An example derived from Lorenz attractor

In the study of differentiable dynamical systems, a central problem is to understand most of the systems. From a topological viewpoint, this could mean to characterize an open and dense subset, or at least a residual subset, of all systems. It was once conjectured by Smale in the early 1960’s that an open and dense subset of all systems should be comprised of uniformly hyperbolic ones. Recall that a dynamical system is uniformly hyperbolic if its limit set consists of finitely many transitive hyperblic sets and without cycles in between. The first counterexamples to this conjecture were given by Abraham and Smale [AS] in the $C^{1}$ topology, and by Newhouse [N1, N2] in the $C^{2}$ topology. These examples give the following two obstructions to hyperbolicity:

1. 1.

   heterodimensional cycle : there exist two hyperbolic periodic orbits of different indices such that the stable manifold of one periodic orbit intersects the unstable manifold of the other and vice versa;

2. 2.

   homoclinic tangency : there exists a hyperbolic periodic orbit whose stable and unstable manifolds have a nontransverse intersection.

Bifurcations of these two mechanisms can exhibit very rich dynamical phenomena. For example, a heterodimensional cycle associated to a pair of periodic saddles with index $i$ and $j=i+1$ can be perturbed to obtain a robust heterodimensional cycle [BD]: there exist transitive hyperbolic sets $\Lambda_{1}$ and $\Lambda_{2}$ with different indices such that the stable manifold of $\Lambda_{1}$ meets the unstable manifold of $\Lambda_{2}$ in a $C^{1}$ robust way, and vice versa. For more results, see e.g. [PT, D, BDPR]. Trying to give a global view of all dynamical systems, Palis [Pa1, Pa2, Pa3, Pa4] conjectured in the 1990’s that the uniformly hyperbolic systems and the systems with heterodimensional cycles or homoclinic tangencies form a dense subset of all diffeomorphisms. The Palis conjecture has been proved by Pujals and Sambarino [PS] for $C^{1}$ diffeomorphisms on surfaces. Moreover, a weaker version, known as the weak Palis conjecture, was shown to hold for $C^{1}$ diffeomorphisms on any closed manifold [BGW, C1]. When it comes to flows, or vector fields, the Palis conjecture has to be recast so as to take into consideration the singularities. There have been many progresses on this topic, see e.g. [ARH, GY, CY1]. Also, related conjectures have been given by Bonatti [Bo] in his program for a global view of $C^{1}$ systems.

The conjectures by Palis and Bonatti have lead to the study of systems away from homoclinic tangencies and/or heterodimensional cycles. In the $C^{1}$ topology, a diffeomorphism (or a vector field) is said to be away from homoclinic tangencies if every system in a $C^{1}$ neighborhood exhibits no homoclinic tangency. It is now well understood [W, Go] that for a diffeomorphism away from homoclinic tangencies there exist dominated splittings of the tangent bundle over preperiodic sets. More delicate results can be obtained in the $C^{1}$ generic setting, see e.g. [CSY]. Also, it is known [LVY] that every $C^{1}$ diffeomorphism away from homoclinic tangencies is entropy expansive, and it follows that there exists a measure of maximum entropy. When considering diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, it is shown in [CP] that a $C^{1}$ generic system is essentially hyperbolic. See also [C2].

While there have been plenty of results on diffeomorphisms away from homoclinic tangencies (and heterodimensioinal cycles), parallel studies on vector fields are rare. Regarding the Palis conjecture, it is shown [CY1] that on every 3-manifold the singular hyperbolic vector fields form a $C^{1}$ open and dense subset of all vector fields away from homoclinic tangencies. Also in dimension 3, the weak Palis conjecture has been proved for $C^{1}$ vector fields [GY]. For higher dimensional (dim $\geq 3$) vector fields, studies are mostly limited to either the singular hyperbolic vector fields [MPP] or the star vector fields, see e.g. [SGW, PYY, CY2, dL, BdL]. General vector fields away from homoclinic tangencies (not star) on a higher dimensional manifold are rarely studied. In [GYZ], for general higher dimensional flows away from homoclinic tangencies, it is shown that $C^{1}$ generically a nontrivial Lyapunov stable chain recurrence class contains periodic orbits and hence is a homoclinic class. Their result leaves the possibility that the periodic orbits in the chain recurrence class can have different indices. When this happens robustly, such a chain recurrence class shall exhibit a robust heterodimensional cycle [BD] and the flow can not be star. Apart from this result and possibly a handful others, the world of general higher dimensional vector fields away from homoclinic tangencies is far from being understood.

