Robustly shadowable chain transtive sets and hyperbolicity

TL;DR

This paper establishes that for $C^1$-vector fields on manifolds, chain transitive sets are hyperbolic precisely when they are robustly shadowable, linking stability and hyperbolicity in dynamical systems.

Contribution

It proves an equivalence between hyperbolicity and robust shadowability for chain transitive sets in $C^1$-vector fields.

Findings

Chain transitive sets are hyperbolic iff they are robustly shadowable.

Robust shadowability characterizes hyperbolic structure in dynamical systems.

The result applies to $C^1$-vector fields on compact manifolds.

Abstract

We say that a compact invariant set $\Lambda$ of a $C^1$-vector field $X$ on a compact boundaryless Riemannian manifold $M$ is robustly shadowable if it is locally maximal with respect to a neighborhood $U$ of $\Lambda$, and there exists a $C^1$-neigborhood $\mathcal{U}$ of $X$ such that for any $Y \in \mathcal{U}$, the continuation $\Lambda_Y$ of $\Lambda$ for $Y$ and $U$ is shadowable for $Y_t$. In this paper, we prove that any chain transitive set of a $C^1$-vector field on $M$ is hyperbolic if and only if it is robustly shadowable.

Equations

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

$$d(X_{h(t)}(y),x_0\ast t)<\epsilon ,$$

$$S_i=\{\begin{array}{ll}{\sum }_{j=0}^{i-1}{t}_j& \text{mbox}fori>0,\\ 0& \text{mbox}fori=0,\\ -{\sum }_{j=i}^{-1}{t}_j& \text{mbox}fori<0.\end{array}$$

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

$$W^s(\gamma )=\{x\in M:X_t(x)\to \gamma \mathrm{as}t\to \infty \}\text{and}$$

$$W^u(\gamma )=\{x\in M:X_t(x)\to \gamma \mathrm{as}t\to -\infty \}$$

$$H_X(\gamma )=\overline{W^s(\gamma )⋔W^u(\gamma )}.$$

$$W^s({\gamma }_1)⋔W^u({\gamma }_2)\ne \varnothing \text{and}{W}^s({\gamma }_2)⋔W^u({\gamma }_1)\ne \varnothing .$$

$$H_X(\gamma )=\overline{\{{\gamma }^{\prime }:{\gamma }^{\prime }\sim \gamma \}}.$$

$$Sing(X)\cup PO(X_t)\subset \Omega (X_t)\subset CR(X_t).$$

$${\Lambda }_Y=\bigcap _{t\in \mathbb{R}}{Y}_t(U)$$

$${\exp }_p:T_{p}M(1)\to M$$

$$N_x=(\mathrm{span}X(x){)}^{\perp }\subset T_{x}M,$$

$$f_{x,t}:{\hat{N}}_{x,r}\to {\hat{N}}_{X_t(x),1},f_{x,t}(y)=X_{\tau (y)}(y)$$

$${\hat{U}}_{x,t_0,\delta }=\{tX(x)+y:0\le t\le t_0,y\in N_x(\delta )\}\subset T_{x}M,$$

$$supp(\varphi )\subset {\hat{N}}_{x,r/2}\text{and}{d}_{C^1}(\varphi ,id)<\epsilon .$$

$${\Psi }_t:\mathcal{N}\to \mathcal{N},{\Psi }_t{|}_{N_x}={\pi }_{N_x}\circ D_x{X}_t{|}_{N_x},$$

$${\Psi }_t{|}_{N_x}=D_x{f}_{x,t}\text{mbox}and{f}_{x,t}\circ {\exp }_x={\exp }_{X_t(x)}\circ {\Psi }_t.$$

$$\Vert {\Psi }_t{|}_{{\Delta }^s(x)}\Vert \cdot \Vert {\Psi }_{-t}{|}_{{\Delta }^u(X_t(x))}\Vert \le \frac{1}{2}$$

$$\Vert {\Psi }_t{|}_{{\Delta }^s(x)}\Vert \le C{\lambda }^t\text{mbox}and\Vert {\Psi }_{-t}{|}_{{\Delta }^u(x)}\Vert \le C{\lambda }^t$$

$$\frac{1}{t}[\text{log }m({\Psi }_t^Y{\mid }_{E_x^u})-\text{log}\Vert {\Psi }_t^Y{\mid }_{E_x^s}\Vert ]\ge 2\tilde{\eta };$$

$$t_k-t_{k-1}\ge \tilde{T},k=1,2,...,l,$$

$$\frac{1}{mT}\sum _{k=0}^{l-1}\text{log}\Vert {\Psi }_{t_{k+1}-t_k}^Y{\mid }_{E_{X_{t_{k-1}}(x)}^s}\Vert \le -\tilde{\eta },$$

$$\frac{1}{mT}\sum _{k=0}^{l-1}\text{log }m({\Psi }_{t_{k+1}-t_k}^Y{\mid }_{E_{X_{t_{k-1}}(x)}^u})\ge \tilde{\eta }.$$

$$0=T_0<T_1<...<T_l=t$$

$$\frac{1}{T_k}\sum _{j=1}^k\text{log}\parallel {\Psi }_{T_j-T_{j-1}}{\mid }_{{\Delta }^s(X_{T_{j-1}})(x)}\parallel \le -\eta ,$$

$$\frac{1}{T_l-T_{k-1}}\sum _{j=k}^l\text{log}m({\Psi }_{T_j-T_{j-1}}{\mid }_{{\Delta }^u(X_{T_{j-1}})(x)})\ge \eta ,$$

$$\text{log}\parallel {\Psi }_{T_k-T_{k-1}}{\mid }_{{\Delta }^s(X_{T_{k-1}})(x)}\parallel -\text{log}m({\Psi }_{T_k-T_{k-1}}{\mid }_{{\Delta }^u(X_{T_{k-1}})(x)})\le -2\eta ,$$

$$\dot{x}=Ax+K(x),$$

$$\underset{x\to 0}{\lim }\frac{K(x)}{\parallel x{\parallel }^2}=0.$$

Full text

1. Introduction

The main goal of the study of differentiable dynamical systems is to understand the structure of the orbits of vector fields (or diffeomorphisms) on a compact boundaryless Riemannian manifold. To descirbe the dynamics on the underlying manifold, it is usual to use the dynamic properties on the tangent bundle such as hyperbolicity and dominated splitting. A fundamental problem in recent years is to study the influence of a robust dynamic property (i.e., property that holds for a given system and all $C^{1}$-nearby systems) on the behavior of the tangent map on the tangent bundle (e.g., see [GLT, LS, LTW, LGW, PT]).

