On the intersection of sectional-hyperbolic sets

We analyse the intersection of positively and negatively sectional-hyperbolic sets for flows on compact manifolds. First we prove that such an intersection is hyperbolic if the intersecting sets are both transitive (this is false without such a hypothesis). Next we prove that, in general, such an intersection consists of a nonsingular hyperbolic set, finitely many singularities and regular orbits joining them. Afterward we exhibit a three-dimensional star flow with two homoclinic classes, one being positively (but not negatively) sectional-hyperbolic and the other negatively (but not positively) sectional-hyperbolic, whose intersection reduces to a single periodic orbit. This provides a counterexample to a conjecture by Shy, Zhu, Gan and Wen (\cite{sgw}, \cite{zgw}).


Introduction
Dynamical systems is concerned with the study of the asymptotic behavior of the orbits of a given system. Certain hypothesis like Smale's hyperbolicity guarantee the knowledge of this behavior. Indeed, the celebrated Smale spectral decomposition theorem asserts that every hyperbolic system on a compact manifold comes equipped with finite many pairwise disjoint compact invariant sets (homoclinic classes or singularities) to which every trajectory converge. Although present in a number of interesting examples, such a hypothesis is far from being abundant in the dynamical forrest. This triggered several attempts to extend it including the sectional-hyperbolicity [17], committed to merge the hyperbolic theory to the socalled geometric and multidimensional Lorenz attractors [1], [8], [13]. A number of results from the hyperbolic theory have been carried out to the sectional-hyperbolic context. This is nowadays matter of study in a number of works, see [2] and references therein. One of these results was motivated by the well-known fact that two different homoclinic classes contained in a common hyperbolic set are disjoint. It was quite natural to ask if this statement is also true in the sectional-hyperbolic context too. In other words, are two different homoclinic classes contained in a common sectional-hyperbolic set disjoint? But recent results dealing with this question say that the answer is negative [20], [21]. Moreover, [21] studied the dynamics of nontransitive sectional-Anosov flows with dense periodic orbits nowadays called venice masks. It was proved that three-dimensional venice masks with a unique singularity exists [7] and that their maximal invariant set consists of two different homoclinic classes with nonempty intersection [21]. Venice mask with n singularities can be constructed for n ≥ 3 whereas ones with just two singularities have not been constructed yet.
These fruitful results motivate a related problem which is the analysis of the intersection of a sectional-hyperbolic set for the flow and a sectional-hyperbolic set for the reversed flow. For simplicity we keep the terms positively and negatively sectional-hyperbolic for these sets (respectively) which was coined by Shy, Gan and Wen in their recent paper [24]. After observing that every hyperbolic set can be realized as such an intersection, we show an example where such an intersection is not hyperbolic. Next we show that such an intersection is hyperbolic if the intersecting sets are both transitive. In general the intersection consists of a nonsingular hyperbolic set (possibly empty), finitely many singularities and regular orbits joining them. Finally, we construct a three-dimensional star flow exhibiting two homoclinic classes, one being positively (but not negatively) sectional-hyperbolic and the other being negatively (but not positively) sectional-hyperbolic, whose intersection reduces to a single periodic orbit. This will provide a counterexample to a conjecture by Zhu, Gan and Wen [25] (as amended by Shy, Gan and Wen [24]).

Statement of the results
Let M be a differentiable manifold endowed with a Riemannian metric ·, · an induced norm · . We call flow any C 1 vector field X with induced flow X t of M . If dim(M ) = 3, then we say that X is a three-dimensional flow. We denote by Sing(X) the set of singularities (i.e. zeroes) of X. By a periodic point we mean a point x ∈ M for which there is a minimal t > 0 satisfying X t (x) = x. By an orbit we mean O(x) = {X t (x) : t ∈ R} and by a periodic orbit we mean the orbit of a periodic point. We say that Λ ⊂ M is invariant if X t (Λ) = Λ for all t ∈ R. In such a case we write Λ * = Λ \ Sing(X). We say that Λ ⊂ M is transitive if there is x ∈ Λ such that ω(x) = Λ, where ω(x) is the ω-limit set, ω(x) = y ∈ M : y = lim n→∞ X tn (x) for some sequence t n → ∞ .
The α-limit set α(x) is the ω-limit set for the reversed flow −X. If the set of periodic points of X in Λ is dense in Λ, we say that Λ has dense periodic points.
