu. Levchenko, V. Medvedev, O. Pochinka [4] proved that two-dimensional attractor-repeller 3-diffeomorphisms there are only on mapping torus and obtained complete topological classification of such rough systems.

Particular case is a surfaces diffeomorphism with one-dimensional attractor and repeller. Such dynamics is achieved, for example, by taking of the connected sum of two DA-models on 2-tori. However R. Robinson, R. Williams [5] proved that among such diffeomorphisms there are no structural stable one.

B. Jiang, Y. Ni and S. Wang [2] proved that a 3-manifold M admits an axiom A diffeomorphism f whose non-wandering set consists of solenoid attractors and repellers if and only if M is a lens space L(p, q) with p = 0. They also shown that such f are not structural stable. C. Wang and Y. Zhang [6] got infinitely many genus two 3-manifolds, each admits a diffeomorphism whose non-wandering set consists of two Williams solenoids, one attractor and one repeller. These manifolds contain half of Prism manifolds, Poincare's homology 3-sphere and many other Seifert manifolds, all integer Dehn surgeries on the figure eight knot, also many connected sums.

On the other hand, due to Ch. Bonatti, N. Guelman [1], Shi Yi [3], there are examples of rough 3-diffeomorphisms with one-dimensional attracor and repeller. But all examples have very complicated descriptions.

In this paper we suggest a method of a construction of an axiom A 3-diffeomorphisms whose non-wandering set consists of exactly one-dimensional surface attractor and one-dimensional surface repeller. All known examples were constructed in [1] and [3]. Constructed in this paper diffeomorphisms are not structurally stable, however suggested method gives rather simple construction of new 3-manifolds, admitting "hyperbolic sink-hyperbolic source" dynamics, different from the manifolds constructed in [1] and [3].

2. Construction

The diffeomorphism will be constructed step by step in this section as following:

• take an Anosov diffeomorphism of a 2-torus T2;

• make a Smale "surgery operation" to obtain the system with one fixed source and one-dimensional attractor on the torus;

• multiply T2 by R with contraction to 0, hence one-dimensional surface attractor on T2 x R will be obtained;

• construct a fundamental domain of the attractor;

• take an analogical sample with one-dimensional surface repeller;

• "glue" the fundamental domains of the diffeomorphisms in the basins of attractor and repeller in accordance with dynamics.

2.1. Anosov diffeomorphism of a 2-torus

Let C E GL(2, Z) be a hyperbolic matrix with the eigenvalues Ai,A2 so that A = | Ai| > 1 and |A2| = 1/A. As the matrix C has the determinant equals 1 then it generates the hyperbolic automorphism C : T2 ^ T2 with the fixed point O. This automorphism is Anosov diffeomorphism, so there are two transversal foliations (stable and unstable) which are dense on the 2-torus, and a set of periodical points is also dense.

2.2. Smale "surgery operation"

Let U(O) C T2 be a some neighbourhood of the fixed point O of the diffeomorphism C and x, y be local coordinates such that the diffeomorphism C in these coordinates has a form

C(x,y) = (x/A,Ay).

Then Ox C WOO and Oy C WO, also {y = const} and {x = const} are stable and unstable foliations. A diffeomorphism B : T2 ^ T2 with properties described below will be constructed in this section:

• B will be identity on T2 \ U(O);

• B will keep unstable manifolds of C everywhere;

• BC will add additional expansion along stable manifolds of CC inside some neighbourhood of O;

• the composition ^ = B o C is DA-diffeomorphism with a fixed source O and one-dimensional attractor A.

Let v : [0,1] ^ [0,1] be a diffeomorphism defined by the graph on Fig. 1. Then vt(x) = (1 — t)x + tv(x), t E [0,1] is an isotopy between the identity map v0(x) on [0,1] and v1(x) = v(x). Let a(x, a, b) : R ^ R be a sigmoid function of the form

a < x < b,

a(x,a,b) = ^ 1+exp( )2)

0, otherwise.

It monotonically sends [a,b] to [0,1].

For every t E [0,1] let us define a diffeomorphism Bt : [0,1]2 ^ [0,1]2 which possess the symmetries with respect to both axis Ox, Oy and in the first quadrant given by the formula

(vt(x),y) , 0 ^ y< 2^3,

Bt(x,y)= { (v(У, 2^3,1 )x + (1 — v(У, 2I3,1 ))vt(x),y), 213 ^ У< 2,

,(x,У), 2 ^ У < 1-

1 X

Рис. 1. Graph of funcion v(x)

By the construction Bt is an isotopy between the identity map B0(x,y) on [0,1]2 and B\(x,y) = B(x,y). As B preserves y-coordinate, we will use the following designation: B(x, y) = (j(x, y),y). Let D = {(x,y) G U(O) : x2 + y2 ^ (2\3)2} By the construction

B is identity out of [0,1/2]2 and for points (x,y) G D we have y(x,y) = X2x.

Let Bt be a Bt inside [0,1]2 and is identity out of it. By arguments like to [7] it is possible to prove that Ф = B о C is a DA-diffeomorphism, whose non-wandering set consists of a one-dimensional attractor and a source.

2.3. One-dimensional surface attractor of T2 x R

Consider a smooth function ф : R ^ R by the formula <+p(z) = f. Define a diffeomorphism of T2 x R in coordinates w G T2, z G R by the formula

&(w,z ) = (Ф (w),p(z)).

The diffeomorphism Ф is an A-diffeomorphism whose non-wandering set contains one saddle point {O} x {0} and an one-dimensional attractor A which is placed on a 2-torus T2 x {0}.