Recently, several results dealing with the influence of a robust dynamics property of a $C^{1}$-vector field were appeared. For instance, Lee and Sakai [LS] proved that a nonsingular vector field $X$ is robustly shadowable (i.e., $X$ and its $C^{1}$-nearby systems are shadowable) if and only if it satisfies both Axiom $A$ and the strong transversality condition (i.e., it is structurally stable). Afterwards, Pilyugin and Tikhomirov [PT] gave a description of robustly shadowable oriented vector fields which are structurally stable. In particular, it is proved in [LTW] that any robustly shadowable chain component $C_{X}(\gamma)$ of $X$ containing a hyperbolic periodic orbit $\gamma$ does not contain a hyperbolic singularity, and it is hyperbolic if $C_{X}(\gamma)$ has no non-hyperbolic singularity. Here we say that the chain component $C_{X}(\gamma)$ is robustly shadowable if there is a $C^{1}$-neighbohood $\mathcal{U}$ of $X$ such that for any $Y\in\mathcal{U}$, the continuation $C_{Y}(\gamma_{Y})$ of $Y$ containing $\gamma_{Y}$ is shadowable for $Y_{t}$, where $\gamma_{Y}$ is the continuation of $\gamma$ with respect to $Y$. Very recently, Gan et al. [GLT] showed that the set of all robustly shadowable oriented vector fields is contained in the set of vector fields with $\Omega$-stability. In this direction, the following question is still open: *if the chain component $C_{X}(\gamma)$ of a $C^{1}$-vector field $X$ on a compact boundaryless Riemannian manifold $M$ containing a hyperbolic periodic orbit $\gamma$ is robustly shadowable, then is it hyperbolic? *

In this paper, we study the dynamics of robustly shadowable chain transitive sets. More precisely, we prove that any chain transitive set of a vector field $X$ is hyperbolic if and only if it is robustly shadowable. For this, we first show that if a compact invariant set $\Lambda\subset M$ is robustly shadowable then every singularity and periodic orbit in $\Lambda_{Y}$ are hyperbolic for $Y_{t}$, where $\Lambda_{Y}$ is the continuation of $\Lambda$ with repect to a $C^{1}$-nearby vector field $Y$. Moreover, we see that any robustly shadowable chain transitive set $\Lambda$ does not contain a singularity. Finally we show that $\Lambda$ admits a dominated splitting, and it is indeed a hyperbolic splitting.

Now we round out the introduction with some notations, definitions and main theorem which we will use throughout the paper. Let $M$ be a compact boundaryless Riemannian manifold with dimension $n$. Denote by $\mathcal{X}^{1}(M)$ the set of all $C^{1}$ vector fields of $M$ endowed with the $C^{1}$ topology. Then every $X\in\mathcal{X}^{1}(M)$ generates a $C^{1}$ flow $X_{t}:M\times\mathbb{R}\to M$, that is, a family of diffeomorphisms on $M$ such that $X_{s}\circ X_{t}=X_{t+s}$ for all $t,s\in\mathbb{R}$, $X_{0}=Id$ and $\frac{dX_{t}}{dt}|_{t=0}=X(p)$ for any $p\in M$. Throughout the paper, for $X,Y,\ldots\in\mathcal{X}^{1}(M)$, we always denote the generated flows by $X_{t},Y_{t},\ldots$, respectively. For $x\in M$, let us denote the orbit $\{X_{t}(x),t\in\mathbb{R}\}$ of the flow $X_{t}$ (or $X$) through $x$ by $orb(x,X_{t})$, or $O(x)$ if no confusion is likely. We say that a point $x\in M$ is a singularity of $X$ if $X(x)=0$; and an orbit $O(x)$ is closed (or periodic) if it is diffeomorphic to a circle $S^{1}$. Let $d$ be the distance induced from the Riemannian structure on $M$. A sequence $\{(x_{i},t_{i}):x_{i}\in M;t_{i}\geq 1;a<i<b\}$ ($-\infty\leq a<b\leq\infty$) is called a $\delta$-pseudo orbit (or a $\delta$-chain) of $X_{t}$ if for any $a<i<b-1$,

[TABLE]

Roughly speaking, a pseudo orbit is composed by a set of segments of real orbits. We need the restriction $t_{i}\geq 1$ because without this, for any $\delta>0$, all points $x,y\in M$ can be connected by a $\delta$-pseudo orbit.

Let Rep be the set of all increasing homeomorphisms (called reparametrizations) $h:\mathbb{R}\rightarrow\mathbb{R}$ such that $h(0)=0.$ We say that a compact invariant set $\Lambda$ of $X_{t}$ is shadowable if for any $\varepsilon>0$, there is $\delta>0$ satisfying the following property: given any $\delta$-pseudo orbit $\{(x_{i},t_{i}):-\infty\leq i\leq\infty\}$ in $\Lambda$, there exist a point $y\in M$ and $h\in{\it Rep}$ such that for all $t\in\mathbb{R}$ we have

[TABLE]

where $x_{0}*t=X_{t-S_{i}}(x_{i})$ for any $t\in[S_{i},~{}S_{i+1}]$, and $S_{i}$ is given by

[TABLE]

Note that the above concept of pseudo orbit is slightly different from that of pseudo orbit in [LS, PT]. However we point out here that a compact invariant set $\Lambda$ is shadowable for $X_{t}$ under the above definition if and only if it is shadowable for $X_{t}$ under the definition in [LS, PT]. A point $x\in M$ is called chain recurrent if for any $\delta>0$, there exists a $\delta$-pseudo orbit $\{(x_{i},t_{i}):0\leq i<n\}$ with $n>1$ such that $x_{0}=x$ and $d(X_{t_{n-1}}(x_{n-1}),x)<\delta$. The set of all chain recurrent points of $X_{t}$ is called the chain recurrent set of $X_{t}$, and denote it by $CR(X_{t})$. For any $x,y\in M$, we say that $x\sim y$, if for any $\delta>0$, there are a $\delta$-pseudo orbit $\{(x_{i},t_{i}):0\leq i<n\}$ with $n>1$ such that $x_{0}=x$ and $d(X_{t_{n-1}}(x_{n-1}),y)<\delta$ and a $\delta$-pseudo orbit $\{(x^{\prime}_{i},t^{\prime}_{i}):0\leq i<m\}$ with $m>1$ such that $x^{\prime}_{0}=y$ and $d(X_{t^{\prime}_{n-1}}(x^{\prime}_{n-1}),x)<\delta$. It is easy to see that $\sim$ gives an equivalence relation on the set $CR(X_{t})$. An equivalence class of $\sim$ is called a chain component of $X_{t}$ (or $X$). We say that a compact invariant set $\Lambda$ of $X_{t}$ is chain transitive if for any $x,y\in\Lambda$ and any $\delta>0$, there is a $\delta$-pseudo orbit $\{(x_{i},t_{i})\in\Lambda\times\mathbb{R}\mid t_{i}\geq 1,0\leq i<n\}$ with $n>1$ such that $x_{0}=x$ and $d(X_{t_{n-1}}(x_{n-1}),y)<\delta$.

A compact invariant set $\Lambda$ of $X_{t}$ is called hyperbolic if there are constants $C>0$ and $\lambda>0$ such that the tangent flow $DX_{t}:T_{\Lambda}M\to T_{\Lambda}M$ leaves a continuous invariant splitting $T_{\Lambda}M=E^{s}\oplus\langle X\rangle\oplus E^{u}$ satisfying

[TABLE]

for any $x\in\Lambda$ and $t>0$, where $\langle X\rangle$ denotes the subspace generated by the vector field $X$. For any hyperbolic closed orbit $\gamma$, the sets

[TABLE]

[TABLE]

are said to be the stable manifold and unstable manifold of $\gamma$, respectively. We say that the dimension of the stable manifold $W^{s}(\gamma)$ of $\gamma$ is the index of $\gamma$, and denoted by $ind(\gamma)$.

