2.1 Hyperbolicity

Recall that the flow and the tangent flow generated by X are denoted as $\phi _t$ and $\Phi _t$ , respectively. Let $\Lambda $ be a compact $\phi _t$ -invariant set and $E\subset T_{\Lambda }M$ a $\Phi _t$ -invariant subbundle. The dynamics $\Phi _t|_E$ is contracting, or E is contracted, if there exists constants $C\geq 1$ , $\unicode{x3bb}>0$ , such that

$$ \begin{align*}\|\Phi_t|_{E(x)}\|\leq C{e}^{-\unicode{x3bb} t}\quad \text{for any}\ x\in \Lambda, \ t\geq 0.\end{align*} $$

One says that $\Phi _t|_E$ is expanding if $\Phi _{-t}|_E$ is contracting. The set $\Lambda $ is called hyperbolic if its tangent bundle admits a continuous splitting $T_{\Lambda }M=E^s\oplus \langle X\rangle \oplus E^u$ , where $\langle X\rangle $ is the subbundle generated by the vector field X, and $E^s$ (respectively $E^u$ ) is contracted (respectively expanded). When $\Lambda $ is hyperbolic, for every $x\in \Lambda $ , the sets

$$ \begin{align*}W^{ss}(x)=\{y\in M: \mathrm{dist}(\phi_t(x), \phi_t(y))\to 0\ \text{as}\ t\to +\infty\}\end{align*} $$

and

$$ \begin{align*}W^{uu}(x)=\{y\in M: \mathrm{dist}(\phi_t(x), \phi_t(y))\to 0\ \text{as}\ t\to -\infty\}\end{align*} $$

are invariant $C^1$ submanifolds tangent to $E^s(x)$ and $E^u(y)$ , respectively, at x, see [Reference Hirsch, Pugh and Shub32]. We denote $W^s(\operatorname {Orb}(x))=\bigcup _{z\in \operatorname {Orb}(x)}W^{ss}(z)$ and $W^u(\operatorname {Orb}(x))=\bigcup _{z\in \operatorname {Orb}(x)}W^{uu}(z)$ , which are the stable and unstable manifolds of $\operatorname {Orb}(x)$ , respectively. For a chain transitive hyperbolic set, all stable manifolds $W^{ss}(x)$ have the same dimension $\operatorname {dim} E^s$ , which will be called the stable index of the hyperbolic set. Sometimes we simply say index with the same meaning of the stable index.

Flows with singularities, or singular flows, can exhibit complex dynamics with regular orbits accumulating on singularities, such as the famous Lorenz attractor [Reference Guckenheimer and Williams31, Reference Lorenz36]. In such cases, the set $\Lambda $ cannot be hyperbolic. Nevertheless, a weaker version of hyperbolicity can be defined. One says that $\Lambda $ is partially hyperbolic if there exists a $\Phi _t$ -invariant splitting of the tangent bundle $T_{\Lambda }M=E^s\oplus E^{cu}$ such that $E^s$ is contracted and it is dominated by $E^{cu}$ . If, moreover, the bundle $E^{cu}$ is sectionally expanded, then $\Lambda $ is said to be singular hyperbolic [Reference Morales, Pacifico and Pujals40]. Note that the partial hyperbolicity or singular hyperbolicity ensures also the existence of stable manifolds.

We will study mainly chain recurrence classes. A chain recurrence class $C(x,X)$ is non-trivial if it is not reduced to a critical element. In [Reference Pacifico, Yang and Yang43], it is shown that a non-trivial, Lyapunov stable, and singular hyperbolic chain recurrence class contains a periodic orbit, and $C^1$ generically, such a chain recurrence class is indeed an attractor. This result has been generalized to $C^1$ open and dense vector fields in [Reference Crovisier and Yang24].

2.2 Linear Poincaré flow and its extension

Given a $\phi _t$ -invariant set $\Lambda \subset M$ , define the normal bundle over $\Lambda $ as

$$ \begin{align*} \mathcal{N}_{\Lambda}=\bigcup_{x\in\Lambda\setminus\operatorname{Sing}(X)}\mathcal{N}_x, \end{align*} $$

where $\mathcal {N}_x$ is the orthogonal complement of the one-dimensional subspace $l^X_x\subset T_xM$ generated by $X(x)$ . In particular, if $\Lambda =M$ , then $\mathcal {N}_M=\bigcup _{M\setminus \operatorname {Sing}(X)}\mathcal {N}_x$ . For any $x\in M\setminus \operatorname {Sing}(X)$ , $v\in \mathcal {N}_x$ , and $t\in {\mathbb R}$ , let $\psi _t(v)$ to be the orthogonal projection of $\Phi _t(v)$ to $\mathcal {N}_{\phi _t(x)}$ . In this way, one defines a flow $\psi _t=\psi ^X_t$ on the normal bundle $\mathcal {N}_M$ , which is called the linear Poincaré flow. Note that the normal bundle is not defined at singularities. Hence, the base of the normal bundle is non-compact for singular flows. We present below a compactification given by [Reference Li, Gan and Wen34].

Define the fiber bundle $G^1=\bigcup _{x\in M}Gr(1,T_xM)$ , where $Gr(1,T_xM)$ is the Grassmannian of lines in the tangent space $T_xM$ through the origin. Let $\beta :G^1\to M$ be the corresponding bundle projection that sends a line $l\in Gr(1,T_xM)$ to x. The tangent flow $\Phi _t$ induces a flow $\hat {\Phi }_t$ on $G^1$ defined by $\hat {\Phi }_t(\langle v\rangle )=\langle \Phi _t(v)\rangle $ , where $v\in TM$ is a non-zero vector and $\langle v\rangle $ is the linear subspace spanned by v.

Let $\xi : TM\to M$ be the bundle projection of the tangent bundle. One then defines a vector bundle with base $G^1$ as

$$ \begin{align*}\beta^*(TM)=\{(l,v)\in G^1\times TM: \beta(l)=\xi(v)\}.\end{align*} $$

The corresponding bundle projection $\iota $ sends every vector $(l,v)\in \beta ^*(TM)$ to l. The tangent flow induces also a flow $\tilde {\Phi }_t$ on $\beta ^*(TM)$ :

$$ \begin{align*}\tilde{\Phi}_t(l,v)=(\hat{\Phi}_t(l),\Phi_t(v))\quad \text{for all } (l,v)\in\beta^*(TM).\end{align*} $$

The flow $\tilde {\Phi }_t$ will be called the extended tangent flow.