Inspired by the result of [GYZ], we notice that there is no known example of a singular flow which is away from homoclinic tangencies and is not singular hyperbolic or star. The aim of this paper is to provide such an example, and to open a door to the world of general higher dimensional vector fields away from homoclinic tangencies. Hopefully it will also attract interests to the study of higher dimensional vector fields.

As a by-product, our example exhibits an interesting property which is not observed before: the tangent bundle of the obtained chain recurrence class is shown to admit robustly an invariant subbundle (containing the flow direction), which is properly included in a subbundle of the finest dominated spitting of the tangent bundle. We shall call such an invariant subbundle exceptional (see Definition 1.1). Note that there can not be any exceptional subbundle for diffeomorphisms away from homoclinic tangencies: robustness of an invariant subbundle implies a corresponding dominated spitting of the tangent bundle. Thus the existence of exceptional subbundles may be counted as an essential difference between diffeomorphisms and (singular) vector fields, and is worth studying.

Let $M$ be a compact Riemannian manifold without boundary. Given a $C^{1}$ vector field on $M$, it generates a $C^{1}$ flow $\phi^{X}_{t}$, or simply $\phi_{t}$, on the manifold. A point $x\in M$ with $X(x)=0$ is called a singularity of $X$. Any point other than singularities is called a regular point. A regular point $x$ satisfying $\phi_{T}(x)=x$ for some $T>0$ is called a periodic point, and its orbit $\operatorname{Orb}(x)=\{\phi_{t}(x):t\in{\mathbb{R}}\}$ will be called a periodic orbit. Singularities and periodic orbits are called critical elements.

Let $\Lambda$ be a compact invariant set of the flow $\phi_{t}$. It is called chain transitive if for any two points $x,y\in\Lambda$ and any constant ${\varepsilon}>0$, there exist a sequence of points $x_{0}=x,x_{1},\ldots,x_{n}=y$ in $\Lambda$ and a sequence of positive numbers $t_{0},t_{1},\ldots,t_{n-1}\geq 1$, such that $d(\phi_{t_{i}}(x_{i}),x_{i+1})<{\varepsilon}$ for all $i=0,1,\ldots,n-1$. If $\Lambda$ is maximally chain transitive, i.e. it is chain transitive and is not a proper subset of any other chain transitive set, then $\Lambda$ is called a chain recurrence class. The union of all chain recurrence classes is called the chain recurrent set, denoted by $\mathrm{CR}(X)$. For any point $x$ in $\mathrm{CR}(X)$, we denote by $C(x,X)$ the chain recurrence class containing $x$.

Denote by $\Phi_{t}$ the tangent flow of $X$, i.e. $\Phi_{t}=D\phi_{t}$. A $\Phi_{t}$-invariant subbundle $E$ of $T_{\Lambda}M$ with $\operatorname{dim}E\geq 2$ is called sectionally expanded if there exists constants $C\geq 1$, $\lambda>0$ such that for any $x\in\Lambda$ and any 2-dimensional subspace $S\subset E(x)$, it holds

$$|\det(\Phi_{-t}|_{S})|\leq C\mathrm{e}^{-\lambda t},\quad\forall t\geq 0.$$

A $\Phi_{t}$-invariant splitting $E\oplus F\subset T_{\Lambda}M$ is called dominated if there exists $C\geq 1$, $\lambda>0$ such that for any $x\in\Lambda$, it holds

$$\|\Phi_{t}|_{E(x)}\|\cdot\|\Phi_{-t}|_{F(\phi_{t}(x))}\|\leq C\mathrm{e}^{-\lambda t},\quad\forall t\geq 0.$$

We also say that a $\Phi_{t}$-invariant splitting $E_{1}\oplus E_{2}\oplus\cdots\oplus E_{k}\subset T_{\Lambda}M$ is dominated if $(E_{1}\oplus\cdots\oplus E_{i})\oplus(E_{i+1}\oplus\cdots\oplus E_{k})$ is dominated for every $i=1,\ldots,k-1$. The finest dominated splitting of $T_{\Lambda}M$ is a $\Phi_{t}$-invariant splitting $T_{\Lambda}M=E_{1}\oplus E_{2}\oplus\cdots\oplus E_{k}$ such that it is dominated and no subbundle $E_{i}$ ($i=1,\ldots,k$) can be decomposed further into a dominated splitting.

