Orbital Shadowing for 𝐶¹-Generic Volume-Preserving Diffeomorphisms

Abstract

We show that 𝐶¹-generically, if a volume-preserving diffeomorphism has the orbital shadowing property, then the diffeomorphism is Anosov.

1. Introduction

In the differentiable dynamical systems, the shadowing theory is a very useful notion for the investigation of the stability condition. In fact, Robinson [1] and Sakai [2] proved that a diffeomorphism belongs to the 𝐶¹-interior of the set of diffeomorphisms having the shadowing property coincides the structurally stable; that is, the diffeomorphism satisfies both Axiom A and the strong transversality condition. In general, if a diffeomorphism is Ω-stable, that is, a diffeomorphism satisfies both Axiom A and the no-cycle condition, then there is a diffeomorphism which does not have the shadowing property. Indeed, let a diffeomorphism 𝑓 of the two-dimensional torus 𝕋². The nonwandering set Ω(𝑓) consists of 4 hyperbolic fixed points, Ω(𝑓)={𝑝₁,𝑝₂,𝑝₃,𝑝₄}, where 𝑝₁ is a sink, 𝑝₄ is a source, and 𝑝₂ and 𝑝₃ are saddles such that 𝑊^𝑠(𝑝₂)∪{𝑝₃}=𝑊^𝑢(𝑝₃)∪{𝑝₂}. It is assumed that the eigenvalues of 𝐷_𝑝2𝑓 are −𝜇,𝜈 with 𝜇>1, 0<𝜈<1 and the eigenvalues of 𝐷_𝑝3𝑓 are −𝜆,𝜅 with 𝜅>1,0<𝜆<1. Then 𝑓 does not have the shadowing property. But it has the orbital shadowing property (see [3]).

In this paper, we study the orbital shadowing property in which it is clear that if a diffeomorphism has the shadowing property, then it has the orbital shadowing property, but the converse is not true. In fact, an irrational rotation map does not have the shadowing property, but it has the orbital shadowing property.

The orbital shadowing property was introduced by Pilyugin et al. [3]. They showed that a diffeomorphism belongs to the 𝐶¹-interior of the set of all diffeomorphisms having the orbital shadowing property if and only if the diffeomorphism is structurally stable.

For a conservative diffeomorphism, Bessa and Rocha proved in [4] that if a conservative diffeomorphism belongs to the 𝐶¹-interior of the set of all topologically stable conservative diffeomorphisms, then it is Anosov. In [5], Bessa proved that a conservative diffeomorphism is in the 𝐶¹-interior of the set of all conservative diffeomorphisms having the shadowing property if and only if it is Anosov. K. Lee and M. Lee [6] proved that a conservative diffeomorphism is in the 𝐶¹-interior of the set of all conservative diffeomorphisms having the orbital shadowing property if and only if it is Anosov. Our result is a generalization of the result in [7].

Let 𝑀 be a closed 𝐶^∞ Riemannian manifold endowed with a volume form 𝜔. Let 𝜇 denote the Lebesgue measure associated to 𝜔, and let 𝑑 denote the metric induced on 𝑀 by the Riemannian structure. Denote by Diff_𝜇(𝑀) the set of diffeomorphisms which preserves the Lebesgue measure 𝜇 endowed with the 𝐶¹ Whitney topology. We know that every volume preserving diffeomorphism satisfying Axiom 𝐴 is Anosov (for more details, see [8]).