For any $l\in G^1$ such that $\beta (l)=x$ , there is a natural identification between $T_xM$ and $\{l\}\times T_xM$ . Thus, we define the extended normal bundle as

$$ \begin{align*} \widetilde{\mathcal{N}}=\{(l,v)\in \beta^*(TM): v\perp l\}. \end{align*} $$

We also define the extended normal bundle over any non-empty subset $\Delta \subset G^1$ as

$$ \begin{align*} \widetilde{\mathcal{N}}_{\Delta}=\{(l,v)\in\iota^{-1}(\Delta): v\perp l\}. \end{align*} $$

Finally, we define the extended linear Poincaré flow $\tilde {\psi }_t$ on $\widetilde {\mathcal {N}}$ as

$$ \begin{align*} \tilde{\psi}_t(l,v)=\tilde{\Phi}_t(l,v)-\langle \Phi_t(u),\Phi_t(v)\rangle\cdot\tilde{\Phi}_t(l,u), \end{align*} $$

where $u\in l$ is a unit vector and $\langle \cdot ,\cdot \rangle $ stands for the inner product on the tangent bundle.

We will also consider invariant measures on $G^1$ . Let $\mu $ be any Borel measure on $G^1$ . The bundle projection $\beta :G^1\to M$ induces a measure $\beta _*\mu $ on M such that $(\beta _*\mu )(A)=\mu (\beta ^{-1}(A))$ for any Borel set $A\subset M$ . When $\mu $ is $\hat {\Phi }_t$ -invariant, the induced measure $\beta _*\mu $ is $\phi _t$ -invariant.

2.3 Fundamental limit and dominated splitting

Given a chain recurrent set $\Gamma $ of the vector field X, let $X_n$ be a sequence of $C^1$ vector fields such that $X_n\to X$ in the $C^1$ topology. Suppose there exists a periodic orbit $\gamma _n$ of $X_n$ such that $\gamma _n$ converges to a compact subset of $\Gamma $ in the Hausdorff topology, then the sequence of pairs $(\gamma _n, X_n)$ is called a fundamental sequence of $\Gamma $ and will be denoted as $(\gamma _n,X_n)\hookrightarrow (\Gamma ,X)$ . The sequence $(\gamma _n,X_n)$ is called an i-fundamental sequence if $\operatorname {Ind}(\gamma _n)=i$ for all n large enough. When there is no possible ambiguity, we will simply say that $\gamma _n$ is a fundamental sequence of $\Gamma $ . We denote by $\mathcal {F}(\Gamma )$ the limit of directions of all fundamental sequences of $\Gamma $ , i.e.

$$ \begin{align*}\mathcal{F}(\Gamma)=\{l\in G^1: \text{ there exists } (\gamma_n,X_n)\hookrightarrow (\Gamma,X),\ p_n\in \gamma_n \ \text{such that}\ l^{X_n}_{p_n} \to l\}.\end{align*} $$

As an immediate consequence of the definition, the map $\mathcal {F}(\cdot )$ is upper semi-continuous in the following sense.

We also denote by $B(\Gamma )$ the limit set of directions of regular points contained in $\Gamma $ , i.e.

$$ \begin{align*}B(\Gamma)=\{l\in G^1: \text{ there exists } x_n\in \Gamma\setminus\operatorname{Sing}(X) \ \text{such that}\ l^X_{x_n} \to l\}.\end{align*} $$

Note that for any $l\in \mathcal {F}(\Gamma )$ , if $x=\beta (l)$ is a regular point, then $l=l^X_x$ . This also holds for $B(\Gamma )$ . Thus, $\mathcal {F}(\Gamma )\setminus B(\Gamma )$ is contained in the tangent spaces of singularities.

A hyperbolic singularity $\rho $ is called Lorenz-like if there is a partially hyperbolic splitting of the tangent bundle $T_{\rho }M=E^{ss}_{\rho }\oplus E^{cu}_{\rho }$ such that $E^{cu}_{\rho }$ is sectionally expanded and it decomposes further into a dominated splitting $E^c_{\rho }\oplus E^u_{\rho }$ with $\operatorname {dim} E^c_{\rho }=1$ and $\Phi _t|_{E^c_{\rho }}$ is contracting. For a Lorenz-like singularity $\rho $ , denote by $W^{ss}(\rho )$ its strong stable manifold tangent to $E^{ss}_{\rho }$ at the singularity.

A fundamental sequence $(\gamma _n,X_n)$ of $\Gamma $ is said to admit an index i dominated splitting if there exist a $\psi ^{X_n}_t$ -invariant splitting $\mathcal {N}_{\gamma _n}=\mathcal {N}_n^{cs}\oplus \mathcal {N}_n^{cu}$ ( $n\in \mathbb {N}$ ) of the normal bundle and constants $T>0$ , $n_0>0$ such that for any $t>T$ and $n>n_0$ , it holds $\operatorname {dim} \mathcal {N}_n^{cs}=i$ and for any $x\in \gamma _n$ ,

$$ \begin{align*}\|\psi^{X_n}_t|_{\mathcal{N}_n^{cs}(x)}\|\cdot\|\psi^{X_n}_{-t}|_{\mathcal{N}_n^{cu}(\phi^{X_n}_t(x))}\|<1/2.\end{align*} $$

Identifying the extended normal bundle $\widetilde {\mathcal {N}}_{B(\gamma _n)}$ with $\mathcal {N}_{\gamma _n}$ (see Remark 2.1), we see that $\widetilde {\mathcal {N}}_{B(\gamma _n)}$ also admits a dominated splitting with respect to $\tilde {\psi }_t^{X_n}$ and of the same index. Note that $B(\gamma _n)$ converges to a compact subset of $\mathcal {F}(\Gamma )$ in the Hausdorff topology of the Grassmannian $G^1$ . Thus, the following lemma is a consequence of the property that a dominated splitting can be extended to the closure and to a neighborhood (see e.g. [Reference Bonatti, Díaz and Viana13, Appendix B]) and its proof will be omitted.

3.1 The Lorenz attractor

Let us begin with the geometric model of the Lorenz attractor, see e.g. [Reference Bonatti, Díaz and Viana13, Reference Guckenheimer and Holmes30, Reference Guckenheimer and Williams31]. Let $(x_1,x_2,x_3)$ be a Cartesian coordinate system in ${\mathbb R}^3$ . Let $B^3$ be a ball in $\mathbb {R}^3$ centered at the origin $O=(0,0,0)$ . We shall consider a $C^1$ vector field $X^0$ on ${\mathbb R}^3$ such that it is transverse to the boundary of $B^3$ and that the ball $B^3$ is an attracting region, that is, $\phi _t^{X^0}(x)\in B^3$ for any $x\in B^3$ and $t>0$ . Moreover, the following properties are assumed (see [Reference Araújo and Pacifico4, §3]).

