Orbital Shadowing for C 1-Generic Volume-Preserving Diffeomorphisms

and Applied Analysis 3 Lemma 4. Let f ∈ G 1 , and let p, q ∈ P h (f). If f has the orbital shadowing property, then W s (p) ∩ W u (q) ̸ = 0, W u (p) ∩W s (q) ̸ = 0, (7) where P h (f) is the set of all hyperbolic periodic points of f. Proof. Let f ∈ G 1 , 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 l 1 > 0 and l 2 > 0 such that d(f1(x), p) < δ and d(f2(x), q) < δ. We may assume that l 2 = l 1 + k for some k > 0. Then we get a finite δ-pseudoorbit {p, f1(x), f1(x), . . . , f2(x), q}. Now we construct a δ-pseudo orbit as follows: put (i) f(p) = x i for i ≤ 0, (ii) f l 1 +i (x) = x 1+i for 0 ≤ i ≤ k − 1, and (iii) f2(q) = x k+i for i ≥ 0. Then ξ = {. . . , p, p, x 0 (= p) , f l 1 (x) , f l 1 +1 (x) , . . . , f l 1 +k−1 (x) , f l 2 (x) (= q) , q, q, . . .} = {. . . , x −2 , x −1 , x 0 (= p) , x 1 , x 2 , . . . , x 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(y) ∈ B ε (x 0 ) and f −i (f j (y)) ∈ B ε (x −i ) for all i ≥ 0; (ii) f(y) ∈ B ε (x k ) and f i (f n (y)) ∈ B ε (x k+i ) for all i ≥ 0. Then O f (y) ⊂ B ε (ξ) , ξ ⊂ B ε (O f (y)) , (9)


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  1 -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  of the two-dimensional torus T 2 .The nonwandering set Ω() consists of 4 hyperbolic fixed points, Ω() = { 1 ,  2 ,  3 ,  4 }, where  1 is a sink,  4 is a source, and  2 and  3 are saddles such that   ( 2 ) ∪ { 3 } =   ( 3 ) ∪ { 2 }.It is assumed that the eigenvalues of   2  are −, ] with  > 1, 0 < ] < 1 and the eigenvalues of   3  are −,  with  > 1, 0 <  < 1.Then  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  1 -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  1 -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  1 -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  1interior 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  be a closed  ∞ Riemannian manifold endowed with a volume form . Let  denote the Lebesgue measure associated to , and let  denote the metric induced on  by the Riemannian structure.Denote by Diff  () the set of diffeomorphisms which preserves the Lebesgue measure  endowed with the  1 Whitney topology.We know that every volume preserving diffeomorphism satisfying Axiom  is Anosov (for more details, see [8]).
For  > 0, a sequence of points {  }  = (−∞ ≤  <  ≤ ∞) in  is called a -pseudo-orbit of  if ((  ),  +1 ) <  for all  ≤  ≤  − 1.We say that  has the shadowing property if, for every  > 0, there is  > 0, such that, for any -pseudo-orbit {  } ∈Z of , there is a point  ∈ , such that, (  (),   ) <  for all  ∈ Z.It is easy to see that  has the shadowing property if and only if   has the shadowing property for  ∈ Z \ {0}.For each  ∈ , let O  () be the orbit of  through ; that is, We say that  has the orbital shadowing property if, for any  > 0, there exists  > 0, such that for any -pseudo-orbit  = {  } ∈Z , we can find a point  ∈  such that where for all  ∈ Λ and  ≥ 0.
In [3], the authors proved that the  1 -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() is residual if G contains the intersection of a countable family of open and dense subsets of Diff(); in this case, G is dense in Diff().A property "" is said to be  1 -generic if "" holds for all diffeomorphisms which belong to some residual subset of Diff().We use the terminology "for  1 -generic " to express that "there is a residual subset G ⊂ Diff() such that for any  ∈ G ⋅ ⋅ ⋅ ." The following is the main result in this paper.
Theorem 1.For  1 -generic , if  has the orbital shadowing property, then  is Anosov.
Let  be a periodic point of  with period ().We say that  is an elementary point if    () eigenvalues are multiplicatively independent over Z. Elementary points have simple spectrum, and none of    () eigenvalues are a root of unity or equal to 1.For a periodic point  of , if we consider dim  = 2, then we have three cases.Firstly,  is a hyperbolic saddle, that is, real eigenvalues  1 ,  2 with  2 =  −1 1 .Secondly,  is an elliptic point; that is, nonreal eigenvalues are conjugated and of norm one.Finally,  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 dim  = 2, there is a residual set in Diff  () such that any elementary in this residual displays all its elliptic points of elementary type.In [10, Theorem 1.3], Newhouse showed that  1 -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  1 generic  ∈ Diff  ( 2 ), if  has the orbital shadowing property, then is it Anosov?

