Generic diffeomorphisms with weak limit shadowing
© Lu et al.; licensee Springer. 2013
Received: 1 November 2012
Accepted: 11 January 2013
Published: 4 February 2013
In this paper, we show that if a -generic diffeomorphism has the weak limit shadowing property on the chain recurrent set, then the diffemorphism satisfies Axiom A and the no-cycle condition.
Let M be a closed manifold, and let be the space of diffeomorphisms of M endowed with the -topology. Denote by d the distance on M induced from a Riemannian metric on the tangent bundle TM. Let and Λ be a closed f-invariant set. For , a sequence of points () in M is called a δ-pseudo orbit of f if for all .
We say that f has the shadowing property on Λ if for every , there is such that for any δ-pseudo orbit of f (), there is a point such that for all . In the dynamical systems, the shadowing theory is a very useful notion. In fact, it deals with the stability theorem (see ). For instance, Robinson  proved that if a diffeomorphism f is structurally stable, then it has the shadowing property. In  Sakai showed that f belongs to the -interior of the shadowing property if and only if f is structurally stable. In this paper, we deal with another shadowing property, that is, the weak limit shadowing property which was studied by .
for all and . If , then f is Anosov. Very recently, Sakai  showed that if a -generic diffeomorphism f has the s-limit shadowing property on , then f satisfies Axiom A and the no-cycle condition. The result is motivation for this study. The main theorem of the paper is as follows.
Theorem 1.1 For -generic f, if f has the weak limit shadowing property on , then f satisfies Axiom A and the no-cycle condition.
2 Proof of Theorem 1.1
are -injectively immersed submanifolds of M. Let be saddles. Let be the set of periodic points of f. Denote by the periodic f-orbit of . We denote if the intersections and . Then we know that if , then . Here is the dimension of the stable manifold of p, that is, .
Proposition 2.1 There is a residual set such that for any , if has the weak limit shadowing property, then for any saddles , .
To prove Proposition 2.1, we need the following lemma.
Lemma 2.2 Let be saddles. If f has the weak limit shadowing property on , then .
and it is clear that . Since f has the weak shadowing property on , there is a point such that as . Then as and as . Hence, and . Thus, . □
The following is called the Kupka-Smale theorem.
Lemma 2.3 There is a residual set such that for any , every periodic point is hyperbolic, and the stable manifolds and the unstable manifolds of periodic points are all transverse.
Proof of Proposition 2.1 Let , and let be saddles. Suppose that f has the weak limit shadowing property on . Let be the number of the weak limit shadowing property of f such that . Then we will drive a contradiction, we may assume that . Then we know that or . In this proof, we consider that (the other case is similar). Since , . Since f has the weak limit shadowing property on , by Lemma 2.2, . This is a contradiction. □
Let be a hyperbolic saddle with a period . Then there are the local stable manifold and the unstable manifold of p for some . It is easily seen that if for all , then , and if for all , then . The following lemma shows that if f has the s-limit shadowing property on , then the numbers of sinks and sources are finite (see [, Lemma 2]). From the above facts, we show that if f has the weak limit shadowing property on , then the numbers of sinks and sources are finite.
Lemma 2.4 Let f have the weak limit shadowing property on , and let be the number of the weak limit shadowing property of f. For any saddle , if is a sink or a source, then .
Proof We will derive a contradiction. Suppose that is a saddle and is a sink with . For the sake of simplicity, we may assume that , . Since q is a saddle, there is such that if for any , as , then , and if , as , then . Then we may assume that . Then we construct a δ-limit pseudo orbit as follows. Put for and for . Then is clearly a δ-limit pseudo orbit of f, and . Since f has the weak limit shadowing property on , there is a point such that as . Since p is a sink, as . Then . Since as , there is such that for . Then . Since , we know that . This is a contradiction. □
Let p be a periodic point of f, and let . We say p has a δ-weak eigenvalue provided has an eigenvalue λ such that . We say that the periodic point has a real spectrum if all of its eigenvalues are real and a simple spectrum if all of its eigenvalues have multiplicity one. The following lemma will play a crucial role in our proof.
Lemma 2.5 [, Lemma 5.1]
for any , if for any -neighborhood of f, there exist and with the same period such that , then there exist with the same period such that ;
for any , if for any -neighborhood of f, there exist and with an η-weak eigenvalue, then there exist with a 2η-weak eigenvalue;
for any , if with an η-weak eigenvalue and a real spectrum, then there exists with an η-weak eigenvalue with a simple real spectrum.
Lemma 2.6 [, Lemma 5.1]
There is a residual set such that for any , for any , if for any -neighborhood of f, there exist and with the same period such that with different indices, then there exist with the same period such that with different indices.
The following so-called Franks lemma will play an essential role in our proof.
Lemma 2.7 Let be any given -neighborhood of f. Then there exists and a -neighborhood of f such that for given , a finite set , a neighborhood U of and linear maps satisfying for all , there exists such that if and for all .
If is hyperbolic, then for any , there is a unique hyperbolic periodic point nearby p such that and , where . Such a is called the continuation of p.
Lemma 2.8 There is a residual set such that for any , if f has the weak limit shadowing property on , then there is such that f has no η-weak eigenvalue.
Proof Let . To derive a contradiction, we may assume that for any , there is a hyperbolic periodic point of g (-nearby f) such that has an η-weak eigenvalue and a simple real spectrum. Let be the number of the weak limit shadowing property of f such that . For the sake of simplicity, we assume that is a fixed point. By Lemma 2.7, there is h -close to g and h -nearby f such that has 1 as an eigenvalue. By Lemma 2.7 and as in the proof of [, Lemma 2.4], we can construct an ()-invariant small arc of containing such that , where , are the end points of , , and , are hyperbolic saddles and different indices. Since , there exist with the same period such that with different indices. Since f has the weak limit shadowing property on , and by Lemma 2.4, are saddles. Since , by Proposition 2.1, we know that . But, since , this is a contradiction. □
Denote by the -interior of the set of diffeomorphisms of M whose periodic points are all hyperbolic. In , Hayashi showed that if , then f satisfies Axiom A and the no-cycle condition. To prove Theorem 1.1, it is enough to show .
End of proof of Theorem 1.1 Let , and let f have the weak limit shadowing property on . If not, then . There is g -closed to f and a non-hyperbolic periodic point such that the point has an weak eigenvalue. Since , there is such that p has an η-weak eigenvalue. By Lemma 2.8, this is a contradiction. □
We wish to thank the referee for carefully reading of the manuscript and for providing us with many good suggestions. GL is supported by the Doctoral Science Foundation of Liaoning Province by Hall of Liaoning province science and technology (No. 2012-1055). KL is supported by the National Research Foundation (NRF) of Korea funded by the Korean Government (No. 2011-0015193). ML is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology, Korea (No. 2011-0007649).
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