(P1) (Lorenz-like singularity). The origin O is a hyperbolic singularity of stable index 2 such that its local stable manifold $W^s_{\mathrm {loc}}(O)$ coincides with the $x_1x_3$ -plane, and its local unstable manifold $W^u_{\mathrm {loc}}(O)$ coincides with the $x_2$ -axis. Moreover, the two-dimensional stable subspace $E^s_O$ decomposes into a dominated splitting $E^s_O=E^{ss}_O\oplus E^{c}_O$ . Assume that the local strong stable manifold (tangent to $E^{ss}_O$ ) coincides with the $x_1$ -axis. The three Lyapunov exponents at the singularity are

$$ \begin{align*} \unicode{x3bb}^s=\log\|D\Phi^{X^0}_1|_{E^{ss}_O}\|,\quad \unicode{x3bb}^c=\log\|D\Phi^{X^0}_1|_{E^{c}_O}\|\quad \text{and}\quad \unicode{x3bb}^u=\log\|D\Phi^{X^0}_1|_{E^{u}_O}\|. \end{align*} $$

Here, $E^u_O$ is the unstable subspace of O. One has $\unicode{x3bb} ^s<\unicode{x3bb} ^c<0<\unicode{x3bb} ^u$ . We assume that $\unicode{x3bb} ^c+\unicode{x3bb} ^u>0$ , which implies sectional expanding property of the subspace $E^{cu}_O=E^{c}_O\oplus E^u_O$ .

(P2) (Cross-section and the first return map). The square $\Sigma _0=\{(x_1,x_2,1): -1\leq x_1, x_2\leq 1\}$ is a cross-section of the flow $\phi ^{X^0}_t$ , meaning that the vector field at every point of $\Sigma _0$ is transverse to $\Sigma _0$ . For simplicity, we assume that the vector field is orthogonal to $\Sigma _0$ and $\|X^0(x)\|=1$ for any $x\in \Sigma _0$ . The cross-section $\Sigma _0$ intersects $W^s_{\mathrm {loc}}(O)$ at a line segment $L_0=\{(x_1,0,1): -1\leq x_1\leq 1\}$ , which cuts $\Sigma _0$ into a left part $\Sigma _0^-$ and a right part $\Sigma _0^+$ , $\Sigma _0\setminus L_0=\Sigma _0^-\cup \Sigma _0^+$ . There is a first return map $R_0: \Sigma _0\setminus L_0\to \Sigma _0$ such that the images $R_0(\Sigma _0^-)$ and $R_0(\Sigma _0^+)$ are each a square pinched at one end and contained in the interior of $\Sigma _0$ , as shown in Figure 1. Let us assume that for any $x\in \Sigma _0\setminus L_0$ , the time $t_x>0$ satisfying $\phi ^X_{t_x}(x)=R_0(x)$ for its first return is at least 2, that is, $t_x>2$ .

Figure 1

The Lorenz attractor.

(P3) (Cone field on the cross-section). For each $\alpha>0$ , there is a cone $C_{\alpha }(x)$ at the point $x=(0,0,1)$ :

$$ \begin{align*}C_{\alpha}(x)=\{(x_1,x_2)\in T_{x}\Sigma_0 : \|x_1\|\leq \alpha\|x_2\|\}.\end{align*} $$

Here, we have identified $T_x\Sigma _0$ with $\Sigma _0$ and use the same coordinates $x_1,x_2$ . By translating $C_{\alpha }(x)$ to every other point on $\Sigma _0$ , one defines a cone field $C_{\alpha }$ on $\Sigma _0$ . We assume that $R_0$ preserves the cone $C_{\alpha }$ with $\alpha =1$ . More precisely, for any $x\in \Sigma _0\setminus L_0$ and any vector $u\in C_1(x)$ , we assume $DR_0(u)\in C_{1/2}(R_0(x))$ .

(P4) (The attractor). Let $\Lambda _0=\mathrm { Cl}(\phi ^{X^0}_{t\in \mathbb {R}}(\Lambda _{\Sigma _0}))$ , where $\Lambda _{\Sigma _0}=\bigcap _{n\geq 0}\overline {R_0}^n(\Sigma _0)$ and $\overline {R_0}^n(\Sigma _0)$ is the closure of the nth return of $\Sigma _0$ , ignoring $L_0$ for each return. Then, $\Lambda _0$ contains O and is the unique attractor in $B^3$ , with an attracting neighborhood $U_{\Lambda _0}\subset B^3$ that contains $\Sigma _0$ .

(P5) (Singular hyperbolicity). The attractor $\Lambda _0$ admits a singular hyperbolic splitting $T_{\Lambda _0}{\mathbb R}^3=E^{ss}\oplus E^{cu}$ , where $E^{ss}$ is uniform contracted and $E^{cu}$ is sectionally expanded. Precisely, we assume that for any two-dimensional subspace $S\subset E^{cu}_x$ , $x\in \Lambda _0$ , it holds

(3.1) $$ \begin{align} \Big|\!\det(\Phi^{X^0}_t|_S)\Big|>e^{\gamma t}\quad \text{for all } t\geq 1, \end{align} $$

where $\gamma>0$ is a constant. The constant $\gamma $ will be assumed to be large so that the inequality (3.1) implies expanding property for the first return map $R_0$ : there exists $\rho>1$ such that for any $x\in \Lambda _0\cap (\Sigma _0\setminus L_0)$ , $v\in E^{cu}_x\cap (T_x\Sigma _0\setminus \{0\})$ , it holds

$$ \begin{align*}\|DR_0(v)\|>\rho\|v\|.\end{align*} $$

As the flow direction $X^0$ is invariant and cannot be uniformly contracted, it is contained in the subbundle $E^{cu}$ , that is, ${X^0}(x)\in E^{cu}_x$ for every $x\in \Lambda _0\setminus \operatorname {Sing}(X)$ (see [Reference Bonatti, Gan and Yang15, Lemma 3.4]). At the singularity O, we have $E^{cu}_O=E^{c}_O\oplus E^u_O$ . Define

$$ \begin{align*}\unicode{x3bb}^s_0=\log\|D\Phi^{X^0}_1|_{E^{ss}_{\Lambda_0}}\|.\end{align*} $$

Note that $\unicode{x3bb} ^s\leq \unicode{x3bb} ^s_0$ . We assume $\unicode{x3bb} ^s_0<\unicode{x3bb} ^c$ .

(P6) (Stable foliation). The stable foliation $\mathcal {W}^s(\Lambda _0)=\{W^s(\operatorname {Orb}(x)): x\in \Lambda _0\}$ is $C^1$ (see [Reference Araújo and Melbourne3]), which induces a $C^1$ foliation $\mathcal {F}^s$ on $\Sigma _0$ such that for any $x\in \Sigma _0$ , $\mathcal {F}^s(x)$ is transverse to $C_1(x)$ . Since $L_0$ is the intersection of $W^s_{\mathrm {loc}}(O)$ with $\Sigma _0$ , it is a leaf of $\mathcal {F}^s$ .

(P7) (Transitivity). The attractor $\Lambda _0$ is a homoclinic class (see [Reference Bautista8, Reference Guckenheimer and Williams31]), so that $\Lambda _0$ is transitive and there is a periodic orbit $Q_0\subset \Lambda _0$ whose stable manifold is dense in $U_{\Lambda _0}$ .

