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Divergencefree vector fields with inverse shadowing
Advances in Difference Equations volume 2013, Article number: 337 (2013)
Abstract
We show that if a divergencefree vector field has the {C}^{1}stably orbital inverse shadowing property with respect to the class of continuous methods {\mathcal{T}}_{d}, then the vector field is Anosov. The results extend the work of Bessa and Rocha (J. Differ. Equ. 250:39603966, 2011).
MSC:37C10, 37C27, 37C50.
1 Introduction
The notion of inverse shadowing property is a dual notion of the shadowing property. It was studied by [1–7]. In fact, Pilyugin [7] showed that every structurally stable diffeomorphism has the inverse shadowing property with respect to the class of continuous methods. In [3], Lee proved that if a diffeomorphism has the {C}^{1}stably inverse shadowing property with respect to the class of continuous methods {\mathcal{T}}_{d}, then the diffeomorphism is structurally stable. For vector fields, Lee and Lee [5] introduced the notion of inverse shadowing for flows and showed that every expansive flow with the shadowing property has the inverse shadowing property with respect to the class of continuous methods. Lee et al. [6] showed that the {C}^{1}interior of the set of vector fields with the orbital shadowing property with respect to the class {\mathcal{T}}_{d} coincides with the set of structurally stable vector fields. From the facts, Lee [4] showed that if a volumepreserving diffeomorphism has the {C}^{1}stably inverse shadowing property with respect to the class {\mathcal{T}}_{d}, then the diffeomorphism is Anosov. Moreover, if a volumepreserving diffeomorphism has the {C}^{1}stably orbital inverse shadowing property with respect to the class {\mathcal{T}}_{d}, then the diffeomorphism is Anosov. In this spirit, we study divergencefree vector fields with the inverse, orbital inverse shadowing property with respect to the class {\mathcal{T}}_{d}.
Let us be more detailed. Let M be a closed, connected, and {C}^{\mathrm{\infty}} Riemannian manifold endowed with a volume form μ, and let μ denote the Lebesgue measure associated to it. Given a {C}^{r} (r\ge 1) vector field X:M\to TM, the solution of the equation {x}^{\prime}=X(x) gives rise to a {C}^{r} flow, {X}^{t}; on the other hand, given a {C}^{r} flow, we can define a {C}^{r1} vector field by considering X(x)=\frac{d{X}^{t}(x)}{dt}{}_{t=0}. We say that X is divergencefree if its divergence is equal to zero. Note that, by the Liouville formula, a flow {X}^{t} is volumepreserving if and only if the corresponding vector field X is divergencefree. Let {\mathfrak{X}}_{\mu}^{r}(M) denote the space of {C}^{r} divergencefree vector fields, and we consider the usual {C}^{r} Whitney topology on this space. Let X\in {\mathfrak{X}}_{\mu}^{1}(M) for any \delta >0 and T>0. We say that a (\delta ,T)pseudo orbit of X\in {\mathfrak{X}}_{\mu}^{1}(M) if
for any {t}_{i}\ge T, i\in \mathbb{Z}. A mapping \mathrm{\Psi}:\mathbb{R}\times M\to M is called a (\delta ,T)method for X if for any x\in M, the map {\mathrm{\Psi}}_{x}:\mathbb{R}\to M defined by
