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Research | Open | Published:

Volume-preserving diffeomorphisms with inverse shadowing

Abstract

Let f be a volume-preserving diffeomorphism of a closed C n-dimensional Riemannian manifold M. In this paper, we prove the equivalence between the following conditions:

(a) f belongs to the C 1 -interior of the set of volume-preserving diffeomorphisms which satisfy the inverse shadowing property with respect to the continuous methods,

(b) f belongs to the C 1 -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 C 1 -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.

MSC:37C50, 34D10.

1 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 TM. Let f:MM be a diffeomorphism, and let ΛM be a closed f-invariant set.

For δ>0, a sequence of points { x i } i = a b (a<b) in M is called a δ-pseudo orbit of f if d(f( x i ), x i + 1 )<δ for all aib1. We say that f has the shadowing property on Λ if for any ϵ>0, there is δ>0 such that for any δ-pseudo orbit { x i } i Z Λ of f, there is yM such that d( f i (y), x i )<ϵ for iZ. Note that in this definition, the shadowing point yM 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 gU(f), g has the shadowing property.

The shadowing property usually plays an important role in the investigation of stability theory and ergodic theory [1].

Now, we introduce the notion of the inverse shadowing property which is a ‘dual’ notion of the shadowing property. The inverse shadowing property was introduced by Corless and Pilyugin in [2], and the qualitative theory of dynamical systems with the property was developed by various authors (see [27]). In this paper, we introduce the various inverse shadowing properties.

Let M Z be the space of all two-sided sequences ξ={ x n :nZ} with elements x n M, endowed with the product topology. For any δ>0, let Φ f (δ) denote the set of all δ-pseudo orbits of f. A mapping φ:M Φ f (δ) M Z is said to be a δ-method for f if φ ( x ) 0 =x, and each φ(x) is a δ-pseudo orbit of f through x, where φ ( x ) 0 denotes the 0th component of φ(x). For convenience, write φ(x) for { φ ( x ) k } k Z . The set of all δ-methods for f will be denoted by T 0 (f,δ). We say that φ is a 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:MM is a homeomorphism with d 0 (f,g)<δ, then g induces a continuous δ-method φ g for f by defining

φ g (x)= { g n ( x ) : n Z } .

Let T h (f,δ) denote the set of all continuous δ-methods φ g for f which are induced by a homeomorphism g:MM with d 0 (f,g)<δ, where d 0 is the usual C 0 -metric. Let T d (f,δ) denote the set of all continuous δ-methods φ g for f which are induced by gDiff(M) with d 1 (f,g)<δ. Then, clearly, we know that

T d (f) T h (f) T c (f) T 0 (f),

T α (f)= δ > 0 T α (f,δ), α=0,c,h,d. We say that f has the inverse shadowing property with respect to the class T α (f), α=0,c,h,d, if for any ϵ>0, there exists δ>0 such that for any δ-method φ T α (f,δ) and any point xM, there is a point yM, such that

d ( f k ( x ) , φ ( y ) k ) <ϵ,kZ.

We say that f has the weak inverse shadowing property with respect to the class T α (f), α=0,c,h,d, if for any ϵ>0, there exists δ>0 such that for any δ-method φ T α (f,δ) and any point xM, there is a point yM such that

φ(y) B ϵ ( O f ( x ) ) ,

where B ϵ (A)={xM:d(x,A)ϵ}. Note that if fDiff(M) has the inverse shadowing property with respect to the class T α (f) (α=0,c,h,d), then by the definition, it clearly has the weak inverse shadowing property with respect to the class T α (f) (α=0,c,h,d). We say that f has the orbital inverse shadowing property with respect to the class T α (f), α=0,c,h,d, if for any ϵ>0, there is a δ>0 such that for any δ-method φ T α (f,δ) and any point xM, there is a point yM such that

O f (x) B ϵ ( φ ( y ) ) andφ(y) B ϵ ( O f ( x ) ) .

Note that if f has the inverse shadowing property with respect to the class T α (f) (α=0,c,h,d), then it has the orbital inverse shadowing property with respect to the class T α (f) (α=0,c,h,d). But the converse does not hold. 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 gU(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 [6], showed that a diffeomorphism belongs to the C 1 -interior inverse shadowing property with respect to the class T d (f) if and only if it is structurally stable. And Pilyugin [7] 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.

Theorem 1.1 Let fDiff(M). A diffeomorphism f belongs to the C 1 -interior inverse shadowing property with respect to the class T d (f) (resp. 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 class T d (f) if and only if it satisfies 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 it satisfies both Axiom A and the strong transversal condition. From the above facts, we get the following.