In particular, if $E\subset T_{\Lambda}M$ is exceptional, it is not a subbundle of any dominated splitting of $T_{\Lambda}M$. Recall that a compact invariant set $\Lambda$ is called Lyapunov stable if for any neighborhood $U$ of $\Lambda$ there exists a neighborhood $V$ of $\Lambda$ such that $\overline{\phi_{t}(V)}\subset U$ for all $t\geq 0$. Let $\mathscr{X}^{1}(M)$ be the set of all $C^{1}$ vector fields on $M$, endowed with the $C^{1}$ topology. We denote by $\mathcal{H}\mathcal{T}$ the set of all $C^{1}$ vector fields on $M$ exhibiting a homoclinic tangency. Then $\mathscr{X}^{1}(M)\setminus\overline{\mathcal{H}\mathcal{T}}$ consists of all $C^{1}$ vector fields away from homoclinic tangencies.

The example (in Theorem A) is derived from Lorenz attractor in the sense that the construction begins with the Lorenz attractor and uses a DA-type surgery. The Lorenz attractor, discovered by the meteorologist E. Lorenz in the early 1960’s [Lo], is among the most important examples of singular flows. It is a chaotic attractor given by a simple group of three ordinary differential equations. We will consider the geometric model of the Lorenz attractor [ABS, GW], which is equivalent to the original attractor for classical parameters [T]. The DA surgery was introduced by Smale [S] as a way of constructing nontrivial basic sets, see also [Wi]. It has been used to give many inspiring and important examples, especially in the study of partially hyperbolic systems, see e.g. [M1, BV].

In the following we give some ideas of the construction.

We begin with a vector field $X^{0}$ on the 3-ball $B^{3}$ having a Lorenz attractor $\Lambda_{0}$. Then we consider a vector field $X(x,s)=(X^{0}(x),-\theta s)$ on the product $B^{3}\times[-1,1]$, where $(x,s)$ are coordinates on $B^{3}\times[-1,1]$ and $\theta>0$ is a constant. In particular, the dynamics is contracting along fibers. Note that $\Lambda=\Lambda_{0}\times\{0\}$ is an attractor and it contains a periodic orbit $P$.

Now the idea is to modify the vector field $X$ in a neighborhood of the periodic orbit $P$ so that one obtains a robust heterodimensional cycle. For this purpose we combine a DA-type surgery with the construction of a blender [BDV]. Roughly speaking, a blender is a hyperbolic set which verifies some specific geometric properties so that it unstable manifold looks like a manifold of higher dimension. It naturally appears in the unfolding of a heterodimensional cycle of two periodic orbits and can be used to obtain robust heterodimensional cycles.

For this construction, we need to show that the obtained vector field is away from homoclinic tangencies. This requires a careful choice of the contraction rate $\theta$ along the fibers. Moreover, we need to show that the singularity and the heterodiemnsional cycle are contained in the same chain recurrence class. This constitutes the most difficult part of our proof.

For the example in Theorem A, we would like to raise the following questions.

Apart from topological properties, one can also studies ergodic properties of the example. In [SYY] it is proved that the entropy function for singular flows away from homoclinic tangencies is upper semi-continuous with respect to both invariant measures and the flows, provided that the limiting measure is not supported on singularities.

One may even consider existence of SRB measures or physical measures. However, let us mention one difficulty in the study of this example. In contrast to the Lorenz attractor or a singular hyperbolic attractor, the usual analysis of the return map to a global cross-section may not be applicable in our example. This is because the chain recurrence class in our example admits a partially hyperbolic splitting $E^{s}\oplus F$, where the 3-dimensional center-unstable subbundle $F$ contains an exceptional 2-dimensional subbundle and is not sectionally expanded.

Apart from the present example, it may also be interesting to consider a DA-type surgery on the two-sided Lorenz attractor constructed by Barros, Bonatti and Pacifico [BBP].