is a (\delta ,T)pseudo orbit of X. Ψ is said to be complete if \mathrm{\Psi}(0,x)=x for any x\in M. Then a (\delta ,T)method for X can be considered a family of (\delta ,T)pseudo orbits of X. A method Ψ for X is said to be continuous if the map {\mathrm{\Psi}}^{\prime}:M\to {M}^{\mathbb{R}}, given by {\mathrm{\Psi}}^{\prime}(x)(t)=\mathrm{\Psi}(t,x) for x\in M and t\in \mathbb{R}, is continuous under the compact open topology on {M}^{\mathbb{R}}, where {M}^{\mathbb{R}} denotes the set of all functions from ℝ to M. The set of all complete (\delta ,1)methods [resp. complete continuous (\delta ,1)methods] for X\in {\mathfrak{X}}_{\mu}^{1}(M) will be denoted by {\mathcal{T}}_{a}(\delta ,X) [resp. {\mathcal{T}}_{c}(\delta ,X)]. We can see that if Y is another vector field which is sufficiently close to X in the {C}^{0} topology, then it induces a complete continuous method for X. Let {\mathcal{T}}_{h}(\delta ,X) [resp. {\mathcal{T}}_{d}(\delta ,X] be the set of all complete continuous (\delta ,1) methods for X which are induced by {C}^{1} vector fields Y with {d}_{0}(X,Y)<\delta [resp. {d}_{1}(X,Y)<\delta], where {d}_{0} is the {C}^{0} metric on {\mathfrak{X}}_{\mu}^{1}(M) such that
and {d}_{1} is the {C}^{1} metric on {\mathfrak{X}}_{\mu}^{1}(M) such that
We say that a vector field X has the inverse shadowing property with respect to the class {\mathcal{T}}_{\alpha} (\alpha =a,c,h,d) if for any \u03f5>0, there is \delta >0 such that for any (\delta ,T)method \mathrm{\Psi}\in {\mathcal{T}}_{\alpha}(\delta ,X) and any point x\in M, there are y\in M and an increasing homeomorphism h:\mathbb{R}\to \mathbb{R} with h(0)=0 such that
We denote by {\mathcal{IS}}_{\mu ,\alpha}(M) the set of divergencefree vector fields on M with the inverse shadowing property with respect to the class {\mathcal{T}}_{\alpha}, where \alpha =a,c,h,d. Let int{\mathcal{IS}}_{\mu ,\alpha}(M) be the {C}^{1}interior of the set of divergencefree vector fields on M with the inverse shadowing property with respect to the class {\mathcal{T}}_{\alpha}, where \alpha =a,c,h,d.
We say that X\in {\mathfrak{X}}_{\mu}^{1}(M) has the {C}^{1}stably inverse shadowing property with respect to the class {\mathcal{T}}_{\alpha} if there is a {C}^{1}neighborhood \mathcal{U}(X)\subset {\mathfrak{X}}_{\mu}^{1}(M) of X such that for any Y\in \mathcal{U}(X), Y has the inverse shadowing property with respect to the class {\mathcal{T}}_{d}, where \alpha =a,c,h,d.
Remark 1.1 Let X\in {\mathfrak{X}}_{\mu}^{1}(M). Then {\mathcal{IS}}_{\mu ,a}(M)\subset {\mathcal{IS}}_{\mu ,c}(M)\subset {\mathcal{IS}}_{\mu ,h}(M)\subset {\mathcal{IS}}_{\mu ,d}(M), where \alpha =a,c,h,d.
We say that X\in {\mathfrak{X}}_{\mu}^{1}(M) has the orbital inverse shadowing property with respect to the class {\mathcal{T}}_{d} if for any \u03f5>0, there is \delta >0 such that for any (\delta ,1)method \mathrm{\Psi}\in {\mathcal{T}}_{d}(\delta ,X) and any point x\in M, there is y\in M such that
where Orb(y,\mathrm{\Psi})=\{\mathrm{\Psi}(t,y):t\in \mathbb{R}\}. Note that if X has the inverse shadowing property with respect to the class {\mathcal{T}}_{d}, then X has the orbital shadowing property with respect to the class {\mathcal{T}}_{d}. But the converse is not true. Indeed, an irrational rotation map does not have the inverse shadowing property. And the map has the orbital shadowing property. Let int{\mathcal{OIS}}_{\mu ,\alpha}(M) be the {C}^{1}interior of the set of divergencefree vector fields on M with the orbital inverse shadowing property with respect to the class {\mathcal{T}}_{\alpha}, where \alpha =a,c,h,d.
Let Λ be a closed {X}_{t}invariant set. We say that Λ is hyperbolic if there are constants C>0, \lambda >0 and a continuous splitting {T}_{\mathrm{\Lambda}}M={E}^{s}\oplus \u3008X(x)\u3009\oplus {E}^{u} such that
for any x\in \mathrm{\Lambda} and t>0. If \mathrm{\Lambda}=M, then X is called Anosov.