The homoclinic class of $X_{t}$ associated to $\gamma$, denoted by $H_{X}(\gamma)$, is defined as the closure of the transversal intersection of the stable and unstable manifolds of $\gamma$, that is;

[TABLE]

By definition, we easily see that the set is closed and $X_{t}$-invariant. Let $C_{X}(\gamma)$ be the chain component of $X_{t}$ containing a hyperbolic periodic orbit $\gamma$. Then we have $H_{X}(\gamma)\subset C_{X}(\gamma)$, but the converse is not true in general. For two hyperbolic closed orbits $\gamma_{1}$ and $\gamma_{2}$ of $X_{t}$, we say $\gamma_{1}$ and $\gamma_{2}$ are homoclinically related, denoted by $\gamma_{1}\sim\gamma_{2}$, if

[TABLE]

By Birkhoff-Smale’s theorem (see [AP]), we know that

[TABLE]

A point $x\in M$ is called nonwandering if for any neighborhood $U$ of $x$, there is $t\geq 1$ such that $X_{t}(U)\cap U\not=\emptyset$. The set of all nonwandering points of $X_{t}$ is called the nonwandering set of $X_{t}$, denoted by $\Omega(X_{t})$. Let $Sing(X)$ be the set of all singularities of $X$, and let $PO(X_{t})$ be the set of all periodic orbits (which are not singularities) of $X_{t}$. Clearly we have

[TABLE]

We say that $X$ satisfies Axiom A if $PO(X_{t})$ is dense in $\Omega(X_{t})\setminus Sing(X)$, and $\Omega(X_{t})$ is hyperbolic for $X_{t}$. A point $y\in M$ is said to be an $\omega$* limit point* of $x$ if there exists a sequence $t_{i}\rightarrow+\infty$ such that $X_{t_{i}}(x)\rightarrow y$. Denote the set of all omega limit points of $x$ by $\omega(x)$. We say that a compact invariant set $\Lambda$ of $X_{t}$ is transitive if there is $x\in\Lambda$ such that $\omega(x)=\Lambda$.

Let $\Lambda$ be a compact invariant set of $X_{t}$. For any $C^{1}$-close $Y$ to $X$ and a neighbourhood $U$ of $\Lambda$, the set

[TABLE]

is called the continuation of $\Lambda$ for $Y$ and $U$. If there exists a neighbourhood $U$ of $\Lambda$ satisfying $\Lambda=\bigcap_{t\in\mathbb{R}}X_{t}(U),$ then we say that $\Lambda$ is locally maximal with respect to $U$, and $U$ is called an isolating block of $\Lambda$. Let $\gamma$ be a hyperbolic closed orbit of $X_{t}$. Then we know that there are a $C^{1}$ neighbourhood $\mathcal{U}$ of $X$ and a neighbourhood $U$ of $\gamma$ such that for any $Y\in\mathcal{U}$, there is a unique hyperbolic closed orbit $\gamma_{Y}$ in $U$ which is equal to the set $\bigcap_{t\in\mathbb{R}}Y_{t}(U)$. Note that every $\gamma_{Y}$ is locally maximal with respect to $U$. The chain component of $Y\in\mathcal{U}$ containing the continuation $\gamma_{Y}$ will be denoted by $C_{Y}(\gamma_{Y})$.

Now we give the definition of robust shadowability for invariant sets of vector fields.

Definition 1.1**.**

We say that a compact invariant set $\Lambda$ of $X_{t}$ is robustly shadowable if it has an isolating block $U$, and there exists a $C^{1}$-neighborhood $\mathcal{U}$ of $X$ such that for any $Y\in\mathcal{U}$, the continuation $\Lambda_{Y}$ for $Y$ and $U$ is shadowable for $Y_{t}$. Here $\mathcal{U}$ is said to be an admissible neighborhood of $X$ with repsect to $\Lambda$.

In this paper, we prove the following main theorem.

Main Theorem**.**

Let $X\in\mathcal{X}^{1}(M)$, and let $\Lambda$ be a compact, invariant and chain transitive set for $X_{t}$. Then $\Lambda$ is hyperbolic if and only if it is robustly shadowable.

2. Linear Poincaré flows and quasi hyperbolic orbit arcs

Hereafter we assume that the exponential map

[TABLE]

is well defined for all $p\in M$, where $T_{p}M(r)$ denotes the $r$-ball $\{v\in T_{p}M:\|v\|\leq r\}$ in $T_{p}M$. For any regular point $x\in M$ (i.e., $X(x)\neq\mathbb{0}$), we let

[TABLE]

and $N_{x}(r)$ the $r$-ball in $N_{x}$. Let $\hat{N}_{x,r}=\exp_{x}(N_{x}(r))$. Given any regular point $x\in M$ and $t\in\mathbb{R}$, we can take a constant $r>0$ and a $C^{1}$ map $\tau:\hat{N}_{x,r}\to\mathbb{R}$ such that $\tau(x)=t$ and $X_{\tau(y)}(y)\in\hat{N}_{X_{t}(x),1}$ for any $y\in\hat{N}_{x,r}$. Now we define the Poincaré map

[TABLE]

for $y\in\hat{N}_{x,r}.$ Let $M_{X}=\{x\in M:X(x)\neq\mathbb{0}\}$. Then it is easy to check that for any fixed $t$ there exists a continuous map $r_{0}:M_{X}\to(0,1)$ such that for any $x\in M_{X}$, the Poincaré map $f_{x,t}:\hat{N}_{x,r_{0}(x)}\to\hat{N}_{X_{t}(x),1}$ is well defined and the respective time function $\tau(y)$ satisfies $2t/3<\tau(y)<4t/3$ for $y\in\hat{N}_{x,r_{0}(x)}$.

Let $t_{0}$ be fixed. At each $x\in M_{X}$, one can consider a flow box chart $(\hat{U}_{x,t_{0},\delta},F_{x,t_{0}})$ at $x$ such that

[TABLE]

where $F_{x,t_{0}}:\hat{U}_{x,t_{0},\delta}\to M$ is defined by $F_{x,t_{0}}(tX(x)+y)=X_{t}(\exp_{x}y)$. Then it is well known that if $X_{t}(x)\neq x$ for any $t\in(0,t_{0}]$, then there is $\delta>0$ such that $F_{x,t_{0}}:\hat{U}_{x,t_{0},\delta}\to M$ is an embedding.

For $\varepsilon>0$ and $r>0$, let $\mathcal{N}_{\varepsilon}(\hat{N}_{x,r})$ be the set of all diffeomorphisms $\phi:\hat{N}_{x,r}\to\hat{N}_{x,r}$ such that

[TABLE]

Here $d_{C^{1}}$ is the usual $C^{1}$ metric, $id$ denotes the identity map and the $supp(\phi)$ is the closure of the set of points where it differs from $id$.