For 𝛿>0, a sequence of points {𝑥_𝑖}^𝑏_𝑖=𝑎(−∞≤𝑎<𝑏≤∞) in 𝑀 is called a 𝛿-pseudo-orbit of 𝑓 if 𝑑(𝑓(𝑥_𝑖),𝑥_𝑖+1)<𝛿 for all 𝑎≤𝑖≤𝑏−1. We say that 𝑓 has the shadowing property if, for every 𝜖>0, there is 𝛿>0, such that, for any 𝛿-pseudo-orbit {𝑥_𝑖}_𝑖∈ℤ of 𝑓, there is a point 𝑦∈𝑀, such that, 𝑑(𝑓^𝑖(𝑦),𝑥_𝑖)<𝜖 for all 𝑖∈ℤ. It is easy to see that 𝑓 has the shadowing property if and only if 𝑓^𝑛 has the shadowing property for 𝑛∈ℤ⧵{0}. For each 𝑥∈𝑀, let 𝒪_𝑓(𝑥) be the orbit of 𝑓 through 𝑥; that is, 𝒪_𝑓(𝑥)=𝑓^𝑛(𝑥)∶𝑛∈ℤ. (1) We say that 𝑓 has the orbital shadowing property if, for any 𝜖>0, there exists 𝛿>0, such that for any 𝛿-pseudo-orbit 𝜉={𝑥_𝑖}_𝑖∈ℤ, we can find a point 𝑦∈𝑀 such that 𝒪_𝑓𝑦⊂𝐵_𝜖(𝜉),𝜉⊂𝐵_𝜖𝒪_𝑓𝑦, (2) where 𝐵_𝜖(𝐴) denotes the 𝜖-neighborhood of a set 𝐴⊂𝑀. It is easy to see that 𝑓 has the orbital shadowing property if and only if 𝑓^𝑛 has the orbital shadowing property for 𝑛∈ℤ⧵{0}. Let Λ be a closed 𝑓-invariant set. We say that Λ is hyperbolic if the tangent bundle 𝑇_Λ𝑀 has a 𝐷𝑓-invariant splitting 𝐸^𝑠⊕𝐸^𝑢 and there exist constants 𝐶>0 and 0<𝜆<1, such that 𝐷_𝑥𝑓^𝑛|_𝐸^𝑠_𝑥≤𝐶𝜆^𝑛,𝐷_𝑥𝑓^−𝑛|_𝐸^𝑢_𝑥≤𝐶𝜆^𝑛, (3) for all 𝑥∈Λ and 𝑛≥0.

In [3], the authors proved that the 𝐶¹-interior of the set of dissipative diffeomorphisms having the orbital shadowing property coincides with the set of structurally stable diffeomorphisms. Note that if a diffeomorphism satisfies structurally stable, then it is not Anosov in general, but the converse is true.

We say that a subset 𝒢⊂Diff(𝑀) is residual if 𝒢 contains the intersection of a countable family of open and dense subsets of Diff(𝑀); in this case, 𝒢 is dense in Diff(𝑀). A property “𝑃” is said to be 𝐶¹-generic if “𝑃” holds for all diffeomorphisms which belong to some residual subset of Diff(𝑀). We use the terminology “for 𝐶¹-generic 𝑓” to express that “there is a residual subset 𝒢⊂Diff(𝑀) such that for any 𝑓∈𝒢⋯.” The following is the main result in this paper.

Theorem 1. For 𝐶¹-generic 𝑓, if 𝑓 has the orbital shadowing property, then 𝑓 is Anosov.

Let 𝑝 be a periodic point of 𝑓 with period 𝜋(𝑝). We say that 𝑝 is an elementary point if 𝐷_𝑝𝑓^𝜋(𝑝) eigenvalues are multiplicatively independent over ℤ. Elementary points have simple spectrum, and none of 𝐷_𝑝𝑓^𝜋(𝑝) eigenvalues are a root of unity or equal to 1. For a periodic point 𝑝 of 𝑓, if we consider dim𝑀=2, then we have three cases. Firstly, 𝑝 is a hyperbolic saddle, that is, real eigenvalues 𝜆₁,𝜆₂ with 𝜆₂=𝜆₁⁻¹. Secondly, 𝑝 is an elliptic point; that is, nonreal eigenvalues are conjugated and of norm one. Finally, 𝑝 is a parabolic point; that is, eigenvalues equal 1 or −1. Note that the first and second cases are robust under small perturbations. Elementary elliptic points are associated with an irrational rotation number. In [9], Robinson showed that if dim𝑀=2, there is a residual set in Diff_𝜇(𝑀) such that any elementary in this residual displays all its elliptic points of elementary type. In [10, Theorem 1.3], Newhouse showed that 𝐶¹-generic volume-preserving diffeomorphisms in surfaces are Anosov, or else the elliptical points are dense. Actually, Newhouse’s proof is strongly supported in the symplectic structure. By Newhouse [10] and Robinson [9], we give a problem as follows: For 𝐶¹ generic 𝑓∈Diff_𝜇(𝑀²), if 𝑓 has the orbital shadowing property, then is it Anosov?