A compact invariant set Λ is hyperbolic if there is a continuous invariant splitting and t ≥ 0. A singularity or periodic orbit is hyperbolic if it does as a compact invariant set of X. The elements of a (resp. hyperbolic) periodic orbit will be called (resp. hyperbolic) periodic points. A singularity or periodic orbit is a sink (resp. source) if its unstable subbundle E u (resp. stable subbundle E s ) vanishes. Otherwise we call it saddle type.
The invariant manifold theory [14] asserts that through any point x of a hyperbolic set it passes a pair of invariant manifolds, the so-called stable and unstable manifolds W s (x) and W u (x), tangent at x to the subbundles E s x and E u x respectively. Saturating them with the flow we obtain the weak stable and unstable manifolds W ws (x) and W wu (x) respectively.
On the other hand, a compact invariant set Λ has a dominated splitting with respect to the tangent flow if there are an invariant splitting T Λ M = E ⊕ F and positive numbers K, λ such that Notice that this definition allows every compact invariant set Λ to have a dominated splitting with respect to the tangent flow: Just take E x = T x M and F x = 0 for every x ∈ Λ (or E x = 0 and F x = T x M for every x ∈ Λ). However, such splittings need not to exist under certain constraints. For instance, not every compact invariant set has a dominated splitting T Λ M = E ⊕ F with respect to the tangent flow which is nontrivial, i.e., satisfying E x = 0 = F x for every x ∈ Λ.
A compact invariant set Λ is partially hyperbolic if it has a partially hyperbolic splitting, i.e., a dominated splitting T Λ M = E ⊕ F with respect to the tangent flow whose dominated subbundle E is contracting in the sense of (1) above.
The Riemannian metric ·, · of M induces a 2-Riemannian metric [22], This in turns induces a 2-norm [12] (or areal metric [15]) defined by Geometrically, u, v represents the area of the paralellogram generated by u and v in T p M . If a compact invariant set Λ has a dominated splitting T Λ M = E ⊕ F with respect to the tangent flow, then we say that its central subbundle F is sectionally expanding (resp. sectionally contracting) if By a sectional-hyperbolic splitting for X over Λ we mean a partially hyperbolic splitting T Λ M = E ⊕ F whose central subbundle F is sectionally expanding. Now we define sectional-hyperbolic set.
Definition 2.1. A compact invariant set Λ is sectional-hyperbolic for X if its singularities are hyperbolic and if there is a sectional-hyperbolic splitting for X over Λ.
Following [24] we use the term positively (resp. negatively) sectional-hyperbolic to indicate a sectional-hyperbolic set for X (resp. −X). The corresponding sectionalhyperbolic splitting will be termed positively (resp. negatively) sectional-hyperbolic splitting.
This definition is slightly different from the original one given in Definition 2.3 of [17] (which requires, for instance, that the central subnbundle be two-dimensional at least). Such a difference permits every hyperbolic set Λ to be both positively and positively and negatively sectional-hyperbolic splittings respectively over Λ. In particular, every hyperbolic set is the intersection of a positively and a negatively sectional-hyperbolic set.
One can ask if the hyperbolic sets are the sole possible intersection between a positively and a negatively sectional-hyperbolic set, but they aren't. In fact, there are nonhyperbolic compact invariant sets which, nevertheless, are both positively and negatively sectional-hyperbolic. This is the case of the example described in Figure 1. In such a figure O(x) represents the orbit of x ∈ W s (σ 1 ) ∩ W u (σ 2 ) whereas a singularity of a three-dimensional flow is Lorenz-like for X if it has three real eigenvalues λ 1 , λ 2 , λ 3 satisfying  The similar results replacing transitivity by denseness of periodic orbits hold. By looking at Figure 1 we observe that this example consists of two singularities and a regular point x whose ω-limit and α-limit set is a singularity. This observation is the motivation for the result below.
Since the intersection of a positively and a negatively sectional-hyperbolic set is both positively and negatively sectional-hyperbolic, we obtain the following corollary.
Corollary 2.5. The intersection of a positively and a negatively sectional-hyperbolic set is a disjoint union of a (possibly empty) nonsingular hyperbolic set H, a (possibly empty) finite set of singularities S and a (possibly empty) set of regular points Our next result is an example of nontrivial transitive sets which are positively and negatively sectional-hyperbolic (resp.) whose intersection is the simplest possible, i.e., a single periodic orbit.
Denote by Cl(·) the closure operation. We say that H ⊂ M is a homoclinic class if there is a hyperbolic periodic point x of saddle type such that It follows from the Birkhoff-Smale Theorem that every homoclinic class is a transitive set with dense periodic orbits.