The rest of this paper is organized as follows. We present some preliminaries of flows in Section 2 and give the construction of the example in Section 3. The proof of Theorem A is also given in Section 3, assuming some robust properties that will be proved in subsequent sections: in Section 4 we show that the example is away from homoclinic tangencies; then in Section 5 we show existence of an exceptional subbundle; finally in Section 6 we show that the singularity and the heterdimensional cycle are contained in the same chain recurrence class.

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 $C\geq 1$, $\lambda>0$, such that

$$\|\Phi_{t}|_{E(x)}\|\leq C\mathrm{e}^{-\lambda t},\quad\text{for any}\ x\in\Lambda,\ t\geq 0.$$

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 $E^{s}$ (resp. $E^{u}$) is contracted (resp. expanded). When $\Lambda$ is hyperbolic, for every $x\in\Lambda$, the sets

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

and

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

are invariant $C^{1}$ submanifolds tangent to $E^{s}(x)$ and $E^{u}(y)$ respectively at $x$, see [HPS]. 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 [Lo, GW]. In such cases, the set $\Lambda$ can not 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 [MPP]. 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 nontrivial if it is not reduced to a critical element. In [PYY], it is shown that a nontrivial, 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 [CY2].

Given a $\phi_{t}$-invariant set $\Lambda\subset M$. Define the normal bundle over $\Lambda$ as

$$\mathcal{N}_{\Lambda}=\bigcup_{x\in\Lambda\setminus\operatorname{Sing}(X)}\mathcal{N}_{x},$$

where $\mathcal{N}_{x}$ is the orthogonal complement of the one-dimensional subspace $l^{X}_{x}\subset T_{x}M$ generated by $X(x)$. In particular, if $\Lambda=M$, then $\mathcal{N}_{M}=\bigcup\limits_{M\setminus\operatorname{Sing}(X)}\mathcal{N}_{x}$. 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 [LGW].

Define the fiber bundle $G^{1}=\bigcup\limits_{x\in M}Gr(1,T_{x}M)$, where $Gr(1,T_{x}M)$ is the Grassmannian of lines in the tangent space $T_{x}M$ through the origin. Let $\beta:G^{1}\to M$ be the corresponding bundle projection that sends a line $l\in Gr(1,T_{x}M)$ 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 nonzero 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

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

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 $\beta^{*}(TM)$:

$$\tilde{\Phi}_{t}(l,v)=(\hat{\Phi}_{t}(l),\Phi_{t}(v)),\quad\forall(l,v)\in\beta^{*}(TM).$$

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 $T_{x}M$ and $\{l\}\times T_{x}M$. Thus we define the extended normal bundle as

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

We also define the extended normal bundle over any nonempty subset $\Delta\subset G^{1}$:

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

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

$$\tilde{\psi}_{t}(l,v)=\tilde{\Phi}_{t}(l,v)-\langle\Phi_{t}(u),\Phi_{t}(v)\rangle\cdot\tilde{\Phi}_{t}(l,u),$$

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.

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.

$$\mathcal{F}(\Gamma)=\{l\in G^{1}:\exists(\gamma_{n},X_{n})\hookrightarrow(\Gamma,X),\ p_{n}\in\gamma_{n},\ \text{such that}\ l^{X_{n}}_{p_{n}}\to l\}.$$

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.

$$B(\Gamma)=\{l\in G^{1}:\exists x_{n}\in\Gamma\setminus\operatorname{Sing}(X),\ \text{such that}\ l^{X}_{x_{n}}\to l\}.$$

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 spitting $E^{c}_{\rho}\oplus E^{u}_{\rho}$ 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

$$\|\psi^{X_{n}}_{t}|_{\mathcal{N}_{n}^{cs}(x)}\|\cdot\|\psi^{X_{n}}_{-t}|_{\mathcal{N}_{n}^{cu}(\phi^{X_{n}}_{t}(x))}\|<1/2.$$

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. Thus, the following lemma is a consequence of continuity of domination.

