Divergence-free vector fields with orbital shadowing

Abstract

We show that a divergence-free vector field belongs to the $C^1$-interior of the set of divergence-free vector fields satisfying the orbital shadowing property when the vector field is Anosov.

MSC:37C10, 37C50, 37D20.

1 Introduction

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 $C^1$-interior of the set of diffeomorphisms having the shadowing property coincides the structural stability, 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 a no-cycle condition, then there is a diffeomorphism which does not have the shadowing property (see, [3]). However, for another shadowing property, if a diffeomorphism is Ω-stable, then the diffeomorphism has another shadowing property.

In this article, we study another shadowing property which is called the orbital shadowing property. 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 $C^1$-interior of the set of diffeomorphisms having the orbital shadowing property if and only if the diffeomorphism is structurally stable.

For a conservative diffeomorphism, Bessa [4] proved that a conservative diffeomorphism is in the $C^1$-interior of the set of all conservative diffeomorphisms satisfying the shadowing property if and only if it is Anosov. Lee and Lee [5, 6] proved that a conservative diffeomorphism is in the $C^1$-interior of the set of all conservative diffeomorphisms satisfying the orbital shadowing property if and only if it is Anosov. Also, for a conservative vector field, that is, a divergence-free vector field, Ferreira [7] proved that if a conservative vector field belongs to the $C^1$-interior of the set of all conservative vector fields satisfying the shadowing property, then it is Anosov. From the results, we study that if a conservative vector field belongs to the $C^1$-interior of the set of all conservative vector fields having the orbital shadowing property, then it is Anosov. Our result is a generalization of the main theorem in [7].

2 Basic notions, definitions and results

Let M be a closed, connected and smooth $n(\ge 1)$-dimensional Riemannian manifold endowed with a volume form, which has a measure μ, called the Lebesgue measure. Given a $C^r$ ($r\ge 1$), vector field $X:M\to TM$, the solution of the equation $x^{\mathrm{\prime }}=X(x)$ generates a $C^r$ flow, $X^t$; on the other hand, given a $C^r$ flow, we can define a $C^{r-1}$ vector field by considering $X(x)=\frac{d{X}^t(x)}{dt}{|}_{t=0}$. We say that X is divergence-free (or a conservative vector field) if its divergence is equal to zero. Note that, by the Liouville formula, a flow $X^t$ is volume-preserving if and only if the corresponding vector field X is divergence-free. Let ${\mathfrak{X}}_{\mu }^1(M)$ denote the space of $C^r$ divergence-free vector fields, and we consider the usual $C^1$ Whitney topology on this space. Let $X\in {\mathfrak{X}}_{\mu }^1(M)$. For any $\delta >0$ and $T>0$, we say that $\{(x_i,t_i):t_i\ge T,i\in \mathbb{Z}\}$ is a $(\delta ,T)$-pseudo orbit of $X\in {\mathfrak{X}}_{\mu }^1(M)$ if

$$d(X^{t_i}(x_i),x_{i+1})<\delta ,$$

for any $t_i\ge T$, $i\in \mathbb{Z}$. Define Rep as the set of increasing homeomorphisms $h:\mathbb{R}\to \mathbb{R}$ such that $h(0)=0$. Fix $ϵ>0$ and define $\mathsf{Rep}(ϵ)$ as follows:

$$\mathsf{Rep}(ϵ)=\{h\in \mathsf{Rep}:|\frac{h(t)}{t}-1|<ϵ\}.$$

Let $\mathrm{\Lambda }\subseteq M$ be a compact $X^t$-invariant set. We say that $X^t$ has the shadowing property on Λ if for any $ϵ>0$, there is $\delta >0$ such that for any $(\delta ,1)$-pseudo orbit ${\{(x_i,t_i)\}}_{i\in \mathbb{Z}}\subset \mathrm{\Lambda }$, let $T_i=t_0+t_1+\cdots +t_i$ for any $0\le i<b$, and $T_i=-t_{-1}-t_{-2}-\cdots -t_i$ for any $a<i\le 0$, there exist a point $y\in M$ and an increasing homeomorphism $h:\mathbb{R}\to \mathbb{R}$ with $h(0)=0$ such that

