Weak shadowing property in omega-stable diffeomorphisms Текст научной статьи по специальности «Математика»

DIFFERENTIAL EQUATIONS AND

CONTROL PROCESSES N 1, 2005 Electronic Journal, reg. N P23275 at 07.03.97

http://www. neva. ru/journal http://www.imop.csa.ru/ diff e-mail: diff@osipenko.stu.neva.ru

Ordinary differential equations

WEAK SHADOWING PROPERTY IN ^-STABLE DIFFEOMORPHISMS

O.A. Tarakanov

St. Petersburg State University, Department of Mathematics and Mechanics, University av., 28, 198504, St. Petersburg, RUSSIA, e-mail: light_ln2@hotmail.com

Abstract.

The weak shadowing property was introduced by R.M. Corless and S.Yu. Pilyugin and studied by these authors, K. Sakai, O.B. Plamenevskaya and others. It was shown by Plamenevskaya that for omega-stable diffeomorphisms this property may be bount to the numerical properties of the eigenvalues of the hyperbolic saddle points of the diffeomorphisms.

In this paper, we prove that if the phase diagram of an omega-stable dif-feomorphism of a manifold does not contain chains of length more than three, then it has the weak shadowing property.

0This work was supported by RFBR (grant 02-01-00675) and by the Ministry of Education of Russia (grant A03-2.8-322).

1 Introduction

The weak shadowing property of dynamical systems was introduced in [1], where it was shown that this property is C 0 -generic.

The study of the weak shadowing property for ^-stable diffeomorphisms is essentially complicated: it was shown by Plamenevskaya [2] (see below) that this property may be bount to the numerical properties of the eigenvalues of hyperbolic saddle points of the diffeomorphisms.

In this paper we prove Theorem 2.1 stating that ^-stable diffeomorphisms (on manifolds of arbitrary dimension) having only "short" connections in phase diagrams have the weak shadowing property.

2 Definitions and main results

Let M be a closed smooth manifold with Riemannian metric dist. Denote by U(a, A) the a-neighborhood of a set A c M.

Denote by Diff :(M) the space of diffeomorphisms of M with the C1 topology. For a diffeomorphism f, we denote by O(x, f) the trajectory of x.

A sequence £ = {xk : k G Z} c M is called a d-pseudotrajectory of f if

dist(f(xk),xk+i) < d, k G Z. We say that a point x G M e-shadows the pseudotrajectory £ if

dist(fk(x),xk) < e, k G Z. We say that a point x G M weakly e-shadows £ if

£ c U(e, O(x, f)).

Now we give definitions of the main properties which we study.

We say that a diffeomorphism f has the (usual) shadowing property if, given e > 0, there exists d > 0 such that any d-pseudotrajectory is e-shadowed by some point of M.

We say that f has the weak shadowing property if, given e > 0, there exists d > 0 such that any d-pseudotrajectory is weakly e-shadowed by some point of M.

Remark 2.1. Let us note that the property defined above was called the first weak shadowing property in [3], where the second weak shadowing property,

"symmetric" to the first one, was introduced: we say that f has the second weak shadowing property if, given e > 0, there exists d > 0 such that for any d-pseudotrajectory £ of f, there is a point x such that

O(x,f) c U(e,£).

It was shown in [3] that any dynamical system with compact phase space has the second weak shadowing property, hence the study of this property in the context of our paper is senseless. For this reason, we use below the term "weak shadowing property" introduced in [1].

Of course, if a diffeomorphism has the shadowing property, it has the weak shadowing property as well. An example of irrational rotation on the circle shows that the inverse statement does not hold.

The following example constructed by Plamenevskaya [2] gives us useful information concerning weak shadowing in ^-stable systems.