Theorem 1.2 Let fDiff(M). If a diffeomorphism f belongs to the C 1 -interior weak inverse shadowing property with respect to the class T d (f), then f satisfies 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 periodic point p of f is hyperbolic if D f π ( p ) has eigenvalues with absolute values different from the one, where π(p) is the period of p. Denote by F(M) the set of fDiff(M) such that there is a C 1 -neighborhood U(f) of f such that for any gU(f), every pP(g) is hyperbolic, where P(g) is the set of periodic points of g. It is proved by Hayashi [8] that fF(M) if and only if f satisfies both Axiom A and the no-cycle condition.

Let Λ be a closed fDiff(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 exist constants C>0 and 0<λ<1 such that

D x f n | E x s C λ n and D x f n | E x u C λ n

for all xΛ and n0. If Λ=M, then we say that f is an Anosov diffeomorphism.

2 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 [9], Mañé proved that any C 1 structurally stable diffeomorphism is an Axiom A diffeomorphism. And in [10], 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 with ω that we call the 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 case, the Axiom A condition is equivalent to the diffeomorphism being Anosov, since Ω(f)=M by the Poincaré recurrence theorem.

We define the set F μ (M) as the set of diffeomorphisms f Diff μ (M) which has a C 1 -neighborhood U(f) Diff μ (M) such that for any gU(f), every periodic point of g is hyperbolic. Note that F μ (M)F(M) (see [[11], Corollary 1.2]).

Very recently, Arbieto and Catalan [11] proved that if a volume-preserving diffeomorphism is contained in F μ (M), then it is Anosov. Indeed, at first they used the Mañé results ([[9], Proposition II.1]). Then they showed that P ( f ) ¯ is hyperbolic, where P(f) is the set of periodic points of f. And they proved that the nonwandering set Ω(f)= P ( f ) ¯ by Pugh’s closing lemma. Finally, by the Poincaré recurrence theorem, Ω(f)=M. From the above facts, we can restate the theorem as follows.

Theorem 2.1 Any diffeomorphism in F μ (M) is Anosov.

In [12], Lee showed that if a volume-preserving diffeomorphism belongs to the C 1 -interior expansive or C 1 -interior shadowing property, then it is Anosov. And [13] proved that if a volume-preserving diffeomorphism belongs to the C 1 -interior weak shadowing property or C 1 -interior weak limit shadowing property, then it is Anosov. From these 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 int IS μ (M) 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 int WIS μ (M) (respect., int OIS μ (M)) denote the set of volume-preserving 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 a position to state the theorem of our paper.

Theorem 2.2 Let f Diff μ (M). We have

int IS μ (M)=int OIS μ (M)=int WIS μ (M)= AN μ (M),

where AN μ (M) is the set of Anosov volume-preserving diffeomorphisms in Diff μ (M).

3 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 give the following well-known Franks lemma for the conservative case, stated and proved in [[14], Proposition 7.4].

Lemma 3.1 Let f Diff μ 1 (M) and U be a C 1 -neighborhood of f in Diff μ 1 (M). Then there exist a C 1 -neighborhood U 0 U of f and ϵ>0 such that if g U 0 , for any finite f-invariant set E={ x 1 ,, x m }, any neighborhood U of E and any volume-preserving linear maps L j : T x j M T g ( x j ) M with L j D x j gϵ for all j=1,,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.

Remark 3.2 From the Moser theorem (see [15]), there is a smooth conservative change of coordinates φ x :U(x) T x M such that , where U(x) is a small neighborhood of xM.

Proposition 3.3 If fint IS μ (M), then every periodic point of f is hyperbolic.

Proof Take fint IS μ (M) and U(f) is a C 1 -neighborhood of fint IS μ (M). Let ϵ>0 and V(f) U 0 (f) be a corresponding number and a C 1 -neighborhood given by Lemma 3.1. Suppose that there exists a nonhyperbolic periodic point pP(g) for some gV(f). To simplify the notation of 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. Denote by E p c the eigenspace corresponding to λ. Then we see that if λR, then dim E p c =1, and if λC, then dim E p c =2.

First, we consider dim E p c =1. For simplicity, we may assume that λ=1 (the other case is similar). By making use of Lemma 3.1, we linearize g at p with respect to the Moser theorem; that is, by choosing α>0 sufficiently small, we construct g 1 C 1 -nearby g such that

g 1 (x)= { φ p 1 D p g φ p ( x ) if  x B α ( p ) , g ( x ) if  x B 4 α ( p ) .

