Volume Preserving Diffeomorphisms with Inverse Shadowing

Let f be a volume-preserving diffeomorphism of a closed C^\infty n-dimensional Riemannian manifold M: In this paper, we prove the equivalence between the following conditions: (a) f belongs to the C1-interior of the set of volume-preserving diffeoeomorphisms which satisfy the inverse shadowing property with respect to the continuous methods. (b) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the weak inverse shadowing property with respect to the continuous methods. (c) f belongs to the C1-interior of the set of volume-preserving diffeomorphisms which satisfy the orbital inverse shadowing property with respect to the continuous methods, (d) f is Anosov.


Introduction
Let M be a closed C ∞ n-dimensional Riemannian manifold, and let Diff(M) be the space of diffeomorphisms of M endowed with the C 1 -topology. Denote by d the distance on M induced from a Riemannian metric · on the tangent bundle T M. Let f : M → M be a diffeomorphism, and let Λ ⊂ M be a closed f -invariant set.
For δ > 0, a sequence of points We say that f has the shadowing property on Λ if for any ǫ > 0 there is δ > 0 such that for any δ-pseudo orbit {x i } i∈Z ⊂ Λ of f there is y ∈ M such that d(f i (y), x i ) < ǫ, for i ∈ Z. Note that in this definition, the shadowing point y ∈ M is not necessarily contained in Λ. We say that f belongs to the C 1 -interior shadowing property if there is a C 1 -neighborhood U(f ) of f such that for any g ∈ U(f ), g has the shadowing property.
The shadowing property usually plays an important role in the investigation of stability theory and ergodic theory( [14]). Now, we introduce the notion of the inverse shadowing property which is a "dual" notion of the shadowing property. Inverse shadowing property was introduced by Corless and Pilyugin in [4], and the qualitative theory of dynamical systems with the property was developed by various authors(see [3,4,5,7,8,15]. In this paper, we introduce the various inverse shadowing property.
Let M Z be the space of all two sided sequences ξ = {x n : n ∈ Z} with elements x n ∈ M, endowed with the product topology. For a fixed The set of all δ-methods for f will be denoted by T 0 (f, δ). Say that ϕ is continuous δ-method for f if ϕ is continuous. The set of all continuous δ-methods for f will be denoted by T c (f, δ). If g : M → M is a homeomorphism with d 0 (f, g) < δ then g induces a continuous δ-method ϕ g for f by defining Let T h (f, δ) denote the set of all continuous δ-methods ϕ g for f which are induced by a homeomorphism g : M → M with d 0 (f, g) < δ, where d 0 is the usual C 0 -metric. Let T d (f, δ) denote by the set of all continuous δ-methods ϕ g for f which are induced by g ∈ Diff(M) with d 1 (f, g) < δ. Then clearly we know that We say that f has the inverse shadowing property on Λ with respect to the class T α (f ), α = 0, c, h, d, if for any ǫ > 0 there exists δ > 0 such that for any δ-method ϕ ∈ T α (f, δ), and for a point x ∈ Λ there is a point y ∈ M such that We say that f has the weak inverse shadowing property on Λ with respect to the class T α (f ), α = 0, c, h, d, if for any ǫ > 0 there exists δ > 0 such that for any δ-method ϕ ∈ T α (f, δ) and any point x ∈ Λ there is a point y ∈ M such that where B ǫ (A) = {x ∈ M : d(x, A) ≤ ǫ}. If Λ = M then f has the inverse, weak inverse shadowing property with respect to the class T α (f ), α = 0, c, h, d.
Note that if f ∈ Diff(M) has the inverse shadowing property with respect to the class T d (f ) then by the definition, it clearly, has the weak inverse shadowing property with respect to the class T d (f ). We say that f has the orbital inverse shadowing property on Λ with respect to the class T α (f ), α = 0, c, h, d, if for any ǫ > 0 there is a δ > 0 such that for any δ-method ϕ ∈ T α (f, δ) and a point x ∈ Λ there is a point y ∈ M such that If Λ = M then f has the orbital inverse shadowing property with respect to the class T α (f ), α = 0, c, h, d.