2. Proof of Theorems 1

Let 𝑀 be as before, and let 𝑓∈Diff_𝜇(𝑀). Let Λ be a closed 𝑓-invariant set. We say that Λ is a transitive set if there is a point 𝑥∈Λ such that 𝑥∈𝜔_𝑓(𝑥), where 𝜔_𝑓(𝑥) is the omega-limit set. If Λ=𝑀, then we called that 𝑓 is transitive. We say that 𝑝∈𝑃(𝑓) is a hyperbolic if 𝐷_𝑝𝑓^𝜋(𝑝)∶𝑇_𝑝𝑀→𝑇_𝑝𝑀 has no eigenvalues of absolute value one. It is well known that if 𝑝 is a hyperbolic periodic point of 𝑓 with period 𝑘, then 𝑊^𝑠𝑝=𝑥∈𝑀∶𝑓^𝑘𝑛(𝑥)⟶𝑝as,𝑊𝑛⟶∞_𝑢𝑝=𝑥∈𝑀∶𝑓^𝑘𝑛(𝑥)⟶𝑝as𝑛⟶−∞ (4) are 𝐶¹-injectively immersed submanifolds of 𝑀. Let 𝑞 be a hyperbolic periodic point of 𝑓. We say that 𝑝 and 𝑞 are homoclinically related if 𝑊^𝑠𝑝⋔𝑊^𝑢𝑞≠∅,𝑊^𝑢𝑝⋔𝑊^𝑠𝑞≠∅. (5) For given hyperbolic periodic points 𝑝 and 𝑞 of 𝑓, we write 𝑝∼𝑞 if 𝑝 and 𝑞 are homoclinically related. It is clear that if 𝑝∼𝑞, then index(𝑝)=index(𝑞). The following result is very useful to prove Theorem 1. Let 𝑝 be a hyperbolic periodic point. For 𝑥∈𝑀, we say that 𝑥 is a homoclinic point if 𝑊^𝑠(𝑝)⋔𝑊^𝑢(𝑝). Denote by 𝐻(𝑝,𝑓)=𝑊^𝑠(𝑝)⋔𝑊^𝑢(𝑝).

Theorem 2 (see [11, Theorem 1.3]). There is a residual set 𝒢₁⊂Diff_𝜇(𝑀) such that, for any 𝑓∈𝒢₁, 𝑓 is transitive. Moreover, 𝑀 is a unique homoclinic class.

We denote ℱ_𝜇(𝑀) by the set of diffeomorphisms 𝑓∈Diff_𝜇(𝑀) which has a 𝐶¹-neighborhood 𝒰(𝑓)⊂Diff_𝜇(𝑀) such that for any 𝑔∈𝒰(𝑓), every periodic point of 𝑔 is hyperbolic.

Very recently, Arbieto and Catalan [8] proved that every volume preserving diffeomorphism in ℱ_𝜇(𝑀) is Anosov.

Theorem 3 (see [8, Theorem 1]). Every volume preserving diffeomorphism in ℱ_𝜇(𝑀) is Anosov.

To prove Theorem 1, it is enough to show that 𝑓∈ℱ_𝜇(𝑀). Let 𝑝 be a hyperbolic periodic point of 𝑓; there exists 𝜖(𝑝)>0 such that, for any 𝑥∈𝑊^𝑠_𝜖(𝑝)(𝑝) and 𝑥∈𝑊^𝑢_𝜖(𝑝)(𝑝), we know that 𝑑𝑓^𝑖(𝑥),𝑓^𝑖𝑝≤𝜖𝑝,𝑑𝑓^−𝑖(𝑥),𝑓^−𝑖𝑝≤𝜖𝑝, (6) for all 𝑖≥0. Then 𝑊^𝑠_𝜖(𝑝)(𝑝) is called the local stable manifold of 𝑝, and 𝑊^𝑢_𝜖(𝑝)(𝑝) is the local unstable manifold of 𝑝. It is clear that 𝑊^𝑠_𝜖(𝑝)(𝑝)⊂𝑊^𝑠(𝑝), and 𝑊^𝑢_𝜖(𝑝)(𝑝)⊂𝑊^𝑢(𝑝).

Lemma 4. Let 𝑓∈𝒢₁, and let 𝑝,𝑞∈𝑃_ℎ(𝑓). If 𝑓 has the orbital shadowing property, then 𝑊^𝑠𝑝∩𝑊^𝑢𝑞≠∅,𝑊^𝑢𝑝∩𝑊^𝑠𝑞≠∅, (7) where 𝑃_ℎ(𝑓) is the set of all hyperbolic periodic points of 𝑓.