Then g 1 (p)=g(p)=p. Since the eigenvalue λ of D p g 1 is one, D p g 1 (v)=v for any v E p c (α). Take v 0 E p c (α) such that v 0 =α/4. Then we set

I v 0 ={t v 0 :1t1} φ p ( B α ( p ) ) .

Take ϵ 1 =α/8. Let 0<δ< ϵ 1 be the number of the inverse shadowing property of g 1 . Then by our construction of g 1 , φ p 1 ( I v 0 ) B α (p). Put J p = φ p 1 ( I v 0 ). Then we see that g 1 ( J p )= J p and it is the identity map. For the above δ>0, we can define T d ( g 1 )-method as follows. For any xM, we set φ p (x)=( v 1 ,, v n ) and D p g( v 1 ,, v n )=( v 1 ,A( v 2 ,, v n )). Here A is corresponding to |λ|1. Then we define

h(x)= ( φ p 1 ( v 1 ) + δ / 2 , φ p 1 ( A ( v 2 , , v n ) ) ) = ( x 1 + δ / 2 , x ) ,

where φ p 1 (A( v 2 ,, v n ))= x =( x 2 ,, x n ). Clearly, d 1 ( g 1 ,h)<δ, and h T d ( g 1 ). Let p be identified with 0=(0,,0). Then choose x=( x 1 ,0,,0) J p such that d(0,x)=2 ϵ 1 . Then

d ( h i ( 0 ) , g 1 i ( x ) ) =d(0,x)=2 ϵ 1 .

Thus, g 1 does not have the inverse shadowing property.

We take a point y=( y 1 ,0,,0) J p such that d(y,x)< ϵ 1 and h i (y)=( φ p 1 ( w 1 )+iδ/2, φ p 1 ( A i ( w 2 ,, w n )))=( y 1 +iδ/2, y i ), where φ p (y)=( w 1 ,, w n ), y = φ p 1 (A( w 2 ,, w n )) and A corresponding to |λ|1. Then d( g i (x), h i (y))=d(x, y 1 +iδ/2) and for some kZ,

d(x, y 1 +kδ/2)>α/8= ϵ 1 .

This is a contradiction.

Therefore, we can choose a point yM J p such that d(x,y)< ϵ 1 . Since d( g 1 i (x), h i (y))=d(( x 1 ,0,,0),( y 1 +iδ/2, y i )), we can find kZ such that

d ( ( x 1 , 0 , , 0 ) , ( y 1 + k δ / 2 , y k ) ) > ϵ 1 ,

where y k = φ p 1 ( A k ( w 1 ,, w n )). This is a contradiction since fint IS μ (M).

Finally, if λC, then dim E p c =2. 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

g 1 (x)= { φ p 1 D p g φ p ( x ) if  x B α ( p ) , g ( x ) if  x B 4 α ( p ) .

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. Take v 0 φ p ( B α (p)) such that v 0 =α/4, and set

L p = φ p 1 ( { t v 0 : 1 t 1 + α / 4 } ) .

Then L p is an arc such that

  • g 1 i ( L p ) g 1 j ( L p )= for 0ijl1,

  • g 1 l ( L p )= L p , and

  • g 1 l | L p is the identity map.

Note that g 1 has the inverse shadowing property if and only if g 1 k has the inverse shadowing property for all kZ (see [6]). As in the first case, we can show that g 1 does not have the inverse shadowing property, which contradicts the fact that g 1 U(f). Thus, every periodic point of fint IS μ (M) is hyperbolic. □

Proposition 3.4 If fint WIS μ (M), then every periodic point of f is hyperbolic.

Proof Take fint WIS μ (M), and U(f) is a C 1 -neighborhood of fint WIS μ (M). Let ϵ>0 and V(f) U 0 (f) be a corresponding number and a C 1 -neighborhood given by Lemma 3.1. Suppose that there exists a nonhyperbolic periodic point pP(g) for some gV(f). To simplify the notation of 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 ϵ 1 =(length J p )/4. Let 0<δ< ϵ 1 be the number of the weak inverse shadowing property for some g 1 . Then we can construct a map hDiff(M) as in the proof of Proposition 3.3. Let p be identified with 0=(0,,0). Then choose a point x=( x 1 ,0,,0) J p such that d(0,x)=2 ϵ 1 . Since g 1 has the weak inverse shadowing property,

O h (0) B ϵ 1 ( O g 1 ( x ) ) .