Proposition 2.1**.**

Let $X\in\mathcal{X}^{1}(M)$, and let $\mathcal{U}\subset\mathcal{X}^{1}(M)$ be a neighborhood of $X$. For any constant $t_{0}>0$, there are a constant $\varepsilon>0$ and a $C^{1}$-neighborhood $\mathcal{V}$ of $X$ such that for any $Y\in\mathcal{V}$, there exists a continuous map $r:M_{Y}\to(0,1)$ satisfying the following property$:$ for any $x\in M_{Y}$ satisfying $Y_{t}(x)\neq x$ for $0<t\leq 2t_{0}$ and any $\phi\in\mathcal{N}_{\varepsilon}(\hat{N}_{x,r(x)})$, there is $Z\in\mathcal{U}$ such that $Y(z)=Z(z)$ for all $z\in M\backslash F_{x}(\hat{U}_{x})$ and $Z_{t}(y)=Y_{t}(\phi(y))$ for any $y\in\hat{N}_{x,r(x)}$ and $2t_{0}/3<t<4t_{0}/3$, where $F_{x}(\hat{U}_{x})$ is the flow box of $Y$ at $x$.

Proof.

See [PR, p. 293–295]. ∎

Remark 2.2**.**

In the above proposition, it is easy to see that if $\phi(x)=x$, then $f_{x,t_{0}}\circ\phi$ is the Poincaré map of $Z$, where $f_{x,t_{0}}:\hat{N}_{x,r(x)}\to\hat{N}_{X_{t_{0}}(x),1}$ is the Poincaré map of $Y$.

For the study of stability conjecture (see [GW]) posed by Palis and Smale, Liao [L] introduced the notion of linear Poincaré flow for a $C^{1}$-vector field as follows. Let $\mathcal{N}=\bigcup_{x\in M_{X}}N_{x}$ be the normal bundle based on $M_{X}$. Then we can introduce a flow (which is called a linear Poincaré flow for $X$)

[TABLE]

where $\pi_{N_{x}}:T_{x}M\to N_{x}$ is the natural projection along the direction of $X(x)$, and $D_{x}X_{t}$ is the derivative map of $X_{t}$. Then we can see that

[TABLE]

Using Proposition 2.1, we can prove the following lemma which has the same philosophy with the Franks’ Lemma for diffeomorphisms. One can find another proof for the lemma in [BGV].

Lemma 2.3**.**

Let $\mathcal{U}$ be a $C^{1}$ neighborhood of $X\in\mathcal{X}^{1}(M)$. For any $T>0$, there exists a constant $\eta>0$ such that for any tubular neighborhood $U$ of an orbit arc $\gamma=X_{[0,T]}(x)$ of $X_{t}$ and for any $\eta$-perturbation $\mathcal{F}$ of the linear Poincaré flow $\Psi_{T}|_{N_{x}}$, there exists a vector field $Y\in\mathcal{U}$ such that the linear Poincaré flow $\tilde{\Psi}_{T}|_{N_{x}}$ associated to $Y$ coincides with $\mathcal{F}$, and $Y$ coincides with $X$ outside $U$ and along $X_{[-t_{1},t_{2}]}(x)$, where $t_{1}=\min\{t>0,X_{-t}(x)\in\partial{U}\}$ and $t_{2}=\min\{t>0,X_{t}(x)\in\partial{U}\}$.

We introduce the notions of dominated splitting and hyperbolic splitting for linear Poincaré flows as follows.

Definition 2.4**.**

Let $\Lambda$ be an invariant set of $X_{t}$ which contains no singularity. We call a $\Psi_{t}$-invariant splitting $\mathcal{N}_{\Lambda}=\Delta^{s}\oplus\Delta^{u}$ as an $l$-dominated splitting (or $\Lambda$ admits an $l$-dominated splitting) if

[TABLE]

for any $x\in\Lambda$ and any $t\geq l$, where $l>0$ is a constant. Moreover, if $\dim(\Delta^{s}_{x})$ is constant for all $x\in\Lambda$, then we say that the splitting is a homogeneous dominated splitting. Furthermore, a $\Psi_{t}$-invariant splitting $\mathcal{N}_{\Lambda}=\Delta^{s}\oplus\Delta^{u}$ is said to be a hyperbolic splitting if there exist $C>0$ and $\lambda\in(0,1)$ such that

[TABLE]

for any $x\in\Lambda$ and $t>0$.

The following proposition which is crucial to prove the hyperbolicity of invariant sets was proved by Doering and Liao [D, L]. For a detailed proof, see Proposition 1.1 in [D].

Proposition 2.5**.**

Let $\Lambda\subset M$ be a compact invariant set of $X_{t}$ such that $\Lambda\cap Sing(X)=\emptyset$. Then $\Lambda$ is hyperbolic for $X_{t}$ if and only if the linear Poincaré flow $\Psi_{t}$ restricted on $\Lambda$ has a hyperbolic splitting $\mathcal{N}_{\Lambda}=\Delta^{s}\oplus\Delta^{u}$.

Proposition 2.6**.**

Let $\Lambda$ be a locally maximal set of $X_{t}$ with an isolating block $U$. Suppose that $X$ has a $C^{1}$-neighbourhood $\mathcal{U}$ such that for any $Y\in\mathcal{U}$, every periodic orbit and singularity of $Y$ in $U$ are hyperbolic. Then $X$ has a neighbourhood $\tilde{\mathcal{U}}$, together with two uniform constants $\tilde{\eta}>0$ and $\tilde{T}>1$ such that for any $Y\in\tilde{\mathcal{U}}$,

1. (i)

whenever $x$ is a point on a periodic orbit of $Y_{t}$ in $U$ and $\tilde{T}\leq t<\infty$, then

[TABLE] 2. (ii)

whenever $P$ is a periodic orbit of $Y_{t}$ in $U$ with period $T,x\in P$, and whenever an integer $m\geq 1$ and a partition $0=t_{0}<t_{1}<\dots<t_{l}=mT$ of $[0,mT]$ are given that satisfy

[TABLE]

then

[TABLE]

and

[TABLE]

Proof.

See Theorem 2.6 in [LGW]. ∎

Let $\Lambda\subset M_{X}$ be a closed invariant set of $X_{t}$ that has a continuous $\Psi_{t}$-invariant splitting $\mathcal{N}_{\Lambda}=\Delta^{s}\oplus\Delta^{u}$ with dim $\Delta^{s}=p$, $1\leq p\leq dimM-2$. For two real numbers $T>0$ and $\eta>0$, an orbit arc $(x,t)=X_{[0,t]}(x)$ will be called $(\eta,T,p)$-quasi hyperbolic orbit arc of $X_{t}$ with respect to the splitting $\Delta^{s}\oplus\Delta^{u}$ if $[0,t]$ has a partition

[TABLE]

such that $T\leq T_{i}-T_{i-1}<2T$, $i=1,2,...,l,$ and the following three conditions are satisfied:

[TABLE]

[TABLE]

[TABLE]

for $k=1,2,...,l$.

Liao [L] proved the following shadowing result which says that any quasi hyperbolic orbit arc with close enough end points can be shadowed by a hyperbolic periodic orbit.