Example. Represent T2 as the square [-2, 2] x [-2, 2] with identified opposite sides. Let g : T2 ^ T2 be a diffeomorphism with the following properties:

(1) the nonwandering set ^(g) of g is the union of 4 hyperbolic fixed points; that is, Q(g) = {pi, p2, pa, p4}, where p1 is a source, p4 is a sink, and p2, pa are saddles;

(2) with respect to coordinates (v,w) £ [-2, 2] x [-2, 2], the following conditions hold:

(2.1) pi = (1, 2), p2 = (1, 0), pa = (-1,0), p4 = (-1, 2),

(2.2) Wu(p2) U {pa} = Ws(pa) U fe} = [-2, 2] x {0},

Ws(p2) = {1} x (-2, 2), Wu(pa) = {-1} x (-2, 2),

where Ws(pi) and Wu(p^) are the stable and unstable manifolds, respectively, defined as usual;

(2.3) there exist neighborhoods U2, Ua of p2, pa such that

g(x) = pi + DPig(x - pi) if x £ Ui,

(2.4) there exists a neighborhood U of the point z = (0,0) such that g(U) c Ua, g-1(U) c U2 and g-1 is affine on g(U),

(2.5) the eigenvalues of Dp3 g are —¡i, v with ^ > 1,0 < v < 1, and the eigenvalues of Dp2 g are —A, k with 0 <A< 1, k > 1.

It was proved in [2] that g has the weak shadowing property if and only if the number log A/ log n is irrational. Note that g satisfies Axiom A and the no-cycle condition (i.e., it is Q-stable) but does not have the shadowing property.

Let f be an Axiom A diffeomorphism of M. By the Smale Spectral Decomposition Theorem, the nonwandering set Q(f) can be represented as a finite union of basic sets Q^ Denote by Ws(^) and Wthe stable and unstable "manifolds" of Q^ For two different basic sets Qi and Qj, we write Qi ^ Qj if

Wu(tti) n Ws(Qj) = 0.

Let us say that the phase diagram of the diffeomorphism f contains a chain of length m if there exist m different basic sets Qi1,..., Qim such that

Qi Qi .

Theorem 2.1. Assume that a diffeomorphism f satisfies Axiom A and the no-cycle condition. If its phase diagram does not contain chains of length m > 3, then f has the weak shadowing property.

Note that the restriction on the lengths of chains in Theorem 2.1 is sharp: the Q-stable diffeomorphism in the Plamenevskaya example has a chain p\ ^ p2 ^ p3 ^ p4 of length 4 in its phase diagram (and may fail to have the weak shadowing property).

3 Proof of Theorem 2.1

Let us first introduce some notation.

Denote by O+(x, f ) and O-(x, f ) the positive and negative semitrajectories of x, respectively. Let £ = {xk : k G Z} be a pseudotrajectory and let l, m be indices with l < m. We denote

£l>m = {xk : l < k < m}, = {xk : l < k}, = {xk : k < l},

= £+, and £_ = £-.

The following three propositions are well known (Proposition 3.1 is the classical Birkhoff theorem, for proofs of statements similar to Propositions 3.2 and 3.3, see [4], for example).

Proposition 3.1. Let f be a homeomorphism of a compact topological space X and U be a neighborhood of its nonwandering set. Then there exists a positive number N such that

card{k : fk (x) / U}< N

for any x E X, where card A is the cardinality of a set A.

In Propositions 3.2, 3.3, 3.2p and 3.3p, we assume that f is an ^-stable diffeomorphism of a closed smooth manifold (below we apply these propositions both to f and f

Proposition 3.2. If Qi is a basic set, then for any neighborhood U of Qi we can find its neighborhood V with the following property: if for some x E V and m> 0, fm(x) E U, then fm+k(x) / V for k > 0.

Proposition 3.2. There exist neighborhoods Ui of the basic sets Qi such that if fm(Ui) fl Uj = 0 for some m > 0, then there exist basic sets Qll,..., Qlk such that

Qi ^ Qi1 ^ ••• ^ Qik ^ Qj.

Obviously, these propositions have the following analogs for pseudotrajec-tories.

Proposition 3.1p. Let f be a homeomorphism of a compact metric space X and U be a neighborhood of its nonwandering set. Then there exist positive numbers d, N such that if £ = {xk} C X is a d-pseudotrajectory and £l,m HU = 0 for some l, m with l < m, then m — l < N.

Proposition 3.2p. If Qi is a basic set, then for any neighborhood U of Qi we can find its neighborhood V and a number d > 0 with the following property: if £ = {xk} is a d-pseudotrajectory of f, x0 E V, and xm / U for some m > 0, then £+m H V = 0.