However, for any y J p ,

g 1 i (y)=yand h i (y)= ( y 1 + i δ / 2 , y i ) ,

where y i = φ p 1 ( A i ( w 2 ,, w n )) is as in the proof of Proposition 3.3. Thus, it is easily seen that

O h (0) B ϵ 1 ( O g 1 ( x ) ) .

If y=( y 1 ,0,,0) J p {p}, then

h k (y)= ( y 1 + k δ / 2 , y k ) =( y 1 +kδ/2,0),

where y k = φ p 1 ( A k ( w 2 ,, w n )). Thus, we know that

O h (y) B ϵ 1 ( O g 1 ( x ) ) .

This is a contradiction.

Finally, we can choose a point yM J p such that d(x,y)< ϵ 1 . Then we know that

h k (y)= ( y 1 + k δ / 2 , y k ) ,

where y k = φ p 1 ( A k ( w 2 ,, w n )). Therefore,

h l (y) B ϵ 1 ( J p ),

for some lZ. Then

O h (y) B ϵ 1 ( O g 1 ( x ) ) = B ϵ 1 (x).

Thus, g 1 does not have the weak inverse shadowing property. This is a contradiction.

If λC, then as in the proof of Proposition 3.3, for g 1 V(f), we can take l>0 such that D p g 1 l (v)=v for any v φ p ( B α (p)) T p M. As in the previous argument, we get a contradiction. Thus, every periodic point of fint WIS μ (M) is hyperbolic. Consequently, if fint WIS μ (M), then f F μ (M). □

Proposition 3.5 If fint OIS μ (M), then every periodic point of f is hyperbolic.

Proof The proof is almost the same as that of Proposition 3.4. Indeed, let fint OIS μ (M) and U(f) be a C 1 -neighborhood of fint OIS μ (M). Let ϵ>0 and V(f) U 0 (f) be a corresponding number and a C 1 -neighborhood given by Lemma 3.1. Suppose that there exists a nonhyperbolic periodic point pP(g) for some gV(f). To simplify the notation of 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 ϵ 1 =(length J p )/4. Let 0<δ< ϵ 1 be the number of the orbital inverse shadowing property for some g 1 . Then we can construct a map hDiff(M) as in the proof of Proposition 3.4. Let p be identified with 0=(0,,0). Then choose a point x=( x 1 ,0,,0) J p such that d(0,x)=2 ϵ 1 . Since g 1 has the orbital inverse shadowing property,

O h (0) B ϵ 1 ( O g 1 ( x ) ) and O g 1 (x) B ϵ 1 ( O h ( 0 ) ) .

However, for any y J p ,

g 1 i (y)=yand h i (y)= ( y 1 + i δ / 2 , y i ) ,

where y i = φ p 1 ( A i ( w 2 ,, w n ))=( y 2 ,, y n ) is as in the proof of Proposition 3.3. Thus, it is easily seen that

O h (0) B ϵ 1 ( O g 1 ( x ) ) .

This is a contradiction since fint OIS μ (M).

If y=( y 1 ,0,,0) J p {p}, then

h k (y)= ( y 1 + k δ / 2 , y k ) =( y 1 +kδ/2,0),

where y k = φ p 1 ( A k ( w 2 ,, w n )). Thus, we know that

O h (y) B ϵ 1 ( O g 1 ( x ) ) .

This is a contradiction.

Finally, we can choose a point yM J p such that d(x,y)< ϵ 1 . Then we know that

h k (y)= ( y 1 + k δ / 2 , y k ) ,

where y k = φ p 1 ( A k ( w 2 ,, w n )). Therefore,

h l (y) B ϵ 1 ( J p ),

for some lZ. Then

O h (y) B ϵ 1 ( O g 1 ( x ) ) = B ϵ 1 (x).

Thus, g 1 does not have the orbital inverse shadowing property. This is a contradiction.

If λC, then as in the proof of Proposition 3.4, for g 1 V(f), we can take l>0 such that D p g 1 l (v)=v for any v φ p ( B α (p)) T p M. As in the previous argument, in order to reach the same contradiction. Thus, every periodic point of fint OIS μ (M) is hyperbolic. □

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Acknowledgements

The author wishes to express their appreciation to the referee for their careful reading of the manuscript 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).

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Correspondence to Manseob Lee.

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Keywords

  • shadowing
  • inverse shadowing
  • weak inverse shadowing
  • orbital inverse shadowing
  • Anosov
  • volume-preserving