Proof. Let 𝑓∈𝒢₁, and let 𝑝,𝑞∈𝑃_ℎ(𝑓) be hyperbolic periodic points of 𝑓. Take 𝜖(𝑝)>0 and 𝜖(𝑞)>0 as before with respect to 𝑝 and 𝑞. For simplicity, we may assume that 𝑓(𝑝)=𝑝 and 𝑓(𝑞)=𝑞. Take 𝜖=min{𝜖(𝑝),𝜖(𝑞)}. Let 0<𝛿=𝛿(𝜖)<𝜖 be the number of the orbital shadowing property of 𝑓 for 𝜖. Since 𝑓 is transitive, there exists 𝑥∈𝑀 such that 𝜔(𝑥)=𝑀. Then there exist 𝑙₁>0 and 𝑙₂>0 such that 𝑑(𝑓^𝑙1(𝑥),𝑝)<𝛿 and 𝑑(𝑓^𝑙2(𝑥),𝑞)<𝛿. We may assume that 𝑙₂=𝑙₁+𝑘 for some 𝑘>0. Then we get a finite 𝛿-pseudo-orbit {𝑝,𝑓^𝑙1(𝑥),𝑓^𝑙1+1(𝑥),…,𝑓^𝑙2−1(𝑥),𝑞}. Now we construct a 𝛿-pseudo orbit as follows: put (i) 𝑓^𝑖(𝑝)=𝑥_𝑖 for 𝑖≤0, (ii) 𝑓^𝑙1+𝑖(𝑥)=𝑥_1+𝑖 for 0≤𝑖≤𝑘−1, and (iii) 𝑓^𝑙2+𝑖(𝑞)=𝑥_𝑘+𝑖 for 𝑖≥0. Then 𝜉=…,𝑝,𝑝,𝑥₀=𝑝,𝑓^𝑙1(𝑥),𝑓^𝑙1+1(𝑥)𝑓,…,_{𝑙1+𝑘−1}(𝑥),𝑓^𝑙2(𝑥)==𝑞,𝑞,𝑞,……,𝑥₋₂,𝑥₋₁,𝑥₀=𝑝,𝑥₁,𝑥₂𝑥,…,_𝑘−1,𝑥_𝑘=𝑞,𝑥_𝑘+1,…,. (8) Since 𝑓 has the orbital shadowing property, there are points 𝑦∈𝑀 and 𝑗≥0,𝑛>0 such that (i) 𝑓^𝑗(𝑦)∈𝐵_𝜖(𝑥₀) and 𝑓^−𝑖(𝑓^𝑗(𝑦))∈𝐵_𝜖(𝑥_−𝑖) for all 𝑖≥0; (ii) 𝑓^𝑛(𝑦)∈𝐵_𝜖(𝑥_𝑘) and 𝑓^𝑖(𝑓^𝑛(𝑦))∈𝐵_𝜖(𝑥_𝑘+𝑖) for all 𝑖≥0. Then 𝒪_𝑓𝑦⊂𝐵_𝜖(𝜉),𝜉⊂𝐵_𝜖𝒪_𝑓𝑦, (9)𝑓^𝑗(𝑦)∈𝑊^𝑢_𝜖(𝑝), and 𝑓^𝑛(𝑦)∈𝑊^𝑠_𝜖(𝑞). Thus, 𝒪_𝑓(𝑦)∩𝑊^𝑢(𝑝)∩𝑊^𝑠(𝑞)≠∅.

The study of the Kupka-Smale systems within volume preserving maps was developed by Robinson (see [1]). We say that 𝑓 is Kupka-Smale if every periodic point is hyperbolic and each invariant manifold has transverse intersections. Denote by 𝒦𝒮(𝑀) the set of Kupka-Smale volume-preserving diffeomorphisms. It is well-known that the 𝒦𝒮(𝑀) is residual in Diff_𝜇(𝑀).

Lemma 5. There is a residual set 𝒢₂⊂Diff_𝜇(𝑀) such that, for any 𝑓∈𝒢₂, if 𝑓 has the orbital shadowing property, then for any 𝑝,𝑞∈𝑃_ℎ(𝑓), index(𝑝)=index(𝑞).