Given points x, y ∈ M , if for every > 0 there are sequences of points {x i } n i=0 and times then we say that x is in the chain stable set of y. If x is in the chain stable set of y and viceversa, then one says that x and y are chain related. If x is chain related to itself, one says that x is a chain recurrent point. The set of chain recurrent points is the chain recurrent set denoted by CR(X). It is clear that the chain related relation is in equivalence on CR(X). By using this equivalence, one splits CR(X) into equivalence classes denominated chain recurrent classes.
A flow is star if it exhibits a neighborhood U (in the space of C 1 flows) such that every periodic orbit or singularity of every flow in U is hyperbolic.
With these definitions we obtain the following result.
Theorem 2.6. There is a star flow X in the sphere S 3 whose chain recurrent set is the disjoint union of two periodic orbits O 1 (a sink), O 2 (a source); two singularities s − (a source), s + (a saddle); and two homoclinic classes H − , H + with the following properties: • H − is negatively (but not positively) sectional-hyperbolic; • H + is positively (but not negatively) sectional-hyperbolic; • H − ∩ H + is a periodic orbit.
Recall that the nonwandering set of a flow X is defined as the set of points x ∈ M such that for every neighborhood U of x and T > 0 there is t ≥ T satisfying X t (U ) ∩ U = ∅. Given a certain subset O of the space of C 1 flows, we say that a C 1 generic flow in O satisfies another property (Q) if there is a residual subset of flows R of O such that every flow in R satisfying (P) also satisfies (Q).
There are two current conjectures relating star flows and sectional-hyperbolicity. These are based on previous results in the literature e.g. [11], [19].
Conjecture 2.7 (Zhu-Shy-Gan-Wen [24], [25]). The chain recurrent set of every star flow is the disjoint union of a positively sectional-hyperbolic set and a negatively sectional-hyperbolic set.
Conjecture 2.8 (Arbieto [4]). The nonwandering set of a C 1 generic star flow is the disjoint union of finitely many transitive sets which are positively or negatively sectional-hyperbolic.
However, the union H − ∪ H + of the homoclinic classes H − and H + in Theorem 2.6 is a chain recurrent class of the corresponding flow X (because H − ∩ H + = ∅). Therefore, Theorem 2.6 gives a counterexample for Conjecture 2.7 in dimension 3. Similar counterexamples can be obtained in dimension ≥ 3.
Corollary 2.9. There is a star flow in S 3 whose chain recurrent set is not the disjoint union of a positively sectional-hyperbolic set and a negatively sectionalhyperbolic set.
Another interesting feature regarding this counterexample is the existence of a chain recurrent class without any nontrivial dominated splitting with respect to the tangent flow. Moreover, every ergodic measure supported on this class is hyperbolic saddle. These features are related to [9] or [18]. Notice also that the star flow in Corollary 2.9 can be C 1 approximated by ones exhibiting the heteroclinic cycle obtained by joinning the unstable manifold W u (σ 1 ) of σ 1 to the stable manifold W s (σ 2 ) of σ 2 in Figure 1. Such a cycle was emphasized in the figure after the statement of Lemma 3.3 in p.951 of [25]. This put in evidence the role of robust transitivity in the proof of such a lemma.  (1) Every singularity in Λ is hyperbolic.
(2) There are continuous invariant subbundles E s , E u of T Λ M such that E s is contracting, E u is expanding and Notice that this definition is symmetric with respect to the reversing-flow operation. Moreover, hyperbolic sets are almost hyperbolic but not conversely by the example in Figure 1. Likewise sectional-hyperbolic sets, the almost hyperbolic sets satisfy Lemma 3.2 (Hyperbolic Lemma). Every compact invariant subset without singularities of an almost periodic set is hyperbolic.
Let T σ M = F s σ ⊕ F u σ be the hyperbolic splitting of σ. By definition T xn M = E s xn ⊕ E X xn ⊕ E u xn so dim(E s xn ) + dim(E u xn ) = dim(M ) − 1, ∀n.