Given a vector field X, we denote by PO(X) the set of closed orbits of X and by Sing(X) the set of singularities of X, i.e., those points x\in M such that X(x)=\overrightarrow{0}. Denote by Crit(X)=Sing(X)\cup PO(X). Let R:=M\setminus Sing(X) be the set of regular points. We assume that the exponential map {exp}_{p}:{T}_{p}M(1)\to M is well defined for all p\in M, where {T}_{p}M(1)=\{v\in {T}_{p}M:\parallel v\parallel \le 1\}. Given x\in R, we consider its normal bundle {N}_{x}={\u3008X(x)\u3009}^{\perp}\subset {T}_{x}M, and let {N}_{x}(r) be the rball in {N}_{x}. Let {\mathcal{N}}_{x,r}={exp}_{x}({N}_{x}(r)). For any x\in R and t\in \mathbb{R}, there are r>0 and a {C}^{1} map \tau :{\mathcal{N}}_{x,r}\to \mathbb{R} with \tau (x)=t such that {X}^{\tau (y)}(y)\in {\mathcal{N}}_{{X}^{t}(x),1} for any y\in {\mathcal{N}}_{x,r}. We say \tau (y) is the first return time of y. Then we define the Poincarè map f by
Let \mathcal{N}={\bigcup}_{x\in R}{\mathcal{N}}_{x} be the normal bundle based on R. We can define the associated linear Poincaré flow by
where {\mathrm{\Pi}}_{{X}^{t}(x)}:{T}_{{X}^{t}(x)}M\to {N}_{{X}^{t}(x)} is the projection along the direction of X({X}^{t}(x)). We say that a vector field X\in {\mathfrak{X}}_{\mu}^{1}(M) is topologically stable if for any \u03f5>0, there is \delta >0 such that for any Y\in {\mathfrak{X}}_{\mu}^{1}(M) with {d}_{0}(X,Y)<\delta, there is a semiconjugacy (h,\tau ) from Y to X satisfying d(h(x),x)<\u03f5 for all x\in M.
In [8], the authors proved that if a vector field is in the {C}^{1}interior of the set of topologically stable vector fields (not divergencefree vector fields), then X satisfies Axiom A and the strong transversality condition. Robinson [9] proved that if a vector field satisfies Axiom A and the strong transversality condition, then the vector field is structurally stable. For divergencefree vector fields, Bessa and Rocha [10] showed that if a divergencefree vector field is in the {C}^{1}interior of the set of topological stable vector fields X\in {\mathfrak{X}}_{\mu}^{1}(M), then X is Anosov.
Remark 1.2 We have the following implication: topological stability ⇒ inverse shadowing property with respect to the continuous method {\mathcal{T}}_{d} ⇒ orbital inverse shadowing property with respect to the continuous method {\mathcal{T}}_{d}.
From the above remark, we know that our result is a slight generalization of the main theorem in [10]. In this paper, we omit the phrase ‘with respect to the class {\mathcal{T}}_{d}’ for simplicity. So, we say that X has the inverse, orbital inverse shadowing property means that X has the inverse, orbital inverse shadowing property with respect to the class {\mathcal{T}}_{d}. Note that if X\in int{\mathcal{IS}}_{\mu ,\alpha}^{1}(M) or X\in int{\mathcal{OIS}}_{\mu ,\alpha}^{1}(M), then it means that X has the {C}^{1}stably inverse, orbital inverse shadowing property with respect to the class {\mathcal{T}}_{\alpha}, \alpha =a,c,h,d. The following is the main result of this paper.
Theorem 1.3 Let X\in {\mathfrak{X}}_{\mu}^{1}(M). Then
where {\mathcal{A}}_{\mu}^{1}(M) is the set of divergencefree Anosov vector fields.
2 Proof of Theorem 1.3
Let M be as before, and let X\in {\mathfrak{X}}_{\mu}^{1}(M). The perturbations (Lemma 2.1) for volumepreserving vector fields allows to realize them as perturbations of a fixed volumepreserving flow. Fix X\in {\mathfrak{X}}_{\mu}^{1}(M) and \tau >0. A oneparameter areapreserving linear family {\{{A}_{t}\}}_{t\in \mathbb{R}} associated to \{{X}^{t}(p);t\in [0,\tau ]\} is defined as follows:

{A}_{t}:{N}_{p}\to {N}_{p} is a linear map for all t\in \mathbb{R},

{A}_{t}=id for all t\le 0 and {A}_{t}={A}_{\tau} for all t\ge \tau,

{A}_{t}\in SL(n,\mathbb{R}), and

the family {A}_{t} is {C}^{\mathrm{\infty}} on the parameter t.
The following result, proved in [[11], Lemma 3.2], is now stated for X\in {\mathfrak{X}}_{\mu}^{1}(M) instead of X\in {\mathfrak{X}}_{\mu}^{4}(M) because of the improved smooth {C}^{1} pasting lemma proved in [[12], Lemma 5.2].
Lemma 2.1 Given \u03f5>0 and a vector field X\in {\mathfrak{X}}_{\mu}^{1}(M), there exists {\xi}_{0}={\xi}_{0}(\u03f5,X) such that for all \tau \in [1,2], for any periodic point p of period greater than 2, for any sufficiently small flowbox T of \{{X}_{t}(p);t\in [0,\tau ]\} and for any oneparameter linear family {\{{A}_{t}\}}_{t\in [0,\tau ]} such that \parallel {A}_{t}^{\prime}{A}_{t}^{1}\parallel <{\xi}_{0} for all t\in [0,\tau ], there exists Y\in {\mathfrak{X}}_{\mu}^{1}(M) satisfying the following properties:

(a)
Y is ϵ{C}^{1}close to X;

(b)
{Y}^{t}(p)={X}^{t}(p) for all t\in \mathbb{R};