Let us begin with the geometric model of the Lorenz attractor, see e.g. [GH, GW, BDV]. 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, i.e. $\phi_{t}^{X^{0}}(x)\in B^{3}$ for any $x\in B^{3}$ and $t>0$. Moreover, the following properties are assumed (see [AP, Section 3]):

1. (P1)

   (Lorenz-like singularity) The origin $O$ is a hyperbolic singularity of stable index 2 such that its local stable manifold $W^{s}_{loc}(O)$ coincides with the $x_{1}x_{3}$-plane, and its local unstable manifold $W^{u}_{loc}(O)$ 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

   $$\lambda^{s}=\log\|D\Phi^{X^{0}}_{1}|_{E^{ss}_{O}}\|,\quad\lambda^{c}=\log\|D\Phi^{X^{0}}_{1}|_{E^{c}_{O}}\|,\quad\text{and}\ \lambda^{u}=\log\|D\Phi^{X^{0}}_{1}|_{E^{u}_{O}}\|.$$

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

2. (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 $W^{s}_{loc}(O)$ 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, i.e. $t_{x}>2$.

   

3. (P3)

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

   $$C_{\alpha}(x)=\{(x_{1},x_{2})\in T_{x}\Sigma_{0}:\|x_{1}\|\leq\alpha\|x_{2}\|\}.$$

   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 $DR_{0}(u)\in C_{1/2}(R_{0}(x))$.

4. (P4)

   (The attractor) Let $\Lambda_{0}={\rm Cl}(\phi^{X^{0}}_{t\in\mathbb{R}}(\Lambda_{\Sigma_{0}}))$, where $\Lambda_{\Sigma_{0}}=\cap_{n\geq 0}\overline{R_{0}}^{n}(\Sigma_{0})$ and $\overline{R_{0}}^{n}(\Sigma_{0})$ is the closure of $n$-th 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 $\Sigma_{0}$.

5. (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

   $$\left|\det(\Phi^{X^{0}}_{t}|_{S})\right|>e^{\gamma t},\quad\forall t\geq 1,$$

   (3.1)

   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

   $$\|DR_{0}(v)\|>\rho\|v\|.$$

   As the flow direction $X^{0}$ is invariant and can not be uniformly contracted, it is contained in the subbundle $E^{cu}$, i.e. ${X^{0}}(x)\in E^{cu}_{x}$ for every $x\in\Lambda_{0}\setminus\operatorname{Sing}(X)$ (see [BGY, Lemma 3.4]). At the singularity $O$, we have $E^{cu}_{O}=E^{c}_{O}\oplus E^{u}_{O}$. Define

   $$\lambda^{s}_{0}=\log\|D\Phi^{X^{0}}_{1}|_{E^{ss}_{\Lambda_{0}}}\|.$$

   Note that $\lambda^{s}\leq\lambda^{s}_{0}$. We assume $\lambda^{s}_{0}<\lambda^{c}$.

6. (P6)

   (Stable foliation) The stable foliation $\mathcal{W}^{s}(\Lambda_{0})=\{W^{s}(\operatorname{Orb}(x)):x\in\Lambda_{0}\}$ is $C^{1}$ (see [AM]), 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 $W^{s}_{loc}(O)$ with $\Sigma_{0}$, it is a leaf of $\mathcal{F}^{s}$.

7. (P7)

   (Transitivity) The attractor $\Lambda_{0}$ is a homoclinic class (see [Ba]), so that $\Lambda_{0}$ is transitive and there is a periodic orbit $Q_{0}\subset\Lambda_{0}$ whose stable manifold is dense in $U_{\Lambda_{0}}$.

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 3-dimensional attractor $\Lambda_{0}$ can be easily generalized to $\Lambda$. We now modify the vector field $\hat{X}$ along fibers to obtain a vector field $X$ so that $\Lambda$ remains an attractor but not singular hyperbolic. This is why we say the example is derived from the Lorenz attractor. 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 $V_{P}$, i.e. $\eta(x,s)=0$ 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:

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

(3.2)

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 fiber. Thus, the Lyapunov exponents of $\sigma$ are $\lambda^{s},\lambda^{c},\lambda^{u}$ and $-\theta$. Note that $-\theta$ is the Lyapunov exponent of the singularity along the fiber direction. We assume further that

$$\lambda_{0}^{s}<-\theta<\lambda^{c}.$$

(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 2-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 (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 3-dimensional topologically stable manifold and its strong stable manifold $W^{ss}(P,X)$ (2-dimensional) has nonempty intersection with the unstable manifold of $Q$. Also, its unstable manifold has a transverse intersection with the stable manifold of $Q$.

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. 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.