Note that if f has the inverse shadowing property with respect to the class T d (f ), then it has the orbital inverse shadowing property with respect to the class T d (f ). But, the converse does not holds. indeed, an irrational rotation on the unit circle has the orbital inverse shadowing property but does not have the inverse shadowing property with respect to the class T d (f ). We say that f belongs to the C 1 -interior inverse(weak inverse, or orbital inverse) shadowing property with respect to the class T α (f ), α = 0, c, h, d, if there is a C 1 -neighborhood U(f ) of f such that for any g ∈ U(f ), g has the inverse(weak inverse, or orbital inverse) shadowing property with respect to the class T α (f ), α = 0, c, h, d.
Lee [8], showed that a diffeomorphism belongs to the C 1 -interior inverse shadowing property with respect to the T d (f ) if and only if it is structurally stable. And Pilyugin [15] proved that a diffeomorphism belongs to the C 1 -interior inverse shadowing property with respect to the class T c (f ) if and only if it is structurally stable. Thus we can restate the above facts as follows.
T c (f )] if and only if it is structurally stable.
In [3] Choi, Lee and Zhang showed that a diffeomorphism belongs to the C 1 -interior weak inverse shadowing property with respect to the T d (f ) if and only if both Axiom A and the no-cycle condition. Moreover, they proved that a diffeomorphism belongs to the C 1 -interior orbital inverse shadowing property with respect to the class T d (f ) if and only if both Axiom A and the strong transversal condition. From the above facts, we get the follows.
If a diffeomorphism f belongs to the C 1 -interior weak inverse shadowing property with respect to the class T d (f ) then f satisfying both Axiom A and the no-cycle condition. Moreover, if f belongs to the C 1 -interior orbital inverse shadowing property with respect to the class T d (f ) then it is structurally stable.
By the theorem, even though a diffeomorphism is contained in the C 1 -interior of the set of diffeomorphisms possessing the weak inverse shadowing property with respect to the class T d (f ), it does not necessarily satisfy the strong transversality condition. A if and only if f satisfies both Axiom A and the no-cycle condition.
Let Λ be a closed f ∈ Diff(M)-invariant set. We say that Λ is hyperbolic if the tangent bundle T Λ M has a Df -invariant splitting E s ⊕ E u and there exists constants C > 0 and 0 < λ < 1 such that for all x ∈ Λ and n ≥ 0. If Λ = M then we say that f is an Anosov diffeomorphism.

Statement of the results
A fundamental problem in differentiable dynamical systems is to understand how a robust dynamic property on the underlying manifold would influence the behavior of the tangent map on the tangent bundle. For instance, in [11], Mañé proved that any C 1 structurally stable diffeomorphism is an Axiom A diffeomorphism. And in [13], Palis extended this result to Ω-stable diffeomorphisms.
Let M be a compact C ∞ n-dimensional Riemannian manifold endowed with a volume form ω. Let µ denote the measure associated to ω, that we call Lebesgue measure, and let d denote the metric induced by the Riemannian structure. Denote by Diff µ (M) the set of diffeomorphisms which preserves the Lebesgue measure µ endowed with the C 1 -topology. In the volume preserving, the Axiom A condition is equivalent to the diffeomorphism be Anosov, since Ω(f ) = M by Poincaré Recurrence Theorem. The purpose of this paper is to do this using the robust property.
We define the set F µ (M) as the set of diffeomorphisms f ∈ Diff µ (M) which have a C 1 -neighborhood U(f ) ⊂ Diff µ (M) such that if for any g ∈ U(f ), every periodic point of g is hyperbolic. Note that F µ (M) ⊂ F (M)(see [1,Corollary 1.2]).