Proof. Let 𝒢₂=𝒢₁∩𝒦𝒮(𝑀), and let 𝑓∈𝒢₂. Suppose that 𝑓 has the orbital shadowing property. Let 𝑝and𝑞 be hyperbolic periodic points of 𝑓. To derive a contradiction, we may assume that index(𝑝)≠index(𝑞). Then we know that dim𝑊^𝑠(𝑝)+dim𝑊^𝑢(𝑞)<dim𝑀 or dim𝑊^𝑢(𝑝)+dim𝑊^𝑠(𝑞)<dim𝑀. Assume that dim𝑊^𝑠(𝑝)+dim𝑊^𝑢(𝑞)<dim𝑀. Since 𝑓 is Kupka-Smale, we have 𝑊^𝑠(𝑝)∩𝑊^𝑢(𝑞)=∅. This is a contradiction by Lemma 4.

Remark 6. In dim(𝑀)=2, the index is always constant, and so these arguments cannot used in this low-dimensional case.

To prove our result, we use Franks’ lemma which is proved in [12, Proposition 7.4].

Lemma 7. Let 𝑓∈Diff¹_𝜇(𝑀), and let 𝒰(𝑓) be a 𝐶¹-neighborhood of 𝑓 in Diff¹_𝜇(𝑀). Then there exist a 𝐶¹-neighborhood 𝒰₀(𝑓)⊂𝒰(𝑓) of 𝑓 and 𝜖>0 such that if 𝑔∈𝒰₀(𝑓), any finite 𝑓-invariant set 𝐸={𝑥₁,…,𝑥_𝑚}, any neighborhood 𝑈 of 𝐸, and any volume-preserving linear maps 𝐿_𝑗∶𝑇_𝑥𝑗𝑀→𝑇_{𝑔(𝑥𝑗)}𝑀 with ‖𝐿_𝑗−𝐷_𝑥𝑗𝑔‖≤𝜖 for all 𝑗=1,…,𝑚, there is a conservative diffeomorphism 𝑔₁∈𝒰(𝑓) coinciding with 𝑓 on 𝐸 and out of 𝑈, and 𝐷_𝑥𝑗𝑔₁=𝐿_𝑗 for all 𝑗=1,…,𝑚.

Denote by 𝑃(𝑓) the set of all periodic points of 𝑓. The following was proved by [7]. Since the paper is still not published yet, we give the proof for completeness.

Lemma 8. Let dim𝑀≥3, and let 𝒰(𝑓) be a 𝐶¹-neighborhood of 𝑓. If 𝑝∈𝑃(𝑓) is not hyperbolic, then there is 𝑔∈𝒰(𝑓) such that 𝑔 has two periodic points 𝑝₁,𝑝₂∈𝑃_ℎ(𝑓) with different indices.

Proof. Let 𝑝∈𝑃(𝑓) be the nonhyperbolic periodic orbit of period 𝜋 and 𝜖>0. By Pugh-Robinson’s closing lemma [13] there is 𝑓₁∈Diff_𝜇(𝑀), such that 𝑓₁ is arbitrarily 𝐶¹-close to 𝑓, with 𝑞₁∈𝑃(𝑓₁) close to 𝑝 by closing some recurrent orbit, since Poincaré recurrence almost every point is recurrent. Moreover, since hyperbolicity holds open and is densely even in the volume-preserving setting, 𝑝₂ can be chosen to be hyperbolic. Let index(𝑝₂)=𝑖. After this perturbation (away from the orbit of 𝑝₂), we get 𝑓₁∈Diff_𝜇(𝑀) such that 𝑓₁ has a periodic orbit 𝑝₁ close to 𝑝, with period 𝜋. We observe that 𝑝₁ may not be the analytic continuation of 𝑝 and this is precisely the case when 1 is an eigenvalue of the tangent map 𝐷𝑓^𝜋(𝑝). If 𝑝₁ is not hyperbolic take 𝑓₂=𝑓₁. If 𝑝₁ is hyperbolic for 𝐷𝑓^𝜋₁(𝑝₁), then, since 𝑓₁ is arbitrarily 𝐶¹-close to 𝑓, the distance between the spectrum of 𝐷𝑓^𝜋₁(𝑝₁) and the unitary circle can be taken arbitrarily close to zero. This means that we are in the presence of a very weak hyperbolicity, that is, of a 𝛿-weak eigenvalue thus in a position to apply [12, Proposition 7.4] to obtain 𝑓₂∈Diff_𝜇(𝑀), such that 𝑝₁ is a non-hyperbolic periodic orbit. Moreover, this local perturbation can be done far from the periodic point 𝑝₂. Once again, we use [12, Proposition 7.4] in order to obtain 𝑔∈Diff_𝜇(𝑀), such that 𝑝₁∈𝑃(𝑔) is hyperbolic and index(𝑝₁)≠𝑖.