Proposition 2.7**.**

Let $\Lambda$ be a compact invariant set of $X_{t}$ without singularities. Assume that there exists a continuous invariant splitting $\mathcal{N}_{\Lambda}=\Delta^{s}\oplus\Delta^{u}$ with dim $\Delta^{s}=p$, $1\leq p\leq\text{dim\,}M-2$. Then for any $\eta>0$, $T>0$, and $\varepsilon>0$, there exists $\delta>0$ such that if $(x,\tau)$ is an $(\eta,T,p)$-quasi hyperbolic orbit arc of $X_{t}$ with respect to the splitting $\Delta^{s}\oplus\Delta^{u}$ and $d(X_{\tau}(x),x)<\delta$ then there exists a hyperbolic periodic point $y\in M$ and an orientation preserving homeomorphism $g:[0,\tau]\rightarrow\mathbb{R}$ with $g(0)=0$ such that $d(X_{g(t)}(y),X_{t}(x))<\varepsilon$ for any $t\in[0,\tau]$ and $X_{g(\tau)(y)}=y.$

3. From robust shadowing to dominated splitting

In this section, we prove that if a nontrivial chain transitive subset $\Lambda$ of $X_{t}$ is robustly shadowable, then it admits a dominated splitting. For this, we first show that any continuition $\Lambda_{Y}$ of $\Lambda$ does not contain both a non-hyperbolic sigularity and a non-hyperbolic periodic orbit. Next we show that $\Lambda$ does not contain a singularity. Finally we prove that $\Lambda$ admits a dominated splitting,

Lemma 3.1**.**

Let $\Lambda$ be a chain transitive set of $X_{t}$. If $\Lambda$ is robustly shadowable, then it is transitive.

Proof.

The proof is straightforward. ∎

Using the perturbation technique developed by Pugh and Robinson [PR], Pilyugin and Tikhomirov [PT] showed that if $M$ is robustly shadowable for $X_{t}$ then there is a $C^{1}$-neighbourhood $\mathcal{U}$ of $X$ such that for any $Y\in\mathcal{U}$, every critical element of $Y_{t}$ is hyperbolic. Here we prove that any continuition $\Lambda_{Y}$ of a robustly shadowable chain transitive set $\Lambda$ does not contain both a non-hyperbolic sigularity and a non-hyperbolic periodic orbit

Proposition 3.2**.**

Let $\Lambda$ be a robustly shdaowable set of $X_{t}$. Then there exists a $C^{1}$-neighbourhood $\mathcal{U}$ of $X$ such that for any $Y\in\mathcal{U}$, every singularity and periodic orbit of $Y_{t}$ in $\Lambda_{Y}$ are hyperbolic for $Y_{t}$.

Proof.

Suppose $\Lambda$ is a robustly shadowable set of $X_{t}$. Then there exist a $C^{1}$-neighborhood $\mathcal{U}$ of $X$ and a neighborhood $U$ of $\Lambda$ such that for any $Y\in\mathcal{U}$, the continuation $\Lambda_{Y}=\displaystyle\cap_{t\in\mathbb{R}}Y_{t}(U)$ is shadowable for $Y_{t}$.

Case $1$: Suppose there is $Y\in\mathcal{U}$ such that $\Lambda_{Y}$ contains a non-hyperbolic singularity $\sigma$. By using the Taylor’s theorem, we may assume that in a neighbourhood of $\sigma$ the dynamical system induced by $Y$ is expressed by the following differential equation:

[TABLE]

where $A\in M_{n\times n}(\mathbb{R})$ and $K:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is a continuous map satisfying

[TABLE]

Since $\sigma$ is not hyperbolic, there is an eigenvalue $\lambda$ of $A$ with zero real part. First we assume that $\lambda=0$. By changing coordinate, if necessary, we may assume that there is a $n\times n$-matrix $D$ close enough to $A$ such that

[TABLE]

where $B$ is a $(n-1)\times(n-1)$-matrix with real entries. We represent the coordinates of a point $x$ in a neighbourhood of $\sigma$ by $x=(y,z)$ with respect to $D$. Let $\varepsilon>0$, and choose a real valued $C^{\infty}$ bump function $\beta:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following conditions:

[TABLE]

Define $\rho:\mathbb{R}^{n}\rightarrow\mathbb{R}$ by $\rho(x)=\beta(\parallel x\parallel)$. By taking $\varepsilon$ small enough, one can see that the vector field $Z$ obtained from the following differential equation

[TABLE]

is $C^{1}$-close to $Y$. Moreover, we have $B_{\frac{\varepsilon}{4}}(\sigma)\subset U$. Consequently we see that $Z\in\mathcal{U}$, $\sigma\in Sing(Z)\cap\Lambda_{Z}$ and $\Lambda_{Z}$ is shadowable for $Z$. Since $\rho(x)=1$ for $\parallel x\parallel<\frac{\varepsilon}{4}$ , in the $\frac{\varepsilon}{4}$ neighbourhood of $\sigma$, the differential equation associated to $Z$ is given by

[TABLE]

By considering coordinates represented in (1), for any $x=(y,z)\in B_{\frac{\varepsilon}{4}}(\sigma),$ we have

[TABLE]

This implies that if $\mid y\mid\leq\frac{\varepsilon}{4}$ then $(y,0)\in Sing(Z)\cap U$, and so $\{(y,0):\,\mid y\mid<\frac{\varepsilon}{4}\}\subset\Lambda_{Z}.$ Let $\delta>0$ be a corresponding constant from the definition of shadowing of $\Lambda_{Y}$ for $\frac{\varepsilon}{8}$. Choose $\alpha_{0}=0<\alpha_{1}<...<\alpha_{n}=\frac{\varepsilon}{2}$ such that  $\mid\alpha_{i}-\alpha_{i-1}\mid<\delta$ for $i=1,...n$. Let

[TABLE]

Clearly $\{(x_{i},t_{i})\mid i=0,\ldots,n\}$ is a finite $\delta$-pseudo orbit of $Z_{t}$ in $\Lambda_{Z}$. Since $x_{0}$ and $x_{n}$ are singularities we can put

[TABLE]

Then $\{(x_{i},t_{i})\mid i\in\mathbb{Z}\}$ is a $\delta$-pseudo orbit of $Z_{t}$ in $\Lambda_{Z}$. Since $\Lambda_{Z}$ is shadowable, there are $(y,z)\in M$ and a reparametrization $h$ such that

[TABLE]

for all $t\in\mathbb{R}$. This implies $O(y)\subset B_{\frac{\varepsilon}{4}}(0)$. Since the intersections of planes formulated by $\{(y,z)\mid y=c\}$ with $B_{\frac{\varepsilon}{4}}(0)$ are invariant ($c$ is a constant), there is $c_{0}\in(-\frac{\varepsilon}{4},-\frac{\varepsilon}{4})$ such that $O(y)\subset\{(y,z)\mid y=c_{0}\}$. Without loss of generality, we may assume $c_{0}=0$. Then we get a contradiction since $d(X_{t}(y,z),x_{1})\geq\frac{\varepsilon}{4}$ for all $t\in\mathbb{R}$.

Suppose that $\lambda=ib$ for some nonzero $b\in\mathbb{R}$. By the same techniques as above, we can construct a vector field $Z$ which is $C^{1}$-close to $Y$ and in a neighbourhood of $\sigma$, the differential equation associated to $Z$ is given by

[TABLE]

where C=\left[\begin{array}[]{ll}\text{cos }(b)&\text{sin }(b)\\ \text{-sin }(b)&\text{cos }(b)\end{array}\right]. By considering the coordinates obtained from (2) in the $\frac{\varepsilon}{4}$ neighbourhood of $\sigma$, we can see that every point $x=(y_{1},y_{2},0)$ is periodic. Since the intersections of cylinders formulated by $\{(y_{1},y_{2},z)\mid y_{1}^{2}+y_{2}^{2}=c,c\in\mathbb{R}\}$ and $B_{\frac{\varepsilon}{4}}(\sigma)$ are invariant, we can derive a contradiction by using the same techniques as above.