Very recently, Arbieto and Catalan [1] proved that if a volume preserving diffeomorphism contained in F µ (M) then it is Anosov. Indeed, the first they used the Mañé's results([11, Proposition II.1]). Then they showed that P (f ) is hyperbolic. And, they proved that nonwandering set Ω(f ) = P (f ) by Pugh's closing lemma. Finally, by Pincaré 's Recurrence Theorem, Ω(f ) = M. From the above facts, we can restate as follows.
In [9], Lee showed that if a volume preserving diffeomorphisms belongs to the C 1 -interior expansive or C 1 -interior shadowing property, then it is Anosov. And [10] proved that if a volume preserving diffeomorphisms belongs to the C 1 -interior weak shadowing property or C 1 -interior weak limit shadowing property, then it is Anosov. Form this results, we study the cases when a volume preserving diffeomorphism f is in C 1 -interior various inverse shadowing property with respect to the class T d (f ), then it is Anosov. Let intIS µ (M) be denote the set of volume preserving diffeomorphisms in Diff µ (M) satisfying the inverse shadowing property with respect to the class T d , and let intWIS µ (M)[respect. intOIS µ (M)] be denote the set of volumepreserving diffeomorphisms in Diff µ (M) satisfying the weak inverse shadowing property with respect to the class T d [respect. the orbital inverse shadowing property with respect to the class T d ] . From now, we only consider the class T d when we mention the inverse shadowing property; that is, the "inverse shadowing property" implies the " inverse shadowing property with respect to the class T d ". Now we are in position to state the theorem of our paper.

Proof of Theorem 2.2
Let M be a compact C ∞ n-dimensional Riemannian manifold endowed with a volume form ω, and let f ∈ Diff µ (M). To prove the results, we will use the following is the well-known Franks' lemma for the conservative case, stated and proved in [2,Proposition 7.4].
. Then there exist a C 1 -neighborhood U 0 ⊂ U of f and ǫ > 0 such that if g ∈ U 0 , any finite f -invariant set E = {x 1 , . . . , x m }, any neighborhood U of E and any volume-preserving linear maps L j : . . , m, there is a conservative diffeomorphism g 1 ∈ U coinciding with f on E and out of U, and D x j g 1 = L j for all j = 1, . . . , m.
. From the Moser's Theorem(see [12]), there is a smooth conservative change of coordinates ϕ x : Proof. Take f ∈ intIS µ (M), and U(f ) a C 1 -neighborhood of f ∈ intE µ (M). Let ǫ > 0 and V(f ) ⊂ U 0 (f ) corresponding number and C 1neighborhood given by Lemma 3.1. To derive a contradiction, we may assume that there exists a nonhyperbolic periodic point p ∈ P (g) for some g ∈ V(f ). To simplify the notation in the proof, we may assume that g(p) = p. Then there is at least one eigenvalue λ of D p g such that |λ| = 1.
Thus g 1 does not have the inverse shadowing property. Therefore, we can choose a point y ∈ M \ J p such that d(x, y) < ǫ 1 .
Thus g 1 does not have the inverse shadowing property. This is a contradiction since f ∈ intIS µ (M).