Proof of Theorems 1
Let  be as before, and let  ∈ Diff  ().Let Λ be a closed -invariant set.We say that Λ is a transitive set if there is a point  ∈ Λ such that  ∈   (), where   () is the omegalimit set.If Λ = , then we called that  is transitive.We say that  ∈ () is a hyperbolic if    () :    →    has no eigenvalues of absolute value one.It is well known that if  is a hyperbolic periodic point of  with period , then are  1 -injectively immersed submanifolds of .Let  be a hyperbolic periodic point of .We say that  and  are homoclinically related if For given hyperbolic periodic points  and  of , we write  ∼  if  and  are homoclinically related.It is clear that if  ∼ , then index() = index().The following result is very useful to prove Theorem 1.Let  be a hyperbolic periodic point.For  ∈ , we say that  is a homoclinic point if   () ⋔   ().Denote by (, ) =   () ⋔   ().
We denote F  () by the set of diffeomorphisms  ∈ Diff  () which has a  1 -neighborhood U() ⊂ Diff  () such that for any  ∈ U(), every periodic point of  is hyperbolic.
Very recently, Arbieto and Catalan [8] proved that every volume preserving diffeomorphism in F  () is Anosov.
where  ℎ () is the set of all hyperbolic periodic points of .
The study of the Kupka-Smale systems within volume preserving maps was developed by Robinson (see [1]).We say that  is Kupka-Smale if every periodic point is hyperbolic and each invariant manifold has transverse intersections.Denote by KS() the set of Kupka-Smale volume-preserving diffeomorphisms.It is well-known that the KS() is residual in Diff  ().Lemma 5.There is a residual set G 2 ⊂ Diff  () such that, for any  ∈ G 2 , if  has the orbital shadowing property, then for any ,  ∈  ℎ (), index(p) = index(q).Proof.Let G 2 = G 1 ∩ KS(), and let  ∈ G 2 .Suppose that  has the orbital shadowing property.Let  and  be hyperbolic periodic points of .To derive a contradiction, we may assume that index() ̸ = index().Then we know that dim   () + dim   () < dim  or dim   () + dim   () < dim .Assume that dim   () + dim   () < dim .Since  is Kupka-Smale, we have   () ∩   () = 0.This is a contradiction by Lemma 4. Remark 6.In dim() = 2, the index is always constant, and so these arguments cannot used in this low-dimensional case.
Denote by () the set of all periodic points of .The following was proved by [7].Since the paper is still not published yet, we give the proof for completeness.Lemma 8. Let dim  ≥ 3, and let U() be a  1neighborhood of .If  ∈ () is not hyperbolic, then there is  ∈ U() such that  has two periodic points  1 ,  2 ∈  ℎ () with different indices.
Proof.Let  ∈ () be the nonhyperbolic periodic orbit of period  and  > 0. By Pugh-Robinson's closing lemma [13] there is  1 ∈ Diff  (), such that  1 is arbitrarily  1 -close to , with  1 ∈ ( 1 ) close to  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,  2 can be chosen to be hyperbolic.Let index( 2 ) = .After this perturbation (away from the orbit of  2 ), we get  1 ∈ Diff  () such that  1 has a periodic orbit  1 close to , with period .We observe that  1 may not be the analytic continuation of  and this is precisely the case when 1 is an eigenvalue of the tangent map   ().If  1 is not hyperbolic take  2 =  1 .If  1 is hyperbolic for   1 ( 1 ), then, since  1 is arbitrarily  1 -close to , the distance between the spectrum of   1 ( 1 ) 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 -weak eigenvalue thus in a position to apply [12,Proposition 7.4] to obtain  2 ∈ Diff  (), such that  1 is a non-hyperbolic periodic orbit.Moreover, this local perturbation can be done far from the periodic point  2 .Once again, we use [12,Proposition 7.4] in order to obtain  ∈ Diff  (), such that  1 ∈ () is hyperbolic and index( 1 ) ̸ = .
The following is a volume-preserving diffeomorphism version of [14, Lemma 2.2].Lemma 9.There is a residual set G 3 ⊂ Diff  () such that, for any  ∈ G 3 , if for any  1 -neighborhood U() of  there is  ∈ U() such that  has two hyperbolic periodic points   ,   ∈  ℎ () with different indices, then  has two hyperbolic periodic points ,  ∈  ℎ () with different indices.
For any  > 0, we say that  ∈ () has a -weak eigenvalue if there is an eigenvalue  of  () (), such that, (1 − ) () < || < (1 + ) () , where () is the period of .Lemma 10.There is a residual set G 4 ⊂ Diff  () such that for any  ∈ G 4 , if  has the orbital shadowing property then there is  > 0 such that for any  ∈ (),  does not have a -weak eigenvalue.
Proof.Let G 4 = G 2 ∩ G 3 , and let  ∈ G 4 have the orbital shadowing property.To derive a contradiction, we may assume that there is  ∈ () such that, for any  > 0,  has a -weak eigenvalue.Then by Lemma 7, we can find ℎ  1 -close to , such that  ℎ is not hyperbolic, where  ℎ is the continuation of .By Lemma 8, again using the Lemma 7, we take   1 -close to ℎ and also  1 -close to  such that  has two hyperbolic periodic points   ,   ∈  ℎ () with index(  ) ̸ = index(  ).Since  ∈ G 6 , by Lemma 9,  has two hyperbolic periodic points, , , with index() ̸ = index().This is a contradiction by Lemma 5.
Proof of Theorem 1.Let G 6 = G 4 ∩ G 5 , and let  ∈ G 6 have the orbital shadowing property.The proof is by a contradiction; we may assume that  ∉ F().Then there is a non-hyperbolic periodic point   for some   1 -nearby , such that   is /2-weak eigenvalue.Then by Lemma 11,  ∈ () has -weak eigenvalue.This is a contradiction by Lemma 10.
() denotes the -neighborhood of a set  ⊂ .It is easy to see that  has the orbital shadowing property if and only if   has the orbital shadowing property for  ∈ Z \ {0}.Let Λ be a closed -invariant set.We say that Λ is hyperbolic if the tangent bundle  Λ  has a -invariant splitting   ⊕   and there exist constants  > 0 and 0 <  < 1, such that          | ≤   ,         − |         ≤   ,