Proposition 3.3p. There exist neighborhoods Ui of the basic sets Qi and a number d > 0 with the following property: if £ = {xk} is a d-pseudotrajectory of f such that xo E Ui and xm E Uj for some m > 0, then there exist basic sets Qll,..., Qik such that

Qi ^ Qi1 ^ ••• ^ Qik ^ Qj.

In what follows, we assume that f is an Q-stable diffeomorphism. We need the following auxiliary statement. Let us say that f has the usual shadowing property on a set A if, given e > 0, there exists d > 0 such that if £ = {xk} is a d-pseudotrajectory of f with £1,m c A, then there exists x such that dist(xk, fk(x)) < e for l < k < m. Since any basic set Qi is hyperbolic, we may assume that f has the usual shadowing property on all neighborhoods of Qi considered below.

Lemma. Let Qi be a basic set and let Ui be a neighborhood of Qi such that Ui R Qj = 0 for i = j. For any positive a, there exists d > 0 with the following property: if £ = {xk} is a d-pseudotrajectory of f with £+ c Ui, then there exists a point z and an open set D containing z such that

(1) dist(x0, z) < a;

(2) £+ c U(a, O+(z', f)) for any z' £ D.

Proof. Fix arbitrary a > 0. Reducing a, if necessary, we may assume that

U(a, Ui) R Qj = 0

for j = i. Applying the usual shadowing property on Ui, let us find d > 0 such that if £ = {xk} is a d-pseudotrajectory of f with £+ c Ui, then there exists y such that dist(xk, fk(y)) < a/4 for k > 0. By the choice of a, O+ (y,f) c U(a, Ui), hence y £ Ws(Qi). Thus, there exists p £ Qi such that y £ Ws(p). In any neighborhood of p, there is a point q such that its trajectory is dense in Qi. Stable manifolds of points of a hyperbolic set depend continuously on the point, hence any neighborhood of y contains a point z such that O+(z,f) is dense in Qi.

There exists a number K > 0 such that fk(y) £ U(a/4, Qi) for k > K. Find a point z such that

(1) dist(fk(y),fk(z)) < a/2 for 0 < k < K;

(2) O+(z,f) is dense in Qi.

There exists a number L > 0 such that for any point p £ Qi there is a point r £ {fk(z) : 0 < k < L} with dist(p, r) < a/4. By the continuity of f, there is an open set D containing z such that Qi c U(a/2, O+(z', f)) for any z' £ D.

To complete the proof, it remains to take D so small that dist(fk(y), fk(z')) < a/2 for 0 < k < K and z' £ D.

Remark 3.1. Let Qi be an attractor. Fix e > 0 and find a neighborhood U of Qi such that

Ui c U(e/2, Qi) (1)

and f (Ui) c Ui. There exist numbers d,a > 0 (depending only on Ui) such that if £ = {xk} is a d-pseudotrajectory of f with x0 G Ui, then £+ c Ui, there is a point y G Ws(Qi) such that dist(fk(y),xk) < e/4, and W = U(a,x1) c Ui. Since points z for which O+(z,f) is dense in Qi are dense in W, the same reasoning as in the proof of the lemma above shows that the set

W' = {x G W : £+ c U(e,O+(x,f))}

is open and dense in W.

Of course, a similar statement holds for a repeller Qi.

In the proof of Theorem 2.1, we have to consider d-pseudotrajectories with decreasing values of d. We use the same notation of points of these pseudotra-jectories, of their neighborhoods, etc; this will lead to no confusion.

Let m be the maximal length of chains in the phase diagram of the considered Q-stable diffeomorphism f. If there are no chains of length 2, then the statement of our theorem is trivial - in this case, f is an Anosov diffeomorphism.

Let us consider the case where m = 2. In this case, any basic set is either a repeller or an attractor. Consider a repeller Q1 and an attractor Q2. Fix an arbitrary e > 0. Standard reasons show that there exist neighborhoods Ui of the sets Qi, i = 1, 2, such that inclusions (1) hold, f-1(U 1) c U1, and f (U2) c U2.