Passing to the limit we obtain dim(E s σ ) + dim(E u σ ) = dim(M ) − 1. Since E s σ and E u σ are contracting and expanding respectively, we obtain E s σ ⊂ F s Now suppose by contradiction that x n ∈ W s (σ)∪W u (σ) for all n (say). Then, by flowing the orbit of x n nearby σ, as described in Figure 2, we obtain two sequences x s n , x u n in the orbit of x n such that x s n → y s and x u n → y u for some y s ∈ W s (σ) \ {σ} and y u ∈ W u (σ) \ {σ} close to σ. If dim(E s σ ) = dim(F s σ ) then E s y s = T y s W s (σ) but also E X y s ⊂ T y s W s (σ) since W s (σ) is an invariant manifold. Therefore, E X y s ⊂ E s y s and then E X y s = 0 since the sum T y s M = E s y s ⊕ E X y s ⊕ E u y s is direct. This is a contradiction. Analogously we obtain a contradiction if dim(E u σ ) = dim(F u σ ) and the proof follows. Now we relate sectional and almost hyperbolicity. Proof. Let Λ be a compact invariant set which is both positively and negatively sectional-hyperbolic. Then, every singularity in Λ is hyperbolic. Moreover, there are positively and negatively sectional-hyperbolic splittings Taking E u =Ê s and E sc =Ê se we obtain an expanding and a sectional contracting subbundles of T Λ M . Since E s is contracting, we have E X ⊂ E se by Lemma 3.2 in [3]. Similarly, E X ⊂ E sc so E X ⊂ E se ∩ E sc . On the other hand, since E s is contracting and E u expanding, the angle E s , E u is bounded away from zero. Then, the dominating condition implies From this we have T Λ M = E se + E sc and so At regular points we cannot have a vector outside E X contained in E se ∩ E sc . Then, E X = E se ∩ E sc and so dim(E se ∩ E sc ) = 1 in Λ * . Replacing above we get Proof of Theorem 2.3. Let Λ be the intersection of a positively and a negatively sectional-hyperbolic set of a flow X. Then, it is both positively and negatively sectional-hyperbolic and so almost hyperbolic by Lemma 3.4. From this we can select δ > 0 as in Lemma 3.3. Clearly we can take δ such that the balls B(σ, δ) are pairwise disjoint for σ ∈ S, where S = Sing(X) ∩ Λ. Define Clearly S consists of finitely many singularities. Moreover, H is nonsingular hence hyperbolic by the Hyperbolic Lemma. Now take x ∈ R. Then, there is (t, σ) ∈ R × S such that X t (x) ∈ B(σ, δ). By Lemma 3.3 we obtain B(ρ, δ) and so x ∈ W u (ρ). All together yields α(x) ⊂ H ∪ S. Similarly we have α(x) ⊂ H ∪ S and ω(x) ⊂ H ∪ S if x ∈ W u (σ) and the result follows.
To prove Theorem 2.2 we use the following lemma. Recall that an invariant set is nontrivial if it does not reduces to a single orbit. Lemma 3.5. Let Λ be a nontrivial transitive positively sectional-hyperbolic set of a flow X. If σ ∈ Sing(X) ∩ Λ, then the hyperbolic and the respective hyperbolic and positively sectional-hyperbolic splittings Proof. Clearly E s σ ⊂ F s σ . Suppose for a while that E s σ = F s σ . Then, dim(E s y ) = dim(T y W s (σ)) for every y ∈ Λ ∩ W s (σ) close to σ. As clearly E s y ⊂ T y W s (σ) for all such points y, we obtain E s y = T y W s (σ) for every y ∈ Λ ∩ W s (σ) close to σ. On the other hand, we also have that E X y ⊂ T y W s (σ) for all such points y. From this we conclude that E X y ⊂ E s y for every point y ∈ Λ ∩ W s (σ) close to σ. Now we observe that since Λ is transitive we obtain E X ⊂ E se . Using again that Λ is nontrivial transitive (see Figure 2) we obtain y = y s ∈ Λ * ∩ W s (σ) close to σ. For such a point we obtain 0 = E X y ⊂ E s y ∩ E se y which is absurd. Therefore, E s σ = F s σ . Next we observe that dim(E se σ ∩ F s σ ) ≤ 1 by sectional expansivity. Suppose for a while that dim(E se σ ∩ F s σ ) = 0. Clearly E s σ ∩ F u σ = 0 and so F u σ ⊂ E se σ by domination. From this we obtain Proof of Theorem 2.2. Let Λ + and Λ − be transitive sets of a flow X such that Λ + is positively sectional hyperbolic and Λ − is negatively sectional-hyperbolic. If one of these sets reduces to a single orbit, then the intersection Λ − ∩ Λ reduces to that orbit and the result follows. So, we can assume both Λ + and Λ − are nontrivial. Let T Λ+ M = E s ⊕ E se and T Λ− M =Ê s ⊕Ê se be the positively and negatively sectional-hyperbolic splittings of Λ + and Λ − respectively. Denoting E u =Ê s and E sc =Ê se we obtain an expanding subbundle and a sectionally contracting subbundle of T Λ M .