3.2 The skew product construction

Let $\Omega =B^3\times I$ , where $I=[-1,1]$ . Let x, s be the coordinates on $B^3$ and I, respectively. Define on $\Omega $ the vector field $\hat {X}(x,s)=(X^0(x),-\theta s)$ , where $\theta>0$ is a constant. As the fiber direction (along I) is uniformly contracting, there exists a unique attractor $\Lambda =\Lambda _0\times \{0\}$ , which is a trivial construction of the Lorenz attractor in dimension 4. All properties of the three-dimensional attractor $\Lambda _0$ can be easily generalized to $\Lambda $ . In the following, we will modify the vector field $\hat {X}$ along fibers to obtain a vector field X such that $\Lambda $ remains an attractor but not singular hyperbolic. This is why we refer to the example as being derived from the Lorenz attractor. It may be of interest to compare our example with the construction of Lorenz attractors with arbitrary expanding dimensions presented by Bonatti et al [Reference Bonatti, Pumariño and Viana17].

The modification will be done in a small neighborhood of a periodic orbit in $\Lambda $ . Let $P_0\subset \Lambda _0$ be a periodic orbit of $X^0$ other than $Q_0$ such that it is homoclinically related to $Q_0$ . Let $P=P_0\times \{0\}$ and $Q=Q_0\times \{0\}$ . Consider a small neighborhood $V_P$ of P contained in $U_{\Lambda _0}\times (-1,1)$ such that $\overline {V_P}\cap (Q\cup \{\sigma \})=\emptyset $ , where $\sigma =(O,0)$ . Let $\eta $ be a $C^{\infty }$ function on $M_0$ that satisfies the following conditions:

• • $0\leq \eta (x,s)\leq 1$ for all $(x,s)\in \Omega $ ;

• • $\eta $ is supported on $V_P$ , that is, $\eta (x,s)=0$ for any $(x,s)\not \in V_P$ ;

• • $\eta (x,s)=1$ if and only if $(x,s)\in P$ .

The vector field X on $\Omega $ is defined as the following:

(3.2) $$ \begin{align} X(x,s)=(X^0(x), -\theta s(1-\eta(x,s))). \end{align} $$

Observe that the dynamics of the vector field is contracting along the fibers, except only for P where it is neutral. Also, $U_{\Lambda }=U_{\Lambda _0}\times (-1,1)$ is an attracting region of X and the maximal invariant set $\Lambda =\Lambda _0\times \{0\}$ in $U_{\Lambda }$ is an attractor. The attractor contains a unique singularity $\sigma =(O,0)$ . The tangent space at $\sigma $ admits a $\Phi ^X_t$ -invariant splitting $T_{\sigma }\Omega =T_{\sigma }B^3\oplus {\mathbb R}^I$ , in which we identify $B^3$ with $B^3\times \{0\}$ and ${\mathbb R}^I$ is the subspace corresponding to the fiber. Thus, the Lyapunov exponents of $\sigma $ are $\unicode{x3bb} ^s, \unicode{x3bb} ^c, \unicode{x3bb} ^u$ , and $-\theta $ . Note that $-\theta $ is the Lyapunov exponent of the singularity along the fiber direction. We assume further that

(3.3)

This implies in particular that there is a dominated splitting of the stable subspace $E^s_{\sigma }=E^{ss}\oplus {\mathbb R}^I\oplus E^{cs}$ . The following properties can be easily verified.

1. (C1) The cube $\Sigma =\Sigma _0\times I$ is a cross-section of the flow $\phi ^X_t$ . The local stable manifold of $\sigma $ cuts $\Sigma $ along a two-dimensional disk $L=L_0\times I$ . So $\Sigma \setminus L$ consists of a left part $\Sigma ^-=\Sigma _0^-\times I$ and a right part $\Sigma ^+=\Sigma _0^+\times I$ . Let $R:\Sigma \setminus L\to \Sigma $ be the first return map. Then, $R(\Sigma ^-)$ is a cube pinched at one end and contained in the interior of $\Sigma $ , and similarly for $R(\Sigma ^+)$ . For any $p\in \Sigma \setminus L$ , let $t_p>0$ be the smallest time that $\phi ^X_{t_p}(p)=R(p)$ . By assumption on $R_0$ , we have $t_p>2$ for any $p\in \Sigma \setminus L$ .

2. (C2) Q is a hyperbolic periodic orbit contained in $\Lambda $ with stable index 2. The stable manifold $W^s(Q,X)$ is dense in $U_{\Lambda }$ . This follows from the fact that the stable manifold of the periodic orbit $Q_0$ is dense in $U_{\Lambda _0}$ and that the tangent flow along the fibers is topologically contracting. Moreover, $W^s(Q,X)$ intersects $\Sigma $ along a dense family of two-dimensional $C^1$ disks, which are leaves of the foliation $\mathcal {F}^s\times I$ . Here, $\mathcal {F}^s$ is the foliation on $\Sigma _0$ as in Property (P6).

3. (C3) $P\subset \Lambda $ is a non-hyperbolic periodic orbit of X: it has a zero exponent along the fiber direction. Nonetheless, P has a three-dimensional topologically stable manifold and its strong stable manifold $W^{ss}(P,X)$ (two-dimensional) has non-empty intersection with the unstable manifold of Q. Also, its unstable manifold has a transverse intersection with the stable manifold of Q.

3.3 Robust properties of the example

As a first result, we show that $\Lambda $ admits a partially hyperbolic splitting.

Note that the splitting $F=E^{cu}\oplus {\mathbb R}^I$ is invariant but not dominated (in F). This is because the bundle ${\mathbb R}^I$ is neutral on the periodic orbit $P=P_0\times \{0\}$ and cannot be dominated by the bundle $E^{cu}$ which contains the flow direction, and vice versa. Also, F is not sectionally expanded; hence, $\Lambda $ is not singular hyperbolic. Nonetheless, the invariance of the splitting $T_{\Lambda }\Omega =E^{ss}\oplus E^{cu}\oplus {\mathbb R}^I$ suggests also an invariant splitting of the extended normal bundle with one-dimensional subbundles.

Here, a chain recurrence class is called locally away from homoclinic tangencies if there exists a neighborhood U of the class and a $C^1$ neighborhood $\mathcal {U}$ of X such that any $Y\in \mathcal {U}$ admits no homoclinic tangency in U.

Also, the sectionally expanding subbundle $E^{cu}$ in the splitting $T_{\Lambda }\Omega =E^{ss}\oplus E^{cu}\oplus {\mathbb R}^I$ will be shown to exist robustly in the following sense.

The proofs of Propositions 3.2 and 3.3 will be given in §§4 and 5, respectively.

By construction, the chain recurrence class $\Lambda $ is an attractor containing the unique singularity $\sigma =(O,0)$ . We will show that for any $C^1$ vector field Y close enough to X, the chain recurrence class $C(\sigma _Y,Y)$ is non-trivial and Lyapunov stable. Moreover, it is the only Lyapunov stable chain recurrence class in $U_{\Lambda }$ , where $\sigma _Y$ is the continuation of the singularity. Precisely, we have the following proposition.