$$d(X^{h(t)}(y),X^{t-T_i}(x_i))<ϵ,$$

for any $T_i<t<T_{i+1}$. If $\mathrm{\Lambda }=M$, then $X^t$ has the shadowing property. Now, we introduce the notion of the orbital shadowing property. For $x\in M$, we denote ${\mathcal{O}}_X(x)$ to be the orbit of X through x; that is, ${\mathcal{O}}_X(x)=\{X^t(x):t\in \mathbb{R}\}$. We say that $X^t$ has the orbital shadowing property if for any $ϵ>0$ there is $\delta >0$ such that for any $(\delta ,1)$-pseudo orbit $\xi =\{(x_i,t_i):t_i\ge 1,i\in \mathbb{Z}\}$ there is a point $y\in M$ such that

$$\xi \subset B_{ϵ}({\mathcal{O}}_X(y))\text{and}{\mathcal{O}}_X(y)\subset B_{ϵ}(\xi ),$$

where $B_{ϵ}(A)$ is the neighborhood of A. Note that the orbital shadowing property is a weak version of the shadowing property: the difference is that we do not require a point $x_i$ of a pseudo-orbit ξ and the point $X^{t_i}(y)$ of an exact orbit ${\mathcal{O}}_X(y)$ to be close ‘at any time moment’; instead, the sets of the points of X and ${\mathcal{O}}_X(y)$ are required to be close. Let Λ be a closed $X^t$-invariant set. We say that Λ is hyperbolic if there are constants $C>0$ and $\lambda >0$ such that a continuous splitting $T_{\mathrm{\Lambda }}M=E^s\oplus 〈X(x)〉\oplus E^u$ satisfying

$$\parallel D{X}^t{|}_{E^s(x)}\parallel \le C{e}^{-\lambda t}\text{and}\parallel D{X}^{-t}{|}_{E^u(x)}\parallel \le C{e}^{-\lambda t}$$

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 $Sing(X)$ the set of singularities of X, i.e., those points $x\in M$ such that $X(x)=\overrightarrow{0}$. Let $R:=M\setminus Sing(X)$ be the set of regular points. We know 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 $N_x={〈X(x)〉}^{\perp }\subset T_{x}M$ and let $N_x(r)$ be the r-ball 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 $\tau (y)$ the first return time of y. Then we define the Poincarè map f by

$$\begin{array}{r}f:{\mathcal{N}}_{x,r}\to {\mathcal{N}}_{X^t(x),1},\\ y\mapsto f(y)=X^{\tau (y)}(y).\end{array}$$

Let $\mathcal{N}={\bigcup }_{x\in R}{\mathcal{N}}_x$ be the normal bundle based on R. One can define the associated linear Poincaré flow by $P_X^t(x):={\mathrm{\Pi }}_{X_t(x)}\circ D{X}^t(x)$, where ${\mathrm{\Pi }}_{X^t(x)}:T_{X^t(x)}M\to N_{X^t(x)}$ is the projection along the direction of $X(X^t(x))$.

Denote by $int{\mathcal{OS}}_{\mu }(M)$ the set of divergence-free vector fields satisfying the orbital shadowing property.

Theorem 2.1 Let $X\in {\mathfrak{X}}_{\mu }^1(M)$. If $X\in int{\mathcal{OS}}_{\mu }(M)$, then X has no singularity and X is Anosov.

3 Proof of Theorem 2.1

Let $\mathrm{\Lambda }\subset M$ be a compact, $X^t$-invariant and regular set. We say that Λ is hyperbolic for $P_X^t$ if $N_{\mathrm{\Lambda }}$ admits a $P_X^t$-invariant splitting $N_{\mathrm{\Lambda }}={\mathrm{\Delta }}_{\mathrm{\Lambda }}^s\oplus {\mathrm{\Delta }}_{\mathrm{\Lambda }}^u$ such that there is $l>0$ satisfying

$$\parallel P_X^l{|}_{{\mathrm{\Delta }}^s(x)}\parallel \le \frac{1}{2}\text{and}\parallel P_X^{-l}{|}_{{\mathrm{\Delta }}^u(x)}\parallel \le \frac{1}{2}$$

for all $x\in \mathrm{\Lambda }$. The following is well known and one can find a proof in [8].

Theorem 3.1 Λ is a hyperbolic set of $X^t$ if and only if the linear Poincaré flow $P_X^t$ restricted on Λ has a hyperbolic splitting $N_{\mathrm{\Lambda }}={\mathrm{\Delta }}^s\oplus {\mathrm{\Delta }}^u$.