Finally, if λ ∈ C, then to avoid the notational complexity, we may assume that g(p) = p. As in the first case, by Lemma 3.1, there are α > 0 and g 1 ∈ V(f ) such that g 1 (p) = g(p) = p and With a C 1 -small modification of the map D p g, we may suppose that there is l > 0(the minimum number) such that D p g l (v) = v for any v ∈ ϕ p (B α (p)) ⊂ T p M. Then, we can go on with the previous argument in order to reach the same contradiction. Thus, every periodic point of f ∈ intIS µ (M) is hyperbolic. Proof. Take f ∈ intWIS µ (M), and U(f ) a C 1 -neighborhood of f ∈ intWIS µ (M). Let ǫ > 0 and V(f ) ⊂ U 0 (f ) corresponding number and C 1 -neighborhood given by Lemma 3.1. To derive a contradiction, we may assume that there exists a nonhyperbolic periodic point p ∈ P (g) for some g ∈ V(f ). To simplify the notation in the proof, we may assume that g(p) = p. Then as in the proof of Proposition 3.3, we can take α > 0 sufficiently small, and a smooth map ϕ p : B α (p) → T p M. Then we can make an arc J p ⊂ B α (p) and for some g 1 ∈ V(f ). Take ǫ 1 = (lengthJ p )/4. Let 0 < δ < ǫ 1 be the number of the weak inverse shadowing property of g 1 for ǫ 1 . Then we can construct a map h ∈ Diff(M) as in the proof of Proposition3.3. Let p = 0. Then choose a point x = (x 1 , 0, . . . , 0) ∈ J p such that d(0, x) = 2ǫ 1 . Since g 1 has the weak inverse shadowing property, However, for any y ∈ J p , g i 1 (y) = y and h i (y) = (y 1 + iδ, A i y ′ ), where y ′ = (y 2 , . . . , y n ). Thus it easily see that If y = (y 1 , 0, . . . , 0) ∈ J p \ {p}, then h k (y) = (y 1 + kδ, A k y ′ ) = (y 1 + kδ, 0).

Thus we know that
). This is a contradiction.
Finally, we can choose a point y ∈ M \ J p such that d(x, y) < ǫ 1 . Then we know that h k (y) = (y 1 + kδ, A k y ′ ).
Therefore, h l (y) ∈ B ǫ 1 (J p ), for some l ∈ Z. Then Thus g 1 does not have the weak inverse shadowing property. This is a contradiction.
If λ ∈ C, then as in the proof of Proposition 3.3, for g 1 ∈ V(f ), we can take l > 0 such that D p g l 1 (v) = v for any v ∈ ϕ p (B α (p)) ⊂ T p M. Then from the previous argument in order to reach the same contradiction. Thus, every periodic point of f ∈ intWIS µ (M) is hyperbolic.
Consequently, if f ∈ intWIS µ (M) then f ∈ F µ (M). Proof. The proof is almost the same that of Proposition 3.4. Indeed, let f ∈ intOIS µ (M), and U(f ) a C 1 -neighborhood of f ∈ intOIS µ (M). Let ǫ > 0 and V(f ) ⊂ U 0 (f ) corresponding number and C 1 -neighborhood given by Lemma 3.1. To derive a contradiction, we may assume that there exists a nonhyperbolic periodic point p ∈ P (g) for some g ∈ V(f ). To simplify the notation in the proof, we may assume that g(p) = p. Then as in the proof of Proposition 3.4, we can take α > 0 sufficiently small, and a smooth map ϕ p : B α (p) → T p M.
Then we can make an arc J p ⊂ B α (p) and for some g 1 ∈ V(f ). Take ǫ 1 = (lengthJ p )/4. Let 0 < δ < ǫ 1 be the number of the orbital inverse shadowing property of g 1 for ǫ 1 . Then we can construct a map h ∈ Diff(M) as in the proof of Proposition3.4. Let p = 0. Then choose a point x = (x 1 , 0, . . . , 0) ∈ J p such that d(0, x) = 2ǫ 1 . Since g 1 has the orbital inverse shadowing property, However, for any y ∈ J p , g i 1 (y) = y and h i (y) = (y 1 + iδ, A i y ′ ), where y ′ = (y 2 , . . . , y n ). Thus it easily see that Thus g 1 does not have the orbital inverse shadowing property.
Thus we know that O h (y) ⊂ B ǫ 1 (O g 1 (x)). This is a contradiction.
Thus g 1 does not have the orbital inverse shadowing property. This is a contradiction.
If λ ∈ C, then as in the proof of Proposition 3.4, for g 1 ∈ V(f ), we can take l > 0 such that D p g l 1 (v) = v for any v ∈ ϕ p (B α (p)) ⊂ T p M. Then from the previous argument in order to reach the same contradiction. Thus, every periodic point of f ∈ intOIS µ (M) is hyperbolic.