The proof of Proposition 3.4 will be given in §6.

3.4 Robust heterodimensional cycles

Before we continue with the construction of the example, let us recall that a vector field is said to have a heterodimensional cycle associated with two transitive hyperbolic sets $\Gamma $ and $\Gamma '$ if:

• • the indices of $\Gamma $ and $\Gamma '$ are different; and

• • the stable manifold of $\Gamma $ intersects the unstable manifold of $\Gamma '$ and vice versa.

The heterodimensional cycle is robust if there is a $C^1$ -neighborhood $\mathcal {U}$ of the vector field such that every $Y\in \mathcal {U}$ has a heterodimensional cycle associated with the hyperbolic sets $\Gamma _{Y}$ and $\Gamma ^{\prime }_Y$ , where $\Gamma _{Y}$ and $\Gamma ^{\prime }_Y$ are the continuations of $\Gamma $ and $\Gamma '$ , respectively.

We will prove the following result.

The proof of Lemma 3.5 is essentially contained in [Reference Bonatti and Díaz11, §4]. We say that a periodic orbit is a pre-saddle if its normal bundle has an (orientable) one-dimensional center along which the Lyapunov exponent is zero, and all other Lyapunov exponents are non-zero. For the vector field X, the periodic orbit P is a pre-saddle with a one-dimensional center in the fiber direction, along which the Lyapunov exponent is zero. The pre-saddle P has a strong stable manifold $W^{ss}(P,X)$ contained in $B^3\times \{0\}$ (as in the three-dimensional Lorenz attractor). Also, its unstable manifold $W^u(P,X)$ is contained in $B^3\times \{0\}$ . By the skew product construction of our example, $P=P_0\times \{0\}$ and $P_0$ exhibits homoclinic intersections, which are exactly the intersections of $W^{ss}(P,X)$ and $W^u(P,X)$ . Such intersections will be called strong homoclinic intersections associated to P. Moreover, each strong homoclinic intersection r is quasi-transverse: $\operatorname {dim} (T_rW^{ss}(P,X)\oplus T_rW^u(P,X))=3<\operatorname {dim} \Omega =4$ . The next result shows how to obtain robust heterodimensional cycles. See also [Reference Li and Turaev33] for more recent progress on this topic.

It follows from the previous discussions and Lemma 3.7 that there exists vector field Y arbitrarily close to X which exhibits a robust heterodimensional cycle. Thus, to prove Lemma 3.5, it remains to show that the robust heterodimensional cycle is contained in $C(\sigma _{Y},Y)$ . For this, we need to slightly change the proof of Lemma 3.7 [Reference Bonatti and Díaz11, §4], in which a key ingredient is the creation of robust heterodimensional cycles through blenders. Roughly speaking, a blender is a hyperbolic set whose stable (or unstable) set behaves as if it is one dimension higher than it actually is. One can refer to e.g. [Reference Bonatti, Díaz and Viana13, Ch. 6] for more discussions on blenders.

Proof of Lemma 3.5

We have seen that the periodic orbit P is a pre-saddle associated to which there is a quasi-transverse strong homoclinic intersection. In fact, we can take two quasi-transverse strong homoclinic intersections z and w associated to P such that their orbits are disjoint. This is because the strong homoclinic intersections correspond to the homoclinic intersections of $P_0$ , which are known to have infinitely many points with mutually disjoint orbits. Following the proof of Lemma 3.7 [Reference Bonatti and Díaz11, §4], one can make arbitrarily small $C^1$ perturbations in a neighborhood of $\mathrm {Cl}(\operatorname {Orb}(z,X)\cup \operatorname {Orb}(w,X))$ (containing P) to obtain a vector field Y having a blender $\Lambda '$ , together with a hyperbolic periodic orbit $P_{\delta }$ close to $P_Y$ such that:

• • the blender $\Lambda '$ is a hyperbolic set of stable index $1$ containing the periodic orbit $P_Y$ ;

• • the periodic orbit $P_{\delta }$ has stable index $2$ , and its stable manifold intersect transversely the unstable manifold of $P_Y$ ;

• • the unstable manifold of $P_{\delta }$ meets robustly the stable set of the blender $\Lambda '$ (by the property of the blender);

• • thus, the vector field Y has a robust heterodimensional cycle associated with the hyperbolic set $\Lambda '\supset P_Y$ and the hyperbolic saddle $P_{\delta }$ .

One can see that $C(\sigma _Y,Y)$ contains the robust heterodimensional cycle if it contains $P_{\delta }$ .

We can assume that Y is close enough to X so that by Proposition 3.4, $C(\sigma _Y,Y)$ is Lyapunov stable and contains the periodic orbit $Q_Y$ . Thus, to guarantee that $C(\sigma _Y,Y)$ contains $P_{\delta }$ and hence the robust heterodimensional cycle, we need only show that $W^u(Q_Y,Y)\pitchfork W^s(P_{\delta },Y)\neq \emptyset $ . A slightly different procedure should be followed when making the perturbations to obtain the robust heterodimensional cycle.

Recall that in [Reference Bonatti and Díaz11, §4], $P_{\delta }$ is obtained as a twin of P by blowing up the pre-saddle P along the neutral center, as shown in Figure 2(a). Instead of obtaining only a twin $P_{\delta }$ , we blow up the pre-saddle to obtain a set of triplet saddles, P, $P_{\delta }$ , and $P_{-\delta }$ , as shown in Figure 2(b). See also [Reference Mañé37] for a similar construction. The saddles $P_{\delta }$ and $P_{-\delta }$ are both of index 2, while P has index 1. Note that for the vector X, there is a quasi-transverse intersection between $W^u(Q,X)$ and $W^{ss}(P,X)$ . After the blow-up, one has either $W^u(Q_Y,Y)\pitchfork W^s(P_{\delta },Y)\neq \emptyset $ or $W^u(Q_Y,Y)\pitchfork W^s(P_{-\delta },Y)\neq \emptyset $ . By interchanging the notation, we can assume without loss of generality that $W^u(Q_Y,Y)\pitchfork W^s(P_{\delta },Y)\neq \emptyset $ . Then, the construction continues as in [Reference Bonatti and Díaz11, §4].

Figure 2

DA-type surgery at the pre-saddle P.

As argued before, the construction ensures that $C(\sigma _Y,Y)$ contains the robust heterodimensional cycle associated with a blender and the hyperbolic saddle $P_{\delta }$ . Note also that this property for Y is robust. The proof of Lemma 3.5 is now complete as Y can be chosen to be arbitrarily $C^1$ close to X.