The set U2 = f (U2) \ f2(U2) is a compact subset of U2 disjoint from Q2. Hence, there exists a number a2 G (0, e) and a neighborhood V2 of Q2 such that

U(02, x) c U2 \ V2

for any x G U2.

Similarly, there exists a number a1 G (0,e) and a neighborhood V of Q1 such that

U(a1,x) c U1 \ V1 for any x G U1 = f-1(U 1) \ f-2(U1).

We may assume that these numbers and neighborhoods have also the following properties. There exists a number d1 > 0 such that if £ = {xk} is a d1-pseudotrajectory of f and xm G U2, then £m c U2 and, in addition, if xm-1 G U2, then

U(02,xm) c U2 \ V2

(and similar statements hold for U1 etc).

It follows from Propositions 3.1p-3.3p that there exist numbers d2 G (0,d1) and N such that if £ = {xk} is a d2-pseudotrajectory of f, then only one of the following possibilities holds:

(I) there exists a basic set Qi such that £ c U^;

(II) there exists a repeller Q1 and an attractor Q2 such that, for the neighborhoods described above, there exist integers 1,m with 0 < m — I < N such that

U(ai,xi) c Ui \ V1 and U(a2,xm) c U2 \ V2.

Case (I) is trivial since a basic set contains a dense trajectory (and, by condition (3.1), £ belongs to the e-neighborhood of such a trajectory).

To consider case (II), find positive numbers a3 < a1 and d3 < d2 such that for any points x,y with dist(x,y) < a3 and for any d3-pseudotrajectory {yk} with y0 = y, the inequalities

dist(fk(x),yk) < a2

hold for 0 < k < N.

Let £ = {xk} be a d3-pseudotrajectory such that

U(a1, x/) c U1 \ V1 and U(a2, xm) c U2 \ V2

for some l, m with 0 < m — I < N. Denote W1 = U(a3, x/) and W2 = U(a2, xm).

The remark after the lemma implies that a3, a2, d3 can be chosen in such a way that the sets

= {x G W1 : £— c U(e, O—(x, f))}

and

W2 = {x G W2 : c U(e,O+(x,f))} are open and dense subsets of W1 and W2, respectively.

By our choice of d3, fm—1 (W1) c W2. Since fm—1 (W{) is an open and dense subset of fm—1 (W1), there is a point x' G fm—1 (WJ) n W2.

Take x = f/—m(x'). It is easy to see that

£L c N(e/2, O—(x, f)), c N(e/2, O+(x, f)),

and dist(fk 1 (x),xk) < e for l < k < m, hence £ c U(e, O(x,f)). This completes the consideration of the case m = 2.

Finally, we consider the case m = 3. Fix e > 0. It follows from Propositions 3.1p - 3.3p that there exist numbers d0,N > 0 and neighborhoods Ui of the basic sets Qi such that inclusions (1) hold and, for any d0-pseudotrajectory £ = {xk} of f, only one of the following possibilities is realized:

(P1) there exists an index i such that £ c Ui;

(P2) there exist a repeller Q^ an attractor Qj, and indices l,m with l < m such that m - l < N, £- c Ui, and £+™ c Uj;

(P3.1) there exist a repeller Qi, a saddle basic set (i.e., a basic set that is not an attractor or repeller) Qj, and indices l, m with l < m such that m -1 < N,

£- c Ui, and £+" c Uj;

(P3.2) there exist a saddle basic set Qi, an attractor Qj, and indices l,m with l < m such that m - l < N, £- c Ui, and £+™ c Uj;

(P4) there exist a repeller Qi, a saddle basic set Qj, an attractor Qs, and indices

l, m, n, t with l < m < n < t such that m - l < N, t - n < N, £L c Ui,

£m,n c Uj, and £+ c Us.

For possibilities (P1) and (P2), the proof is just the same as in the case m = 2.

Let us consider possibility (P3.1) (the same reasoning is applicable for (P3.2)). Similarly to the proof for the case m = 2, we can find ai,d1 > 0 such that, for any d1-pseudotrajectory £ with x/ £ Ui, Wi = U(ai,x1-1) c Ui. After that, we find aj £ (0, e) and d2 < d1 such that for any d2-pseudotrajectory £ with x/ £ Ui, xm £ Uj, and 0 < m - l < N, the inclusion

Wj = U(aj ,xm) c fm-/+1(Wi)

and the inequalities dist(fk-m(y),xk) < e hold for any y £ Wj and l < k < m.