Consider a splitting $N=N^1\oplus \cdots \oplus N^k$ over Λ, for $1\le k\le n-1$, such that all the subbundles have constant dimensions. This splitting is dominated if it is $P_X^t$-invariant, and there is $l>0$ such that for every $0\le i<j\le k$, we have

$$\parallel P_X^l{|}_{N^i(x)}\parallel \cdot \parallel P_X^{-l}{|}_{N^j(X^l(x))}\parallel \le \frac{1}{2}$$

for any $x\in \mathrm{\Lambda }$.

The following was proved in [9].

Theorem 3.2 [[9], Proposition 4.1]

If $X\in {\mathfrak{X}}^1(M)$ admits a linear hyperbolic singularity of a saddle type, then $P_X^t$ does not admit any dominated splitting over $M\setminus Sing(X)$.

From the Theorem 3.2, we know that if a vector field X admits a dominated splitting, then $Sing(X)=\mathrm{\varnothing }$.

Franks’ lemma for divergence-free vector fields allows to realize the perturbations as perturbations of a fixed volume-preserving flow. Fix $X\in {\mathfrak{X}}_{\mu }^1(M)$ and $\tau >0$. A one-parameter area-preserving 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 $t\in \mathbb{R}$,

• $A_t=\mathit{id}$, 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 is proved in [[10], Lemma 3.2].

Lemma 3.3 Given $ϵ>0$ and a vector field $X\in {\mathfrak{X}}_{\mu }^1(M)$, there exists ${\xi }_0={\xi }_0(ϵ,X)$ such that for all $\tau \in [1,2]$, for any periodic point p of period greater than 2, for any sufficient small flowbox $\mathcal{T}$ of $\{X_t(p);t\in [0,\tau ]\}$ and for any one-parameter linear family ${\{A_t\}}_{t\in [0,\tau ]}$ such that $\parallel A_t^{\mathrm{\prime }}{A}_t^{-1}\parallel <{\xi }_0$ 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 3.4 Let $X\in {\mathfrak{X}}_{\mu }^1(M)$. By Zuppa’s theorem [11], we can find Y $C^1$-closed 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 divergence-free vector field X is a divergence-free 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 divergence-free star vector fields is denoted by ${\mathcal{G}}_{\mu }^1(M)$. Then we get the following.

Theorem 3.5 [[12], Theorem  1] If $X\in {\mathcal{G}}_{\mu }^1(M)$, then $Sing(X)=\mathrm{\varnothing }$ and X is Anosov.

Thus, to prove Theorem 3.7, it is enough to show that if X is in the $int{\mathcal{OS}}_{\mu }(M)$, then $X\in {\mathcal{G}}_{\mu }^1(M)$.

Lemma 3.6 If $X\in int{\mathcal{OS}}_{\mu }^1(M)$, then $X\in {\mathcal{G}}_{\mu }^1(M)$.

Proof Let $X\in int{\mathcal{OS}}_{\mu }(M)$. 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 shadowing property. Let $p\in \gamma \in PO(X_t)$ with $X^{\pi }(p)=p$ and $U_p$ be a small neighborhood of p. We will derive a contradiction. Assume that there is an eigenvalue λ of $P_X^{\pi }(p)$ such that $|\lambda |=1$. By Remark 3.4, there is $Y\in \mathcal{U}(X)$ such that $Y\in {\mathfrak{X}}_{\mu }^{\mathrm{\infty }}(M)$, $Y^{\pi }(p)=p$ and $P_Y^{\pi }(p)$ has an eigenvalue λ with $|\lambda |=1$. We define the map $f:{\phi }_p^{-1}(N_p)\to {\mathcal{N}}_p$ with the map being the Poincarè map associated to $Y^t$. Here ${\phi }_p:U_p\to T_{p}M$ is a smooth conservative map with ${\phi }_p(p)=\overrightarrow{0}$ (see, [13]). Let $\mathcal{V}$ be a $C^1$-neighborhood of f. Here ${\mathcal{N}}_p$ is the Poincarè section through p. By Lemma 3.3, we can find a small flowbox $\mathcal{T}$ 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) $Z^t(p)=Y^t(p)$ for all $t\in \mathbb{R}$, $P_Z^{t_0}(p)=P_Y^{t_0}(p)$ and $Z{|}_{{\mathcal{T}}^c}=Y{|}_{{\mathcal{T}}^c}$,

• (b) $g(x)={\phi }_p^{-1}\circ P_Y^{\pi }(p)\circ {\phi }_p(x)$ for all $x\in B_{\alpha }(p)\cap {\phi }_p^{-1}(N_p)$, and

• (c) $g(x)=f(x)$ for all $x\notin B_{4\alpha }(p)\cap {\phi }_p^{-1}(N_p)$.