Case $2$: Suppose there is $Y\in\mathcal{U}$ such that $\Lambda_{Y}$ contains a non-hyperbolic periodic orbit $\gamma$. Let $p\in\gamma$, and denote the period of $\gamma$ by $\pi(p)$. Then the linear Poincaré map $\Psi_{\pi(p)}:N_{p}\rightarrow N_{p}$ has an eigenvalue of modulus $1$. Hence we can find a linear map $P:N_{p}\rightarrow N_{p}$ arbitrarily close to $\Psi_{\pi(p)}$ that has an eigenvalue $\lambda$ of modulus $1$, the multiplicity of $\lambda$ is $1$, and $\lambda$ is a root of unity (i.e., $\lambda^{n}=1$ for some $n\in\mathbb{N}$). Using Lemma 2.3, we may assume that $\Psi_{\pi(p)}=P$. By changing the coordinates in $N_{P}$, if necessary, we may assume that

[TABLE]

and $C\,w=\lambda w$ for some $(w,0)\in N_{p}$, where $C$ is a $1\times 1$ (or $2\times 2$)-matrix. Choose $r>0$ such that $\hat{N}_{r}\subset U$ and the Poincaré map $f_{p,\pi(p)}:\hat{N}_{x,r}\rightarrow\hat{N}_{p,1}$ is well defined. Since $f_{p,\pi(p)}$ is a $C^{1}$ map, using the same techniques as in Case $1$, we can find a map

[TABLE]

which is arbitrarily $C^{1}$-close to $f_{p,\pi(p)}$ and $exp_{p}^{-1}\circ g\circ exp_{p}\mid_{N_{x,\frac{r}{2}}}=\Psi_{\pi(p)}\mid_{N_{x,\frac{r}{2}}}.$ By Proposition 2.1, we may assume that $f_{p,\pi(p)}=g$.

By the tubular flow theorem for closed orbits in Section 2.5.2 in [AP], we can find constants $s,\delta_{0},l>0$ such that if $x\in\hat{N}_{p}\cap B_{s}(p)$, $y\in M$ and $\varepsilon\in(0,\delta_{0})$ then $d(x,y)<\varepsilon$ implies $y=Y_{t^{\prime}}(y^{\prime}),$ for some $y^{\prime}\in\hat{N}_{p}$ and $\mid t^{\prime}\mid,d(y^{\prime},x)<l\varepsilon.$ Let $\delta>0$ be a corresponding constant for $\varepsilon<\text{min}\{\delta_{0},\frac{s}{4l}\}$ obtained from the shadowing property of $\Lambda_{Y}$. Let $v$ be a scalar multiplication of $w$ which obtained in equation (3) satisfying $\parallel v\parallel=s$. To make a $\delta$-pseudo orbit, fix $N>0$ and define

[TABLE]

and $t_{i}=\tau(x_{i}),$ where $\tau$ is the first return map. Then we get

[TABLE]

for sufficient large $N$. Since $C^{n}v=\lambda^{n}v=v$, we see that each $\{x_{i}\}$ is periodic and $O(x_{i})\subset U$ for all $i\in\mathbb{Z}$. Consequently, we get $x_{i}\in\Lambda_{Y}$ for all $i\in\mathbb{Z}$. Since $\Lambda_{Y}$ satisfies the shadowing property, there are $x\in M$ and $h\in\text{Rep}$ such that

[TABLE]

for all $t\in\mathbb{R}$. Hence there are $t_{1},t_{2}\in\mathbb{R}$ such that

[TABLE]

By the above fact, we can choose $t^{\prime}_{1}$ and $t^{\prime}_{2}$ in $\mathbb{R}$ such that

[TABLE]

Suppose that

[TABLE]

Then (4) implies that

[TABLE]

Moreover, we see that $(v_{1},w_{1})$ and $(v_{2},w_{2})$ belongs to the same orbit of $\Psi_{\pi(p)}$. Hence, without loss of generality, we may assume that there is $j\in\mathbb{N}$ such that $v_{1}=C^{j}v_{2}$. Consequently, we get

[TABLE]

But (5) implies that

[TABLE]

On the other hand, we have $\parallel C^{n}v\parallel=\parallel v\parallel=s$, and so the contradiction completes the proof of our proposition. ∎

Recently, Gan et al. [GLT] showed that if $M$ is robustly shadowable for $X\in\mathcal{X}^{1}(M)$, then there is no singularity $\sigma\in Sing(X)$ exhibiting homoclinic connection. Here the homoclinic connection is the closure of a orbit of a regular point which is contained in both the stable and the unstable manifolds of $\sigma$.

Proposition 3.3**.**

Let $\Lambda$ be a nontrivial chain transitive set of $X_{t}$. If $\Lambda$ is robustly shadowable then it does not contain a singularity of $X$.

Proof.

Let $U$ be an isolating block of $\Lambda$, and suppose $U$ contains a singularity $\sigma$. By Proposition 3.2, it must be hyperbolic.

First we show that there is $z\in W^{s}(\sigma)\cap W^{u}(\sigma)$ such that

[TABLE]

Choose $x\in\Lambda\setminus\{\sigma\}$, and let $\eta>0$ be a constant to ensure that the local stable manifold $W_{\eta}^{s}(\sigma)$ and the local unstable manifold $W_{\eta}^{u}(\sigma)$ of $\sigma$ are embedded submanifolds of $M$. Take $\delta_{0}>0$ satisfying $\displaystyle\bigcup_{y\in\Lambda}B_{\delta_{0}}(y)\subset U$. Let $\delta>0$ be a corresponding constant for $\varepsilon=min\{\frac{\eta}{2},\frac{\delta_{0}}{2},\frac{d(\sigma,x)}{2}\}$ obtained from the shadowability of $\Lambda$. Since $\Lambda$ is transitive, there are two finite $\delta$-pseudo orbits in $\Lambda$

$\{(x^{\prime}_{i},t^{\prime}_{i})\mid t^{\prime}_{i}\geq 1,i=1,\ldots,n\}$ and $\{(x^{\prime\prime}_{i},t^{\prime\prime}_{i})\mid t^{\prime\prime}_{i}\geq 1,i=1,\ldots,m\}$

such that $x^{\prime}_{0}=x^{\prime\prime}_{m}=\sigma$, and $x^{\prime}_{n}=x^{\prime\prime}_{0}=x$. Define an infinite $\delta$-pseudo orbit in $\Lambda$ as follows:

[TABLE]

Then there are $z\in M$ and $h\in\text{Rep}$ such that

[TABLE]

for all $t\in\mathbb{R}$. This implies that there is $T>0$ such that

[TABLE]

By our construction, we see that $z\in W^{s}(\sigma)\cap W^{u}(\sigma).$ Hence we have

[TABLE]

This implies that $O(z)\subset U$ and $z\in\Lambda$.

