# Orbital Shadowing forC1-Generic Volume-Preserving Diffeomorphisms

Authors: Manseob Lee
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We show thatC1-generically, if a volume-preserving diffeomorphism has the orbital shadowing property, then the diffeomorphism is Anosov.


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## Body

## 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 Ws(p2)∪{p3}=Wu(p3)∪{p2}. It is assumed that the eigenvalues of Dp2f are -μ,ν with μ>1, 0<ν<1 and the eigenvalues of Dp3f 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].LetM 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 Diffμ(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]).Forδ>0, a sequence of points {xi}i=ab(-∞≤a<b≤∞) in M is called a δ-pseudo-orbit of f if d(f(xi),xi+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 {xi}i∈ℤ of f, there is a point y∈M, such that, d(fi(y),xi)<ϵ for all i∈ℤ. It is easy to see that f has the shadowing property if and only if fn has the shadowing property for n∈ℤ∖{0}. For each x∈M, let 𝒪f(x) be the orbit of f through x; that is,
(1)𝒪f(x)={fn(x):n∈ℤ}.
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={xi}i∈ℤ, we can find a point y∈M such that
(2)𝒪f(y)⊂Bϵ(ξ),ξ⊂Bϵ(𝒪f(y)),
where Bϵ(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 fn 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 TΛM has a Df-invariant splitting Es⊕Eu and there exist constants C>0 and 0<λ<1, such that
(3)∥Dxfn|Exs∥≤Cλn,∥Dxf-n|Exu∥≤Cλn,
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 subset𝒢⊂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.
ForC1-generic f, if f has the orbital shadowing property, then f is Anosov.Letp be a periodic point of f with period π(p). We say that p is an elementary point if Dpfπ(p) eigenvalues are multiplicatively independent over ℤ. Elementary points have simple spectrum, and none of Dpfπ(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 Diffμ(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∈Diffμ(M2), if f has the orbital shadowing property, then is it Anosov?