By the notion of Lemma 3.3, we can assume that $P_Z^{\pi }(p)$ has an eigenvalue $|\lambda |=1$. Firstly, we assume that $\lambda =1$ (the other case is similar). Then we can choose a vector v associated to λ such that $\parallel v\parallel =\alpha /4$, and we set ${\mathcal{I}}_v=\{t\cdot v:0\le t\le 1\}$. Since ${\phi }_p^{-1}(v)\in {\phi }_p^{-1}(N_p)\setminus \{p\}$,

$$g({\phi }_p^{-1}(v))={\phi }_p^{-1}\circ P_Y^{\pi }(p)\circ {\phi }_p({\phi }_p^{-1}(v))={\phi }_p^{-1}\circ P_Y^{\pi }(p)(v)={\phi }_p(v).$$

For $0<ϵ<\alpha /8$, let $0<\delta <ϵ$ be as in the definition of the orbital shadowing property of $Z^t$. Set ${\mathcal{J}}_p={\phi }_p^{-1}({\mathcal{I}}_v)$. There is $k\in \mathbb{N}$ such that $x_k={\phi }_p^{-1}(v)$, $v_0=p$ and $|v_i-v_{i+1}|<\delta$ for $0\le i\le k-1$, where $v_i=t_i\cdot v$ for $0\le i\le k-1$. We construct a $(\delta ,1)$ pseudo-orbit of $Z^t$ belonging to ${\mathcal{J}}_p$ as follows:

• (a) $x_i={\phi }_p^{-1}(v_0)$, $t_i=\pi$ for $i<0$,

• (b) $x_i=g({\phi }_p^{-1}(v_k))$, $t_i=\pi$ for $0\le i\le k-1$, and

• (c) $x_i=g^{i-k}({\phi }_p^{-1}(v_i))$, $t_i=\pi$ for $i\ge k$.

Then $\xi =\{(x_i,t_i):i\in \mathbb{Z}\}$ is a $(\delta ,1)$-pseudo orbit of $Z^t$ and it is contained in ${\mathcal{J}}_p$. By the orbital shadowing property, we can take a point $z\in M$ such that

$${\mathcal{O}}_Z(z)\subset B_{ϵ}(\xi )\text{and}\xi \subset B_{ϵ}({\mathcal{O}}_Z(z)).$$

If $z\in {\mathcal{J}}_p$, then we know that there is $T_0>0$ such that $Z^{T_0}(z)\in B_{ϵ}(x_0)\cap {\mathcal{J}}_p$, and $d(Z^{T_0}(z),x_k)=\alpha /32$. Then

$$\begin{array}{rl}d(x_0,Z^{T_0}(z))& =d({\phi }_p^{-1}(v_0),{\phi }_p^{-1}(Z^{T_0}(z)))=d({\phi }_p^{-1}(\overrightarrow{0}),{\phi }_p^{-1}(Z^{T_0}(z)))\\ =d(p,{\phi }_p^{-1}(Z^{T_0}(z)))=\frac{\alpha }{8}>ϵ.\end{array}$$

Thus ${\mathcal{O}}_Z(z)\not\subset B_{ϵ}(\xi )$. This is a contradiction.

If $z\in M\setminus {\mathcal{J}}_p$, there is $T_1>0$ such that $Z^{T_1}(z)\in B_{ϵ}(x_0)$. Then for some $j=n\pi$, we have

$$2ϵ<d(x_0,x_k)\le d(x_0,Z^{T_1}(z))+d(Z^{T_1+j}(z),x_k)<2ϵ,$$

which is a contradiction.

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. Then, as in the previous argument, we get a contradiction. □

End of the proof of Theorem 3.7. By Lemma 3.6, $X\in {\mathcal{G}}_{\mu }^1(M)$. Thus by Theorem 3.5, $Sing(X)=\mathrm{\varnothing }$ and X is Anosov. □

By [7] and our main result, we have the following.

Corollary 3.7 Let $X\in {\mathfrak{X}}_{\mu }^1(M)$. Then

$$int{\mathcal{S}}_{\mu }(M)=int{\mathcal{OS}}_{\mu }^1(M)={\mathcal{A}}_{\mu }^1(M),$$

where $int{\mathcal{S}}_{\mu }(M)$ is the set of all divergence-free vector fields satisfying the shadowing property.