The following is a volume-preserving diffeomorphism version of [14, Lemma 2.2].

Lemma 9. There is a residual set 𝒢₃⊂Diff_𝜇(𝑀) such that, for any 𝑓∈𝒢₃, if for any 𝐶¹-neighborhood 𝒰(𝑓) of 𝑓 there is 𝑔∈𝒰(𝑓) such that 𝑔 has two hyperbolic periodic points 𝑝_𝑔,𝑞_𝑔∈𝑃_ℎ(𝑔) with different indices, then 𝑓 has two hyperbolic periodic points 𝑝,𝑞∈𝑃_ℎ(𝑓) with different indices.

For any 𝛿>0, we say that 𝑝∈𝑃(𝑓) has a 𝛿-weak eigenvalue if there is an eigenvalue 𝜆 of 𝐷𝑓^𝜋(𝑝)(𝑝), such that, (1−𝛿)^𝜋(𝑝)<|𝜆|<(1+𝛿)^𝜋(𝑝), where 𝜋(𝑝) is the period of 𝑝.

Lemma 10. There is a residual set 𝒢₄⊂Diff_𝜇(𝑀) such that for any 𝑓∈𝒢₄, if 𝑓 has the orbital shadowing property then there is 𝛿>0 such that for any 𝑝∈𝑃(𝑓), 𝑝 does not have a 𝛿-weak eigenvalue.

Proof. Let 𝒢₄=𝒢₂∩𝒢₃, and let 𝑓∈𝒢₄ have the orbital shadowing property. To derive a contradiction, we may assume that there is 𝑝∈𝑃(𝑓) such that, for any 𝛿>0, 𝑝 has a 𝛿-weak eigenvalue. Then by Lemma 7, we can find ℎ  𝐶¹-close to 𝑓, such that 𝑝_ℎ is not hyperbolic, where 𝑝_ℎ is the continuation of 𝑝. By Lemma 8, again using the Lemma 7, we take 𝑔  𝐶¹-close to ℎ and also 𝐶¹-close to 𝑓 such that 𝑔 has two hyperbolic periodic points 𝑝_𝑔,𝑞_𝑔∈𝑃_ℎ(𝑔) with index(𝑝_𝑔)≠index(𝑞_𝑔). Since 𝑓∈𝒢₆, by Lemma 9, 𝑓 has two hyperbolic periodic points, 𝑝,𝑞, with index(𝑝)≠index(𝑞). This is a contradiction by Lemma 5.

Lemma 11 (see [15, Lemma 5.1]). There is a residual set 𝒢₅⊂Diff_𝜇(𝑀) such that, for any 𝑓∈𝒢₅, for any 𝛿>0, if for any 𝐶¹-neighborhood 𝒰(𝑓), there is 𝑔∈𝒰(𝑓), such that for any 𝑝_𝑔∈𝑃_ℎ(𝑔), 𝑝_𝑔 has a 𝛿-weak eigenvalue, then 𝑝∈𝑃_ℎ(𝑓) has a 2𝛿-weak eigenvalue.

Proof of Theorem 1. Let 𝒢₆=𝒢₄∩𝒢₅, and let 𝑓∈𝒢₆ have the orbital shadowing property. The proof is by a contradiction; we may assume that 𝑓∉ℱ(𝑀). Then there is a non-hyperbolic periodic point 𝑝_𝑔 for some 𝑔  𝐶¹-nearby 𝑓, such that 𝑝_𝑔 is 𝛿/2-weak eigenvalue. Then by Lemma 11, 𝑝∈𝑃(𝑓) has 𝛿-weak eigenvalue. This is a contradiction by Lemma 10.

Acknowledgments

The author wishes to express his deepest appreciation to the referee for his careful reading of the paper, critical comments, and valuable suggestions. This work is supported by 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).