3.5 Proof of Theorem 1.2

Let M be any four-dimensional closed Riemannian manifold. One considers a gradient-like vector field $Z_0$ on M, which has at least a sink. Consider a local chart around a sink and a ball $B^4$ in the chart, centered at the sink. By shrinking the local chart, we assume that the ball $B^4$ is an attracting region of the flow. In particular, the vector field $Z_0$ is transverse to $\partial B^4$ and inwardly pointing. By changing the metric, we assume that $\tfrac 12B^4$ contains $\Omega =B^3\times [-1,1]$ . Let X be a vector field on $\Omega $ as constructed in §§3.1 and 3.2. Let us denote by $\tfrac 12B^4$ the ball with the same center of $B^4$ and half the radius. One then modifies the vector field $Z_0$ in a neighborhood of $\tfrac 12B^4$ to obtain a $C^1$ vector field Z such that it coincides with X on $\Omega \subset \tfrac 12B^4$ . The modification shall be taken outside a neighborhood of $M\setminus B^4$ ; thus, Z coincides with $Z_0$ in a neighborhood of $M\setminus B^4$ . Moreover, since the vector field on $\Omega $ is transverse to each smooth component of $\partial \Omega $ and inwardly pointing, one can require that the obtained vector field Z has no recurrence in $B^4\setminus \Omega $ . Then, the chain recurrence set of Z is composed of finitely many hyperbolic critical elements and the maximal invariant set in $\Omega $ .

The following properties can be verified.

1. (1) There is a non-trivial chain recurrence class $C(\sigma ,Z)$ associated to a hyperbolic singularity $\sigma \in \Omega $ .

2. (2) There exist a neighborhood $U\subset \Omega $ of $C(\sigma ,Z)$ and a neighborhood $\mathcal {U}$ of Z such that for any $Y\in \mathcal {U}$ , the continuation $C(\sigma _Y,Y)$ is well defined and contained in U. In particular, $\sigma _Y$ is the only singularity in U. Moreover, since $C(\sigma ,Z)$ is an attractor, U can be chosen to be an attracting neighborhood of $C(\sigma ,Z)$ . By reducing $\mathcal {U}$ , we assume that U remains an attracting region for any $Y\in \mathcal {U}$ and the chain recurrent set of Y is composed of finitely many hyperbolic critical elements plus the maximal invariant set in U.

3. (3) For any $Y\in \mathcal {U}$ , there exists no homoclinic tangency in U (Proposition 3.2). As the chain recurrent set of Y outside U is composed of finitely many hyperbolic critical elements, one concludes that Y is away from homoclinic tangencies.

4. (4) There is a partially hyperbolic splitting $T_{C(\sigma _Y,Y)}M=E^{ss}\oplus F$ with $\operatorname {dim} E^{ss}=1$ (Lemma 3.1). The subbundle F contains a continuous two-dimensional subbundle $E^Y\subset T_{C(\sigma _Y,Y)}M$ containing the flow direction such that it is invariant for $\Phi ^Y_t$ and sectionally expanded. Moreover, Proposition 3.3 gives the continuity of $E^Y$ as stated at the end of Theorem 1.2.

5. (5) There exists a unique Lyapunov stable chain recurrence class in $U$ , which is $C(\sigma_Y, Y)$ (Proposition 3.4).

By Lemma 3.5, there exists an open set $\mathcal {V}\subset \mathcal {U}$ such that for any $Y\in \mathcal {V}$ , the chain recurrence class $C(\sigma _Y,Y)$ robustly contains a robust heterodimensional cycle. To finish the proof of Theorem 1.2, it remains to show that for any $Y\in \mathcal {V}$ , the subbundle $E^Y$ is tangled. Arguing by contradiction, if $E^Y$ is not tangled, then there exists a dominated splitting $F=E^Y\oplus E'$ or $F=E'\oplus E^Y$ . In the first case, since $E^Y$ is sectionally expanded and is dominated by $E'$ , one deduces that $E'$ is expanded. It follows that $C(\sigma _Y,Y)$ is singular hyperbolic, contradicting to the fact that $C(\sigma _Y,Y)$ contains a robust heterodimensional cycle. In the second case, since $E'$ is dominated by $E^Y$ which contains the flow direction, one deduces that $E'$ should be contracted. Then, $C(\sigma _Y,Y)$ would also be singular hyperbolic, which is a contradiction again.

This completes the proof of Theorem 1.2.

6.1 Dominated splitting and cone field on the normal bundle

For simplicity, we assume that for the dominated splitting $\widetilde {\mathcal {N}}_{\mathcal {F}(\Lambda )}=\mathcal {N}_1\oplus \mathcal {N}_2$ , the domination constant $T=1$ . In other words,

$$ \begin{align*} \|\tilde{\psi}_t|_{\mathcal{N}_1(l)}\|\cdot\|\tilde{\psi}_{-t}|_{\mathcal{N}_2(l_t)} \|<\tfrac{1}{2}\quad\text{for all } t\geq 1, \end{align*} $$

where $l_t=\hat {\Phi }_t(l)$ . We extend the splitting $\widetilde {\mathcal {N}}_{\mathcal {F}(\Lambda )}=\mathcal {N}_1\oplus \mathcal {N}_2$ to a neighborhood $\mathcal {B}_0$ of $\mathcal {F}(\Lambda )$ in a continuous way. Define on $\mathcal {B}_0$ a cone field $\mathcal {C}_{\alpha }$ on the normal bundle around the $\mathcal {N}_2$ subbundle for any $\alpha>0$ :

$$ \begin{align*}\mathcal{C}_{\alpha}=\{v=v_1+v_2\in\mathcal{N}_{\mathcal{B}_0}: v_1\in\mathcal{N}_1, v_2\in \mathcal{N}_2, \|v_1\|\leq\alpha\|v_2\|\}.\end{align*} $$

We define also the cone $\widetilde {\mathcal {C}}_{\alpha }=\mathcal {B}_0\oplus \mathcal {C}_{\alpha }$ . (Technically, $\widetilde {\mathcal {C}}_{\alpha }$ is not a cone, but a ‘wedge’.) In other words, $v\in \widetilde {\mathcal {C}}_{\alpha }$ if and only if there is a decomposition $v=v_1+v_2$ such that $v_1\in \mathcal {B}_0$ and $v_2\in \mathcal {C}_{\alpha }$ . By domination of the splitting $\mathcal {N}_1\oplus \mathcal {N}_2$ and continuity, we have the following lemma.