Second we show that there is $x\in W^{s}(\sigma)\cap W^{u}(\sigma)$ such that

[TABLE]

Let $\varepsilon>0$ be such that $\bigcup_{x\in\Gamma}B_{\varepsilon}(x)\subset U,$ and let $\delta$ be a corresponding constant for $\varepsilon$ obtained from the shadowing property of $\Lambda$. Since $z\in W^{s}(\sigma)\cap W^{u}(\sigma)$, there is $m\in\mathbb{N}$ such that

[TABLE]

for all $n\geq m$. Consider a $\delta$-pseudo orbit in $\Lambda$

[TABLE]

Then there are $x\in M$ and $h\in\text{Rep}$ such that $d(X_{h(t)}(x),x_{0}*t)<\varepsilon.$ We also easily check that

[TABLE]

This implies that $dimE^{s}=dimW^{s}(\sigma)=k\geq 2$. By applying Lemma 3.5 in [GLT], we can assume that there is a dominated splitting $E^{s}=E^{c}\oplus E^{ss}$ such that $dimE^{c}=1$. We also perturb $\Gamma$ and $\Gamma^{\prime}$ to make sure that

[TABLE]

where $W^{ss}(\sigma)$ be the strong stable submanifold of $M$ tangent to $E^{ss}$. Furthermore we may perturb that in a neighbourhood $V$ of $\sigma$, the dynamic induced by $X$ is expressed by the following differential equation

[TABLE]

where $B_{1}$, $B_{2}$ and $C$ are preserving the splitting $E^{c}\oplus E^{ss}\oplus E^{u}$. Here the eigenvalues of $B_{2}$ and $C$ have negative and positive real part, repectively, and the spectrum of $B_{1}=\{\lambda_{1}\}$. For more details on these perturbations, see [GLT]. Since the dynamic on $V$ is induced by the differential equation $\dot{x}=Ax$, we can express every point $y$ in $V$ by $y=(y^{c},y^{ss},y^{u})$ based on the coordinates obtained from $E^{c}\oplus E^{ss}\oplus E^{u}$. Then we get

[TABLE]

Next we are going to get some useful properties for $\Gamma\cup\Gamma^{\prime}$ that helps us to complete the proof. Choose $x\in\Gamma^{\prime}$ and $z_{1},z_{2}\in\Gamma$ satisfying

[TABLE]

Fix $r>0$ and let $y\in\hat{N}_{x,r}$. Assume that there exists $t>0$ such that

[TABLE]

For any $y\in\hat{N}_{x,r},$ denote by $\tau(y)$ the minimum of $t$ with the above property (if such a $t$ exists). Define a map $P_{r}$ by

[TABLE]

We show that there is $r_{0}>0$ such that Dom$(P_{r})\neq\emptyset$ for any $r\in(0,r_{0}]$. Fix $r_{0}>0$ such that

[TABLE]

Let $r\in(0,r_{0}]$, and take $r_{1}\in(0,r]$ such that $d(X_{t}(y),x)<r_{1}$ for $y\in M$ and $t>0$. Then there is $t^{\prime}\in[0,t]$ such that

[TABLE]

If $d(X_{t}(y),z_{2})<r_{1}$, then there is $t^{\prime}\in[t,\infty)$ such that

[TABLE]

Let $\delta>0$ be a corresponding constant for $\frac{r_{1}}{2}$ obtained from the shadowing property of $\Lambda$. Let $m\in\mathbb{N}$ be such that

[TABLE]

Consider the following $\delta$-pseudo orbit

[TABLE]

Then there are $y\in M$ and $h\in Rep$ such that

[TABLE]

This implies that there are $0\leq t_{1}<t_{2}<t_{3}<\infty$ such that

[TABLE]

Hence we have $X_{[h(t_{1}),h(t_{3})]}(y)\subset V$. Let $t^{\prime}_{1},t^{\prime}_{2}$ be constants corresponding to $h(t_{1}),h(t_{3})$, respectively, obtained from the same way we get $r_{1}$. Then we get

$X_{t^{\prime}_{1}}(y)\in\hat{N}_{x,r}$, $X_{t^{\prime}_{2}}(y)\in\hat{N}_{z_{2},r}$, and $X_{[t^{\prime}_{1},t^{\prime}_{2}]}(y)\subset exp_{\sigma}(T_{\sigma}M(1))$.

Consequently, we have $y\in~{}\text{Dom}(P_{r})$ and so Dom$(P_{r})\neq\emptyset$.

Consider the following set

[TABLE]

We will show that for any $\varepsilon>0$ there is $r>0$ satisfying

[TABLE]

Let $y\in~{}\text{Dom}(P_{r})\cap\hat{N}_{x,r}$. Since $P_{r}(y)\in\hat{N}_{z_{2},r}$, we have

[TABLE]

for sufficiently samll $r>0$. Using (7), we get

[TABLE]

Hence $\tau(y)\rightarrow+\infty$ as $\parallel y^{u}\parallel\rightarrow 0$. On the other hand, we have

[TABLE]

Since $x^{c}\neq 0$, we get $y^{c}\not\rightarrow 0$ as $y\rightarrow x$. In addition, because $E^{c}\oplus E^{ss}$ is a dominated splitting, the right side of (10) tends zero as $\tau(y)\rightarrow+\infty$, and (9) is proved.

Next we perturb $X$ so that if $z_{1}=X_{t^{\prime}}(z_{2})$ then $\Psi_{t}(L)\cap\Delta^{s}=\emptyset$, where $\Delta^{s}=N_{z_{1}}\cap T_{z_{1}}W^{s}(\sigma)$. If $\Psi_{t^{\prime}}(L)\not\subset\Delta^{s}$ we have nothing to prove. Otherwise, let $u\in N_{z_{1}}$ be such that $u\not\in\Delta^{s}$. Fix $\alpha>0$, and denote $u_{\alpha}=\alpha u+(1-\alpha)v$, where $\Psi_{t^{\prime}}(L)=\rm{Span}\{v\}$. Then there is a linear map $H_{\alpha}:N_{z_{1}}\rightarrow N_{z_{1}}$ such that

[TABLE]

Define a map

[TABLE]

Choose $\alpha>0$ so small that we can use Lemma 2.3, and replace $\Psi_{t^{\prime}}$ with $\Psi^{\prime}$. Then we get

[TABLE]

Since the Poincaré map $f_{z_{1},t}:\hat{N}_{z_{2},r}\rightarrow\hat{N}_{z_{1},r}$ is continuous, there is $\varepsilon>0$ such that

[TABLE]

where $\mathcal{C}_{\varepsilon}$ is defined in (9). Let $r>0$ be such that $r$ satisfies (9) for $\varepsilon$, and let $\delta>0$ be a corresponding constant for $\varepsilon^{\prime}=min\{r,\varepsilon,\eta\}$ obtained from the shadowing property of $\Lambda$. Consider the $\delta$-pseudo orbit (8) we constructed in the above. Then there are $y\in M$ and $h\in$ Rep such that

[TABLE]

This implies that there are constants $0<t_{1}<t_{2}<t_{3}<t_{4}$ satisfying

[TABLE]

Without loss of generality, we may assume that

$X_{h(t_{1})}(y)\in\hat{N}_{x,r}$, $X_{t_{3}}(y)\in\hat{N}_{z_{2},r}$, and $X_{h(t_{4})}(y)\in\hat{N}{z_{1},r}$.