## 2. Proof of Theorems1

LetM be as before, and let f∈Diffμ(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∈ωf(x), where ωf(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 Dpfπ(p):TpM→TpM 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
(4)Ws(p)={x∈M:fkn(x)⟶pasn⟶∞},Wu(p)={x∈M:fkn(x)⟶pasn⟶-∞}
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
(5)Ws(p)⋔Wu(q)≠∅,Wu(p)⋔Ws(q)≠∅.
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 Ws(p)⋔Wu(p). Denote by H(p,f)=Ws(p)⋔Wu(p)¯.Theorem 2 (see [11, Theorem 1.3]).
There is a residual set𝒢1⊂Diffμ(M) such that, for any f∈𝒢1, f is transitive. Moreover, M is a unique homoclinic class.We denoteℱμ(M) by the set of diffeomorphisms f∈Diffμ(M) which has a C1-neighborhood 𝒰(f)⊂Diffμ(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 ℱμ(M) is Anosov.Theorem 3 (see [8, Theorem 1]).
Every volume preserving diffeomorphism inℱμ(M) is Anosov.To prove Theorem1, it is enough to show that f∈ℱμ(M). Let p be a hyperbolic periodic point of f; there exists ϵ(p)>0 such that, for any x∈Wϵ(p)s(p) and x∈Wϵ(p)u(p), we know that
(6)d(fi(x),fi(p))≤ϵ(p),d(f-i(x),f-i(p))≤ϵ(p),
for all i≥0. Then Wϵ(p)s(p) is called the local stable manifold of p, and Wϵ(p)u(p) is the local unstable manifold of p. It is clear that Wϵ(p)s(p)⊂Ws(p), and Wϵ(p)u(p)⊂Wu(p).Lemma 4.
Letf∈𝒢1, and let p,q∈Ph(f). If f has the orbital shadowing property, then
(7)Ws(p)∩Wu(q)≠∅,Wu(p)∩Ws(q)≠∅,
where Ph(f) is the set of all hyperbolic periodic points of f.Proof.
Letf∈𝒢1, and let p,q∈Ph(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(fl1(x),p)<δ and d(fl2(x),q)<δ. We may assume that l2=l1+k for some k>0. Then we get a finite δ-pseudo-orbit {p,fl1(x),fl1+1(x),…,fl2-1(x),q}. Now we construct a δ-pseudo orbit as follows: put (i) fi(p)=xi for i≤0, (ii) fl1+i(x)=x1+i for 0≤i≤k-1, and (iii) fl2+i(q)=xk+i for i≥0. Then
(8)ξ={…,p,p,x0(=p),fl1(x),fl1+1(x),…,111fl1+k-1(x),fl2(x)(=q),q,q,…}={…,x-2,x-1,x0(=p),x1,x2,…,111xk-1,xk(=q),xk+1,…,}.
Since f has the orbital shadowing property, there are points y∈M and j≥0,n>0 such that (i) fj(y)∈Bϵ(x0) and f-i(fj(y))∈Bϵ(x-i) for all i≥0; (ii) fn(y)∈Bϵ(xk) and fi(fn(y))∈Bϵ(xk+i) for all i≥0. Then
(9)𝒪f(y)⊂Bϵ(ξ),ξ⊂Bϵ(𝒪f(y)),fj(y)∈Wϵu(p), and fn(y)∈Wϵs(q). Thus, 𝒪f(y)∩Wu(p)∩Ws(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 Diffμ(M).Lemma 5.
There is a residual set𝒢2⊂Diffμ(M) such that, for any f∈𝒢2, if f has the orbital shadowing property, then for any p,q∈Ph(f),  index(p)=index(q).Proof.
Let𝒢2=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 dimWs(p)+dimWu(q)<dimM or dimWu(p)+dimWs(q)<dimM. Assume that dimWs(p)+dimWu(q)<dimM. Since f is Kupka-Smale, we have Ws(p)∩Wu(q)=∅. This is a contradiction by Lemma 4.Remark 6.
Indim(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.
Letf∈Diffμ1(M), and let 𝒰(f) be a C1-neighborhood of f in Diffμ1(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,…,xm}, any neighborhood U of E, and any volume-preserving linear maps Lj:TxjM→Tg(xj)M with ∥Lj-Dxjg∥≤ϵ for all j=1,…,m, there is a conservative diffeomorphism g1∈𝒰(f) coinciding with f on E and out of U, and Dxjg1=Lj for all j=1,…,m.Denote byP(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.
LetdimM≥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∈Ph(f) with different indices.Proof.
Letp∈P(f) be the nonhyperbolic periodic orbit of period π and ϵ>0. By Pugh-Robinson’s closing lemma [13] there is f1∈Diffμ(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∈Diffμ(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 Dfπ(p). If p1 is not hyperbolic take f2=f1. If p1 is hyperbolic for Df1π(p1), then, since f1 is arbitrarily C1-close to f, the distance between the spectrum of Df1π(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∈Diffμ(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∈Diffμ(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 set𝒢3⊂Diffμ(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 pg,qg∈Ph(g) with different indices, then f has two hyperbolic periodic points p,q∈Ph(f) with different indices.For anyδ>0, we say that p∈P(f) has a δ-weak eigenvalue if there is an eigenvalue λ of Dfπ(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 set𝒢4⊂Diffμ(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.
Let𝒢4=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 ph is not hyperbolic, where ph 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 pg,qg∈Ph(g) with index(pg)≠index(qg). 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 set𝒢5⊂Diffμ(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 pg∈Ph(g), pg has a δ-weak eigenvalue, then p∈Ph(f) has a 2δ-weak eigenvalue.Proof of Theorem1.
Let𝒢6=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 pg for some g  C1-nearby f, such that pg is δ/2-weak eigenvalue. Then by Lemma 11, p∈P(f) has δ-weak eigenvalue. This is a contradiction by Lemma 10.



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