Lemma 6.1. For any $\alpha>0$ , there exist a neighborhood $\mathcal {U}$ of X and a neighborhood $\mathcal {B}\subset \mathcal {B}_0$ of $\mathcal {F}(\Lambda )$ such that for any any $Y\in \mathcal {U}$ and $l\in \mathcal {B}$ , if $\hat {\Phi }^Y_{[0,t]}(l)\subset \mathcal {B}$ with $t\geq 1$ , then

$$ \begin{align*}\tilde{\psi}^Y_t(\mathcal{C}_{\alpha}(l))\subset\mathcal{C}_{\alpha/2}(\hat{\Phi}^Y_t(l)) \quad \text{and}\quad \tilde{\Phi}^Y_t(\widetilde{\mathcal{C}}_{\alpha}(l))\subset\widetilde{\mathcal{C}}_{\alpha/2}(\hat{\Phi}^Y_t(l)).\end{align*} $$

Note that the vector bundle $\mathcal {F}(\Lambda )\oplus \mathcal {N}_2$ is invariant for the extended tangent flow $\tilde {\Phi }_t$ . By Remark 4.4, $\beta _*(\mathcal {F}(\Lambda )\oplus \mathcal {N}_2)=E^{cu}_{\Lambda }$ , which is sectionally expanded by the tangent flow. Following from the continuity of the map $\beta _*$ and by reducing $\mathcal {B}_0$ , we can fix $\alpha _0>0$ small and a neighborhood $\mathcal {U}_0$ of X such that for any $Y\in \mathcal {U}_0$ and $l\in \mathcal {B}_0$ , if $\hat {\Phi }^Y_{[0,t]}(l)\subset \mathcal {B}_0$ with $t\geq 1$ , then for any two-dimensional subspace $\mathcal {P}\subset \widetilde {\mathcal {C}}_{\alpha _0}(l)$ , the projection $\beta _*(\mathcal {P})$ is close enough to $E^{cu}_{\Lambda }$ and hence,

(6.1) $$ \begin{align} |\!\det(\Phi^Y_t|_{\beta_*(\mathcal{P})})|>e^{\gamma t/2}, \end{align} $$

where $\gamma>0$ is given as in (3.1).

We assume that the results of Lemma 6.1 hold for the fixed $\alpha =\alpha _0$ with $\mathcal {U}=\mathcal {U}_0$ and $\mathcal {B}=\mathcal {B}_0$ . Moreover, by Lemma 2.3, we assume that

(6.2) $$ \begin{align} W^{ss}(\sigma_Y,Y)\cap C(\sigma_Y,Y)=\{\sigma_Y\} \quad \text{for any } Y\in\mathcal{U}_0, \end{align} $$

where $W^{ss}(\sigma _Y,Y)$ is the strong stable manifold of $\sigma _Y$ tangent to $E^{ss}_{\sigma }\oplus {\mathbb R}^I$ .

6.2 A forward invariance of $\mathcal {B}_0$

Recall that there is a cross-section $\Sigma =\Sigma _0\times I$ of the flow $\phi ^X_t$ . One can see that $\Sigma $ remains a cross-section for vector fields close to X. In this section, we prove the following lemma.

For the proof of Lemma 6.2, we need to consider orbit segments arbitrarily close to the singularity. Note that the tangent space at the singularity $\sigma $ admits a dominated splitting $T_{\sigma }\Omega =F^{ss}_{\sigma }\oplus E^{cu}_{\sigma }$ , where $F^{ss}_{\sigma }=E^{ss}_{\sigma }\oplus {\mathbb R}^I$ ; and by construction, it holds $\mathcal {F}(\Lambda )\cap T_{\sigma }\Omega =E^{cu}_{\sigma }$ . One defines on $T_{\sigma }\Omega $ a $cu$ -cone $C^{cu}_{\alpha }$ around the $E^{cu}_{\sigma }$ bundle. By domination, the cone $C^{cu}_{\alpha }$ is forward contracting. We extend the cone continuously to a ball around $\sigma $ , which is denoted by $W_0$ . Let K be the intersection of $\bigcup _{t\geq 0}\phi ^X_t(L)$ with $\partial W_0$ , which is a compact subset of $\partial W_0$ . We assume that the ball $W_0$ is small such that $W_0\cap \Sigma =\emptyset $ and since the singularity is Lorenz-like, there exists $\alpha>0$ such that for any $p\in K$ , it holds $l^X_p\subset C^{cu}_{\alpha }(p)$ . By reducing $\mathcal {U}_0$ if necessary, there exists a neighborhood $N_K\subset \partial W_0$ of K such that $l^Y_p\subset C^{cu}_{\alpha }(p)$ for any $p\in N_K$ and $Y\in \mathcal {U}_0$ . Moreover, we assume that the cone $C^{cu}_{\alpha }$ is forward contracting under the tangent flow $\Phi ^Y_t$ for any $Y\in \mathcal {U}_0$ .

Proof. Since $\Sigma $ can be a cross-section for any vector field $C^1$ close to X and its image under the first return map remains in the interior of $\Sigma $ , we need only consider orbit segments from $p\in \Sigma $ to $q\in \overline {W_1}$ without returns to $\Sigma $ . Since $N_K$ is a neighborhood of K, there exists a neighborhood $W_1\subset W_0$ of $\sigma $ such that any orbit segment S of X from a point $p\in \Sigma $ to a point $q\in \overline {W_1}$ (without returns) crosses $N_K$ . Moreover, let r be the intersection of S with $N_K$ , then the orbit segment from r to q is contained in $\overline {W_0}$ . See Figure 3. Shrinking $W_1$ if necessary, there exists a neighborhood $\mathcal {U}_1\subset \mathcal {U}_0$ of X such that the same property holds for every $Y\in \mathcal {U}_1$ . The following argument shows that $l^Y_q\in \mathcal {B}_0$ if we choose $W_1$ small enough.

Figure 3

Local analysis around the singularity. Note that the strong stable direction is two-dimensional.

Since $E^{cu}_{\sigma }\subset \mathcal {F}(\Lambda )$ and $\mathcal {B}_0$ is a neighborhood of $\mathcal {F}(\Lambda )$ , we can assume that $W_1$ is small enough and there exists a constant $\alpha '>0$ such that for any $q\in \overline {W_1}$ and non-zero vector $v\in T_q\Omega $ , if $v\in C^{cu}_{\alpha '}(q)$ , then $\langle v\rangle \in \mathcal {B}_0$ ; and if the backward orbit of q hits $N_K$ at a point r before leaving $W_0$ , then the time t from r to q is large enough so that $\Phi ^Y_t(C^{cu}_{\alpha }(r))\subset C^{cu}_{\alpha '}(q)$ , by the contracting property of the cone. In particular, since $l^Y_r\subset C^{cu}_{\alpha }(r)$ , we have $l^Y_q=\hat {\Phi }^Y_t(l^Y_r)\subset C^{cu}_{\alpha '}(q)$ . It follows that $l^Y_q\in \mathcal {B}_0$ .

The next result considers orbit segments away from the singularity.

Proof. Reducing $\mathcal {U}_1$ if necessary, we can take a neighborhood $W^{\prime }_1\subset W_1$ of $\sigma $ such that for any $p\in W^{\prime }_1$ and any $Y\in \mathcal {U}_1$ , we have $\phi ^Y_t(p)\in W_1$ for all $t\in [-1,1]$ . Since $\sigma $ is the only singularity in $\Lambda $ , we have

(6.3) $$ \begin{align} \mathcal{F}(\Lambda)\setminus\beta^{-1}(W^{\prime}_1)=\{l^X_p: p\in \Lambda\setminus W^{\prime}_1\}. \end{align} $$

By construction of the vector field X, there exist arbitrarily small attracting neighborhoods of $\Lambda $ . Following from (6.3), we can take an attracting neighborhood $U_1$ of $\Lambda $ and reduce the neighborhood $\mathcal {U}_1$ of X such that $\phi ^Y_1(\overline {U_1})\subset U_1$ and $l^Y_p\in \mathcal {B}_0$ for any $Y\in \mathcal {U}_1$ and for any $p\in \overline {U_1}\setminus W^{\prime }_1$ . This implies that for all $p\in \overline {U_1}\setminus W_1$ and $t\in [0,1]$ , we have $l^{Y}_{p_t}\in \mathcal {B}_0$ , and hence by Lemma 6.1, $\tilde {\psi }^{Y}_1(\mathcal {C}_{\alpha _0}(l^{Y}_p))\subset \mathcal {C}_{\alpha _0/2}(l^{Y}_{p_1})$ .