(c)
{P}_{Y}^{\tau}(p)={P}_{X}^{\tau}(p)\circ {A}_{\tau}, and

(d)
Y{}_{{\mathcal{T}}^{c}}\equiv X{}_{{\mathcal{T}}^{c}}.
Remark 2.2 Let X\in {\mathfrak{X}}_{\mu}^{1}(M). By Zuppa’s theorem [13], we can find Y {C}^{1}close to X such that Y\in {\mathfrak{X}}_{\mu}^{\mathrm{\infty}}(M), {Y}^{\pi}(p)=p and {P}_{Y}^{\pi}(p) has an eigenvalue λ with \lambda =1.
A divergencefree vector field X is a divergencefree star vector field if there exists a {C}^{1}neighborhood \mathcal{U}(X) of X in {\mathfrak{X}}_{\mu}^{1}(M) such that if Y\in \mathcal{U}(X), then every point in Crit(Y) is hyperbolic. The set of divergencefree star vector fields is denoted by {\mathcal{G}}_{\mu}^{1}(M). Then we get the following theorem.
Theorem 2.3 [[14], Theorem 1.1]
If X\in {\mathcal{G}}_{\mu}^{1}(M), then Sing(X)=\mathrm{\varnothing} and X is Anosov.
Thus, to prove Theorem 1.3, it is enough to show that if X has the inverse shadowing property, or the orbital inverse shadowing property, then X\in {\mathcal{G}}_{\mu}^{1}(M).
Proposition 2.4 Let X\in int{\mathcal{IS}}_{\mu ,d}^{1}(M). Then X\in {\mathcal{G}}_{\mu}^{1}(M).
Proof Suppose that X\in {\mathfrak{X}}_{\mu}^{1}(M) has the {C}^{1}stably inverse shadowing property. Then there is a {C}^{1}neighborhood \mathcal{U}(X) of X such that for any Y\in \mathcal{U}(X), Y has the inverse shadowing property. Let p\in \gamma \in PO(X) with {X}^{\pi}(p)=p and {U}_{p} be a small neighborhood of p\in M. We will derive a contradiction. Assume that there is an eigenvalue λ of {P}_{X}^{\pi}(p) such that \lambda =1. By Remark 2.2, there is Y\in \mathcal{U}(X) such that Y\in {\mathfrak{X}}_{\mu}^{\mathrm{\infty}}(M), {Y}_{Y}^{\pi}(p)=p and {P}_{Y}^{\pi}(p) has an eigenvalue λ with \lambda =1. Using Moser’s theorem (see [15]), there is a smooth conservative change of coordinates {\phi}_{p}:{U}_{p}\to {T}_{p}M such that {\phi}_{p}(p)=\overrightarrow{0}. Let f:{\phi}_{p}^{1}({N}_{p})\to {\mathcal{N}}_{p} be the Poincarè map associated to {Y}^{t}, and let V be a {C}^{1}neighborhood of f. Here, {\mathcal{N}}_{p} is the Poincarè section through p. By Lemma 2.1, we can find a small flowbox T of {Y}^{[0,{t}_{0}]}, 0<{t}_{0}<\pi, and there are Z\in {\mathcal{U}}_{0}(Y)\subset \mathcal{U}(X), g\in \mathcal{V} and \alpha >0 such that

(a)
{Z}^{t}={Y}^{t} for all t\in \mathbb{R},

(b)
{P}_{Z}^{{t}_{0}}(p)={P}_{Y}^{{t}_{0}}(p),

(c)
Z{}_{{\mathcal{T}}^{c}}=Y{}_{{\mathcal{T}}^{c}},

(d)
g(x)={\phi}_{p}^{1}\circ {P}_{Y}^{\pi}(p)\circ {\phi}_{p}(x) for all x\in {B}_{\alpha}(p)\cap {\phi}_{p}^{1}({N}_{p}), and