This means that $X_{t_{3}}(y)=P_{r}(X_{h(t_{1})}(y))$, and so we have $X_{h(t_{3})}(y)\in\mathcal{C_{\varepsilon}}$. Consequently, we get

[TABLE]

This is a contradiction to the fact that

[TABLE]

and so completes the proof. ∎

Proposition 3.4**.**

Let $\Lambda$ be a chain transitive set. If $\Lambda$ is robustly shadowable, then it admits a homogeneous dominated splitting for $\Psi_{t}$.

Proof.

If $\Lambda$ is a periodic orbit, then it admits a dominated splitting for $\Psi_{t}$ by Proposition 3.2. Hence we suppose $\Lambda$ is not a periodic orbit, and take a point $x\in\Lambda$ be such that $\omega(x)=\Lambda$. By applying the Pugh’s closing lemma (see [PR]), we can select a sequence $\{Y^{n}\}_{n\in\mathbb{N}}\subset\mathcal{U}$ converging to $X$ such that each $Y^{n}$ has a periodic point $p_{n}$ converging to $x$; and for each $t>0$, the sequence $\phi_{n}:[0,t]\rightarrow M$ given by $\phi_{n}(s)=Y^{n}_{s}(p_{n})$ converges to $\phi:[0,t]\rightarrow M,~{}\phi(s)=X_{s}(x)$. Note that here $O(p_{n})$ is hyperbolic for $Y^{n}_{t}$ for every $n$. Moreover we can see that the period of $p_{n}$ tends to $\infty$ as $n\rightarrow\infty$. By applying Proposition 2.6, we can take $l>0$ such that the linear Poincaré flow of $Y_{n}$ over $O(p_{n})$ admits an $l$-dominated splitting. By taking a subsequence, if necessary, we may assume that there is $k\in\mathbb{N}$ such that $ind(p_{n})=k$ for all $n\in\mathbb{N}$.

Let $\{x_{k}\}$ be a sequence in $\Lambda$ converging to $x$, and let $E(x_{k})$ be an $m$-dimensional subspace of $T_{x_{k}}M$. We say that $E(x_{k})$ converges to $E(x)$ if, for each k, there is a basis $\{e^{1}_{k},\ldots,e^{m}_{k}\}$ of $E(x_{k})$ and a basis $\{e^{1},\ldots,e^{m}\}$ of $E(x)$ such that $e^{i}_{k}\rightarrow e^{i}$ for each $i=1,\cdots,m$.

Put

[TABLE]

For each $t>0$, we denote by

[TABLE]

where $T_{Y^{n}_{t}(p_{n})}M=E^{s}_{n}(Y^{n}_{t}(p_{n}))\oplus E^{u}_{n}(Y^{n}_{t}(p_{n}))$. Then we have

[TABLE]

[TABLE]

where $\Psi^{n}_{t}$ is the linear Poincaré flow for $Y^{n}$. This means that the splitting $\Delta^{s}(x)\oplus\Delta^{u}(x)$ is $\Psi_{t}$ invariant, and we have $\mathcal{N}_{x}=\Delta^{s}(x)\oplus\Delta^{u}(x)$. If $t$ is sufficiently large, then we can see that

[TABLE]

This means that the orbit $O(x)$ admits a dominated splitting for $\Psi_{t}$, and so $\Lambda=\overline{O(x)}$ also has a dominated splitting for $\Psi_{t}$, ∎

4. From dominated splitting to hyperbolicity

Lemma 4.1**.**

If a chain transitive set $\Lambda$ of $X_{t}$ is robustly shadowable, then it admits a hyperbolic periodic orbit.

Proof.

Let $\Delta^{s}\oplus\Delta^{u}$ be the $l$-dominated splitting of $(T_{\Lambda}M,\Psi_{t}|_{N_{\Lambda}})$ obtained in Proposition 3.4. By using lemma $3.4$ in [GW] and Theorem 2.6 we may assume that ${dim\,}(\Delta^{s})\leq{dim\,}M-2$. Denote by

[TABLE]

For any $\varepsilon>0$, choose $\varepsilon^{\prime}\in(0,\frac{\alpha}{2})$, $\delta^{\prime}>0$, and $Y\in\mathcal{U}$ having a periodic point $p$ such that

[TABLE]

where $\Psi$ and $\Psi^{\prime}$ are linear Poincaré flows of $X$ and $Y$, respectively. Since $p$ is a hyperbolic periodic point of $Y_{t}$, there are $C>0$ and $\lambda\in(0,1)$ such that

[TABLE]

for all $t\geq 0$ and $y\in O(p)$. Denote by $C^{\prime}={\rm{max}}\{C,C^{-1}\}$, and let $\delta$ be a constant as in Proposition 2.7 for the triple $(\varepsilon,T,\eta)=(\varepsilon,1,-(log(c^{\prime})+\varepsilon)).$ Because $x$ is a nonwandering point, there is $t^{\prime}>0$ such that $d(X_{t^{\prime}}(x),x)<\delta$. Let $T_{0},...,T_{m}\in\mathbb{R}$ be such that

[TABLE]

is a partition for $[0,t^{\prime}]$ with $T_{i+1}-T_{i}\in[1,2]$. Let $p_{0},...,p_{m}\in O(p)$ be such that

[TABLE]

We show that $X_{[0,t^{\prime}]}(x)$ is an $(\varepsilon,T,\eta)$-quasi hyperbolic arc. By using (11) we have

[TABLE]

For the first and second inequality, we used the properties in (11); for the third inequality, we used the hyperbolicty of $O(p)$; and for the fourth and fifth inequality, we used the property $T_{j}-T_{j-1}\geq 1$.

On the other hand, we have

[TABLE]

Hence we get

[TABLE]

Similarly we obtain

[TABLE]

for all $k\in\{1,...,m\}$. Consequently we can see that $\Lambda$ contains a hyperbolic periodic orbit by Proposition 2.7. ∎

End of proof of main theorem.

Let $\Lambda$ be a chain transitive set, and suppose it is robustly shadowable. Then $\Lambda$ contains a hyperbolic periodic orbit, say $\gamma$, by Lemma 4.1. Since $\Lambda$ is transitive, we see that $\Lambda\subset C_{X}(\gamma)$ and also $\Lambda\subset H_{X}(\gamma)$. Since $\Lambda$ is compact and the periodic points are dense in $\Lambda$, we may assume that for any $T>0$ there is a periodic point $p$ in $\Lambda$ whose period is bigger than $T$. Then by using the results and techniques in Section 5 of [LTW], we can show that the dominated splitting $\mathcal{N}_{\Lambda}=\Delta^{s}\oplus\Delta^{u}$ is a hyperbolic spliting for $\Psi_{t}$. Consequently we can see that $\Lambda$ is hyperbolic for $X_{t}$ by applying Proposition 2.5.

The converse is clear by the robust property of hyperbolic sets and the shadowability of the hyperbolic sets, and so completes the proof of our main theorem. ∎

Acknowledgement. The second author was supported by the NRF grant funded by the Korea government (MSIP) (No. NRF-2015R1A2A2A01002437).