We can now finish the proof of Lemma 6.2.

6.3 Invariant cone on the cross-section and expanding property

Recall that $R:\Sigma \setminus L\to \Sigma $ is the first return map, where L is the intersection of the local stable manifold of $\sigma $ with $\Sigma $ (§3.2). For any $Y\in \mathcal {U}_1$ , let $L_Y=W^s_{\mathrm {loc}}(\sigma _Y,Y)\cap \Sigma $ and $R^Y:\Sigma \setminus L_Y\to \Sigma $ be the first return map. Let $TR^Y$ be the tangent map of $R^Y$ . For simplicity, let us denote

$$ \begin{align*}\Sigma^*=\Sigma\cap \overline{U_1}.\end{align*} $$

We define in this section a cone on $\Sigma ^*$ such that it is invariant under $TR^Y$ and moreover, vectors in the cone will be shown to be expanded by $TR^Y$ . This allows us to consider curves tangent to the cone on $\Sigma ^*$ and show that the length of such curves increase under the iteration of the first return map.

In the following, for any $0<\alpha \leq \alpha _0$ and $p\in \overline {U_1}\setminus \{\sigma _Y\}$ such that $l^Y_p\in \mathcal {B}_0$ , we denote $\mathcal {C}^Y_{\alpha }(p)=\beta _*(\widetilde {\mathcal {C}}_{\alpha }(l^Y_p))$ , which is a cone on the tangent space $T_p\Omega $ . We then define a cone field $\mathcal {D}^Y_{\alpha }$ on $\Sigma ^*$ by letting $\mathcal {D}^Y_{\alpha }(p)=\mathcal {C}^Y_{\alpha }(p)\cap T_p\Sigma $ . By shrinking $U_1$ and $\mathcal {U}_1$ , the following result holds.

Proof. Note that restricted to $\Sigma ^*$ , the cone $\mathcal {C}^Y_{\alpha }(p)$ and hence, $\mathcal {D}^Y_{\alpha }(p)$ are uniformly continuous with respect to the vector field Y in the $C^1$ topology. Let us fix $\alpha _1\in (\alpha _0/2,\alpha _0)$ . Then, by reducing $\mathcal {U}_1$ , we can assume that for any $Y\in \mathcal {U}_1$ , it holds

$$ \begin{align*}\mathcal{D}^Y_{\alpha_0/2}(p)\subset \mathcal{D}^X_{\alpha_1}(p)\subset \mathcal{D}^Y_{\alpha_0}(p).\end{align*} $$

Also, from Property (C1) in §3.2, we can assume that $t_p>2$ for any $p\in \Sigma ^*\setminus L_Y$ . Now, for any $v\in \mathcal {D}^X_{\alpha _1}(p)\subset \mathcal {D}^Y_{\alpha _0}(p)$ , we have $(l^Y_p,v)\in \widetilde {\mathcal {C}}^Y_{\alpha _0}(l^Y_p)$ . From Lemma 6.2, it follows that $l^Y_{p_t}\in \mathcal {B}_0$ for all $t\geq 0$ . Thus, by Lemma 6.1, we have $\widetilde {\Phi }^Y_{t_p}(l^Y_p,v)\in \widetilde {\mathcal {C}}_{\alpha _0/2}(l^Y_{p'})$ , where $p'=R^Y(p)$ . Equivalently, we have $\Phi ^Y_{t_p}(v)\in \mathcal {C}^Y_{\alpha _0/2}(p')$ . Projecting $\Phi ^Y_{t_p}(v)$ to $T_{p'}\Sigma $ along $l^Y_{p'}$ , we obtain that

$$ \begin{align*}TR^Y(v)\in \mathcal{D}^Y_{\alpha_0/2}(p')\subset \mathcal{D}^X_{\alpha_1}(p').\end{align*} $$

This establishes the invariance of the cone $\mathcal {D}^X_{\alpha _1}$ .

To see that $TR^Y$ is expanding on $\mathcal {D}^X_{\alpha _1}$ , we shall refer to the sectional expanding property (6.1). Recall that in the construction of the example, we have assumed the vector field $X^0$ to be orthogonal to the cross-section $\Sigma _0$ and $\|X^0(x)\|=1$ for $x\in \Sigma _0$ , see Property (P2) in §3.1. Then, it follows from (3.2) that the vector field X is orthogonal to the cross-section $\Sigma $ at points $p\in \Sigma _0\times \{0\}$ and $\|X(p)\|=1$ . In particular, this holds for $p\in \Lambda \cap \Sigma $ . By shrinking $U_1$ and $\mathcal {U}_1$ if necessary, we can assume that for any $Y\in \mathcal {U}_1$ and any $p\in \Sigma ^*=\Sigma \cap \overline {U_1}$ , $Y(p)$ is almost orthogonal to the cross-section and $\|Y(p)\|$ is close to 1. Then, the sectional expanding property (6.1) allows us to obtain a constant $\unicode{x3bb}>1$ , independent of Y, such that $\|TR^Y(v)\|\geq \unicode{x3bb} \|v\|$ .

The previous result shows that the cone $\mathcal {D}^X_{\alpha _1}$ on $\Sigma ^*$ is invariant for $TR^Y$ , for any $Y\in \mathcal {U}_1$ . Moreover, $TR^Y$ is expanding on $\mathcal {D}^X_{\alpha _1}$ . Note the first return time $t_p$ of a point $p\in \Sigma ^*\setminus L_Y$ can be arbitrarily large if it is close enough to $L_Y$ . In this case, the sectional expanding property (6.1) will guarantee a large expansion rate for $TR^Y$ .

Lemma 6.6. There exists a neighborhood $N_L$ of L in $\Sigma $ such that for any $Y\in \mathcal {U}_1$ (reducing $\mathcal {U}_1$ if necessary), one has $L_Y\subset N_L$ and for any $p\in (N_L\setminus L_Y)\cap \Sigma ^*$ ,

(6.4) $$ \begin{align} \|TR^Y(v)\|>3\|v\|\quad \text{for all } v\in \mathcal{D}^X_{\alpha_1}(p)\setminus\{0\}. \end{align} $$

Assuming $\mathcal {U}_1$ small enough, the following result holds.