(e)
g(x)=f(x) for all x\notin {B}_{4\alpha}(p)\cap {\phi}_{p}^{1}({N}_{p}).
Then {P}_{Z}^{\pi}(p) has an eigenvalue λ with \lambda =1. For 0<\u03f5<\alpha /8, let 0<\delta <\u03f5 be as in the definition of the inverse shadowing property of {Z}^{t}.
Take a linear map {A}_{t}:{N}_{p}\to {N}_{p} for all t\in \mathbb{R} such that if \parallel {P}_{Z}^{\pi}(p)\circ {A}_{{t}_{0}}{P}_{Z}^{\pi}(p)\parallel <{\delta}_{0}, then {d}_{1}(Z,W)<\delta, and {P}_{Z}^{\pi}(p)\circ {A}_{{t}_{0}} is a hyperbolic linear Poincarè flow. Set {P}_{W}^{t}(p)={P}_{Z}^{\pi}(p)\circ {A}_{{t}_{0}}. Then {W}^{t}\in {\mathcal{T}}_{d}(Z), and p\in \gamma is a periodic point of {W}^{t}. Since Z\in \mathcal{U}(X) for any x\in M, there exist y\in M and an increasing homeomorphism h:\mathbb{R}\to \mathbb{R} with h(0)=0 such that d({Z}^{h(t)}(x),{W}^{t}(y))<\u03f5 for all t\in \mathbb{R}.
First, we assume that \lambda =1 (the other case is similar). Then we can choose a vector v associated to λ such that \parallel v\parallel <\alpha /4. Since {\phi}_{p}^{1}(v)\in {\phi}_{p}^{1}({N}_{p})\setminus \{p\},
Then we can take z\in {\phi}_{p}^{1}({N}_{p}) such that
where {g}_{1} is the Poincarè map associated to {W}^{t}. Since {P}_{W}^{t}(p) is a hyperbolic linear Poincarè flow, we can see that
for some i\in \mathbb{Z}. Thus
for some t\in \mathbb{R}. This is a contradiction by the fact that Z has the inverse shadowing property.
Finally, we assume that λ is complex. By [[10], Lemma 3.2], there is Z\in \mathcal{U}(X) such that {P}_{Z}^{\pi}(p) is a rational rotation. Then there is l>0 such that {P}_{Z}^{l+\pi}(p) is the identity. As in the previous argument, we get a contradiction. □
Proposition 2.5 Let X\in int{\mathcal{OIS}}_{\mu ,d}^{1}(M). Then X\in {\mathcal{G}}_{\mu}^{1}(M).
Proof Suppose that X has the {C}^{1}stably orbital inverse shadowing property. Then there is a {C}^{1}neighborhood \mathcal{U}(X) of X such that for any Y\in \mathcal{U}(X), Y has the orbital inverse shadowing property. Let p\in \gamma \in PO(X) with {X}^{\pi}(p)=p and {U}_{p} be a small neighborhood of p\in M. We will derive a contradiction. Assume that there is an eigenvalue λ of {P}_{X}^{\pi}(p) such that \lambda =1. As in the proof of Proposition 2.4, we get a hyperbolic linear Poincarè flow {P}_{W}^{t}(p), and {W}^{t}\in {\mathcal{T}}_{d}(Z), and p\in \gamma is a periodic point of {W}^{t}. Since Z\in \mathcal{U}(X) for any x\in M, there exist y\in M such that
for all t\in \mathbb{R}. Take {t}^{\prime}=min\{t:{W}^{t}(y)\in {\phi}_{p}^{1}({N}_{p})\}, and let w={W}^{{t}^{\prime}}(y)\in {\phi}_{p}^{1}({N}_{p}).
If \lambda \in \mathbb{R} or \lambda \in \mathbb{C}, then as in the proof of Lemma 2.4, we get a contradiction. Indeed, since {P}_{W}^{t}(p) is a hyperbolic linear Poincarè flow, there is j>0 such that {g}_{1}^{j}(w)\notin {B}_{\alpha /4}(p)\cap {\phi}_{p}^{1}({N}_{p}). Thus
for some t\in \mathbb{R}. Since Z\in \mathcal{U}(X), this is a contradiction. □
End of the proof of Theorem 1.3 By Proposition 2.4 and Proposition 2.5, we have X\in {\mathcal{G}}_{\mu}^{1}(M). Thus by Theorem 2.3, we get Sing(X)=\mathrm{\varnothing} and X is Anosov. □
From the result of [10], we get the following corollary.
Corollary 2.6 Let X\in {\mathfrak{X}}_{\mu}^{1}(M). Then
where int{\mathcal{TS}}_{\mu}^{1}(M) is the {C}^{1}interior of the set of divergencefree vector fields on M which are topologically stable.
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Acknowledgements
The first author is supported by the National Research Foundation (NRF) of Korea funded by the Korean Government (No. 20110015193). The second author is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology, Korea (No. 20110007649).
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Lee, K., Lee, M. Divergencefree vector fields with inverse shadowing. Adv Differ Equ 2013, 337 (2013). https://doi.org/10.1186/168718472013337
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DOI: https://doi.org/10.1186/168718472013337
Keywords
 topological stability
 inverse shadowing
 orbital inverse shadowing
 continuous method
 Anosov