2.4. Fundamental domain of the attractor

To find a fundamental domain of the basin of the attractor A, first of all, notice that T2 x (—212, ) is a trapping neighbourhood of an attractor T2 x {0}. It is not a trapping neighbourhood for A because there are a saddle point O and a segment of

its stable separatrix {O} x (—^, ^) inside. A small neighbourhood of the separatrix will be chosen to remove them.

So U4 = (T2 \ C_1(D)) x (—, 212) is a desired trapping neighbourhood for the

00

attractor A. Indeed, $(Uf) = (T2 \D) x — ^, = U24 C mt Uf and Q $ra(U4) =

n=1

A.

Рис. 2. Fundamental domain of an one-dimensional surface attractor

After that a fundamental domain of a basin of the attractor A is КA = cl (UA \ UA) (see Fig. 2). Notice, that dUf is a pretzel and KA is homeomorphic to the direct product of a pretzel by a segment. We will use the following designation: Kf = dUA,

KA = dU2A. So Ф(КА) = k2a.

2.5. One-dimensional surface repeller of T2 x R

Define a diffeomorphism в : [0,1]2 ^ [0,1]2 by the formula 9(x,y) = (—y,x). For every t E [0,1] let us define a diffeomorphism Qt : [0,1]2 ^ [0,1]2 by the formula Qt = в- 1B- 1в. By the construction Qt is an isotopy between the identity map Qo(x, y) on [0,1]2 and Q1(x, y) = Q(x, y). Let Qt be a Qt inside [0,1]2 and is identity out of it. So Qt keeps stable foliation of (J and add additional contraction along unstable manifolds of C inside some neighbourhood of O. Thus Ф_ = (J оC is a DA-diffeomorphism, whose non-wandering set consists of a one-dimensional repeller and a sink.

Consider a copy of T2 x R with the following diffeomorphisms of it

&-(w,z) = (Ф _(w),p_1(z)).

The diffeomorphism Ф- is an A-diffeomorphism whose non-wandering set contains one saddle point {O} x {0} and an one-dimensional repeller R which is placed on a 2-torus T2 x {0}. Trapping neighbourhoods UR, UR of R and a fundamental domain KR of a basin of the repeller R are the same as for diffeomorphism Ф so Ф-(UR) = UR, KR = dUR, KR = dUR, and Ф-KR) = KR.

2.6. Gluing of the fundamental domains

The goal of this section is to construct a diffeomorphism H : KR — KA with the properties:

• H(KR) = H(KA) and H(KR) = H(KA);

• H = Ф on KR and H = Ф- on KR.

For every t E [0,1] let Ф = Bt о Q-1 : T2 — T2. Notice that Ф = Ф о Ф-1. Then £t о Ф_ is an isotopy between Ф_ and Ф. By the construction ф о Ф_ has a form ф о Ф_(x,y) = (k(t)x,k(t)y) for (x,y) E Ф(D), where

m = (л - {)t+1

Finally let r, q : [0,1] — , 2x2] be functions given by the formulas

r(t) = 1 + ^, q(t) = k(t) • r(t).

Consider the fundamental domain KR of the basin of the repeller R (all reasoning for the attractor A are analogical). Let Dr(t) = {(x,y) E U(O) : x2 + y2 ^ r2(t)}. Then KR can be represented foliated by leaves

{Gr(t) = G x r(t),t E [0,1]}

such that Gr(o) = KR, Gr(X) = KR and Gr(t) coincides with the tori T2 x {±r(t)} out of Dr(t) x R and coincides with the cylinder dDr(t) x [~r(t),r(t)] otherwise (see Fig. 3). Define a map Ht : Gr(t) — Gq(t), t E [0,1] as follows

Ht (w,z) = (§(Ф _Н),щ z}.

Thus the diffeomorphism H : KR — KA, composed by Ht,t E [0,1], glues the dynamics of Ф_ and Ф along the fundamental domains. After a smoothing the corners

T2 x {0}

K.

0

phc. 3. Foliation of the fundamental domain

we get a new 3-manifold M with the desired A-diffeomorphism whose non-wandering set consists of one-dimensional surface repeller and one-dimensional surface attractor.

References

1. BonattiCh., Guelman N. Axiom A diffeomorphisms derived from Anosov flows, J. Mod. Dyn. 4, No.1, 1-63 (2010).

2. JiangB., Ni Y, WangS., 3-manifolds that admit knotted solenoids as attractors, Trans. Amer. Math. Soc. 356, No.11, 4371-4382 (2004).

3. Shi Yi Partially hyperbolic diffeomorphisms on Heisenberg nilmanifolds and holonomy maps, C. R. Math. Acad. Sci. Paris 352, No.9, 743-747 (2014).

4. GrinesV, LevchenkoY., MedvedevV., PochinkaO. The topological classification of structural stable 3-diffeomorphisms with two-dimensional basic sets, Nonlinearity 28, No.11, 4081-4102 (2015).

5. Robinson R. C, Williams R. F. Finite Stability is not generic (Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971)) bT" Academic Press, New York), 451-462 (1973).

6. WangCh., Zhang Y. Alternating Heegaard diagrams and Williams solenoid attractors in 3-manifolds. Topol. Methods Nonlinear Anal. 47, No.2, 769-798 (2016).

7. Williams R. The DA-maps of Smale and structural stability, Global Anal., Proc. Symp. Pure. Math., AMS 14, 329-334 (1970).

Получена 15.10.2018