Divergencefree vector fields with inverse shadowing
 Keonhee Lee^{1} and
 Manseob Lee^{2}Email author
https://doi.org/10.1186/168718472013337
© Lee and Lee; licensee Springer. 2013
Received: 4 March 2013
Accepted: 23 October 2013
Published: 21 November 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.
Keywords
topological stability inverse shadowing orbital inverse shadowing continuous method Anosov1 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 ${\mathcal{T}}_{d}$.
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 $\alpha =a,c,h,d$.
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 $\alpha =a,c,h,d$.
for any $x\in \mathrm{\Lambda}$ and $t>0$. If $\mathrm{\Lambda}=M$, then X is called Anosov.
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.
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].
 (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 $X\in {\mathcal{G}}_{\mu}^{1}(M)$.
 (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 $t\in \mathbb{R}$.
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 $X\in {\mathcal{G}}_{\mu}^{1}(M)$.
for all $t\in \mathbb{R}$. Take ${t}^{\prime}=min\{t:{W}^{t}(y)\in {\phi}_{p}^{1}({N}_{p})\}$, and let $w={W}^{{t}^{\prime}}(y)\in {\phi}_{p}^{1}({N}_{p})$.
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.
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.
Declarations
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).
Authors’ Affiliations
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