Suppose for a while that there is σ ∈ Λ − ∩ Λ + ∩ Sing(X). By Lemma 3.5 applied to X, we have that σ has a real negative eigenvalues λ s corresponding to the one-dimensional eigendirection E se σ ∩ F s σ . Similarly, applying the lemma to −X, we obtain a real positive eigenvalue λ u corresponding to the one-dimensional by sectionally expansiveness. Then by sectionally expansiveness with respect to −X. Then, λ s + λ u < 0 which is absurd. We conclude that Λ − ∩ Λ + ∩ Sing(X) = ∅. Now we can apply the hyperbolic lemma for sectional-hyperbolic sets to obtain that Λ − ∩Λ + is hyperpolic. This finishes the proof.

Proof of Theorem 2.6
Roughly speaking, the proof consists of glueing the so-called singular horseshoe [16] with its time reversed counterpart. We star with the standard Smale horseshoe which is the map in the 2-disk on the left of Figure 3. It turns out that its nonwandering set consists of a sink and a hyperbolic homoclinic class containing the saddle. Its suspension is the flow described in the right-hand picture of the figure. It is a flow in the solid torus whose nonwandering set is also a periodic sink O 1 together with a hyperbolic homoclinic class.
The next Figure 4 describes a procedure of inserting singularities in the suspended Smale horseshoe. We select an horizontal interval I and a point x in the square forming the horseshoe.
The selection is done in order to place I in the stable manifold of a Lorenz-like equilibrium σ + , and x in the stable manifold of a Lorenz-like equilibrium for the reversed flow σ − . This construction requires to add two additional singularities, a source s − to which the unstable branch of σ − not containig x goes; and a saddle s + close to σ + . See Figure 5.
An accurate description of the aforementioned procedure is done in [8] and [23]. Next we observe that the resulting flow's return map presents a cut along I and a blowup circle derived from x.
We now proceed to deform the flow in order to obtain a deformation of the return map by pushing up one branch of the circle, and pushing down the cusped region derived from the cutting as indicated in Figure 6.   Figure 5. Still inserting singularities.
We keep doing this deformation (see Figure 7) up to arrive to the final flow whose return map is described in Figure 8.
This flow is defined in a solid torus, transversal to the boundary and pointing inward there.
The final return map (denoted by R) is described with some detail in Figure 9.
We are in position to describe the homoclinic classes H − and H + in Theorem 2.6. They are precisely the maximal invariant set of R in the upper and lower rectangles Q + and Q − forming the rectangle Q in Figure 9. (for H + ). A rough description of H − and H + is that H + is the singular horseshoe in [16] and H − its time reversal. The proof that H − and H + are nontrivial homoclinic classes is done as in [6], [7]. The analysis in [5] or [16] shows that H + is a sectional-hyperbolic set for the (final) flow and that H + is a sectional-hyperbolic set for the reversed flow. We assume that the horizontal conefield x ∈ Q → C s 1 (x) and the vertical conefield x ∈ Q → C u 1 (x), where C s α (x) = (a, b) ∈ R 2 : |b| |a| ≤ α and C u α (x) = (a, b) ∈ R 2 : |a| |b| ≤ α , ∀α > 0, are contracting and expanding (respectively) for the return map R in the sense that there is ρ > 1 with the following properties: (R(x)) and DR(x)v u ≥ ρ v u , ∀v u ∈ C u 1 (x). (2) If x ∈ R(Q) ∩ Q then (R −1 (x)) and DR −1 (x)v s ≥ ρ v s , ∀v s ∈ C s 1 (x). (See Figure 9.) Since such conefields do not allow the existence of nonhyperbolic periodic points, and are preserved by small perturbations, we obtain that the final flow is star in its solid torus domain.
Next we observe that H − is not hyperbolic, since it contains the singularity σ − and, analogously, H + is not hyperbolic for it contains σ + . Since every homoclinic class is transitive, we conclude from Theorem 2.3 that H − is not positively sectionalhyperbolic and H + is not negatively sectional-hyperbolic.
To complete the proof we extend the final flow from its solid torus domain to the whole S 3 . This is done by glueing it with another solid torus whose core is a periodic source O 2 . This completes the proof.