Applying the lemma (with a = aj), we can find da < d2 with the following property: for any da-pseudotrajectory £ there exists an open subset D of Wj such that £+" c U(e, O+(z, f)) for any z £ D.

Applying the remark after the lemma, we may assume that, for any da-pseudotrajectory £, the set

W/ = {x £ Wi : £L"1 c U(e,O_(x,f))}

is open and dense in W. Its image, fm 1+1 (W), contains Wj (and hence, there is a point x belonging to the intersection of this image with the open subset D

of Wj

It follows from our constructions that £ c U(e, O(x,f)). This completes the consideration of possibility (P3.1).

Finally, we have to consider possibility (P4). Fix a repeller Qi, a saddle basic set Qj, and an attractor Qs for which there exists a d0-pseudotrajectory £ and indices l,m,n,t with l < m < n < t such that m — l < N, t — n < N,

£- c Ui, £m,n c Uj, and £+ c Us.

We may assume that the neighborhoods Ui,Uj, Us satisfy inclusions (1). In addition, we assume that for d2-pseudotrajectories with d2 < di and for numbers ai,as £ (0,e), all of the statements similar to statements in the proof for the case m = 2 (II) are valid (with natural replacement of U1,U2, etc by Ui, Us, etc).

To be exact, we assume that if £ is a d2-pseudotrajectory with £l_ c Ui, £m,n c Uj, and £+ c Us, then the sets Wi = U(ai,xl) and Ws = U(as,xt) are subsets of Ui and Us, respectively, and that the sets

W/ = {x £ Wi : £— c U(e,O—(x,f))}

and

Ws' = {x £ Ws : £+ c U(e,O+(x,f))} are their open and dense subsets.

Now let us find numbers d3 < d2 and aj > 0 such that, for any point xr of a d3-pseudotrajectory £ and for any point y such that dist(y,xr) < aj, the inequalities

dist(fk(y),xr+k) < min(ai,as)

hold if |k| < N.

In addition, since f has the usual shadowing property on Uj and £m'n c Uj, we may assume that there exists a point y such that dist(fk(y),xm+k) < aj for 0 < k < n — m (note that the value n — m may be arbitrarily large, in contrast to the values m — l and t — n not exceeding N).

Denote Wj;1 = U(aj ,xm) and Wj,2 = U(aj, xn). By the choice of aj, V/ = fl—m(Wj,i) c Wi and Vs = ft—n(Wj,2) c Ws. Hence, the intersection V/ = V/ R W/ is open and dense in V/, and the intersection Vs' = Vs R Ws' is open and dense in Vs. It follows that the image V' = fm—l(V/') is open and dense in Wj;1, and the image V'' = fn—t(Vs') is open and dense in Wj,2.

It remains to note that the point y has a small neighborhood D C Wj,1 such that fn-m(D) C Wj,2 and dist(fk(x), xm+k) < e for x G D and 0 < k < n - m. It follows from our considerations that there exists a point x G D H V' such that fn-m(x) C V''. By construction, £ c U(e,O(f,x)).

The theorem is proved.

Remark 3.2. Analyzing the proof of Theorem 2.1, it is easy to see that a similar statement holds for an Q-stable diffeomorphism f under the following condition: if

Qi ^ Qi1 ^ • • • ^ Qik ^ Qj

is a chain in the phase diagram of f such that Qi is a repeller and Qj is an attrac-tor, then stable and unstable manifolds of points of the basic sets Qll,..., Qlk are transverse.

References

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[2] O.B. Plamenevskaya. Weak shadowing for two-dimensional diffeomor-phisms. Vestnik St. Petersburg Univ. Math, 31, 49-56 (1999).

[3] S.Yu. Pilyugin, A.A. Rodionova and K. Sakai. Orbital and weak shadowing properties. Discrete Contin. Dyn. Syst, 9, 287-308 (2003).

[4] I.P. Malta. Hyperbolic Birkhoff centers. Trans. Amer. Math. Soc, 262, 181-193 (1980).