Properties of compact center-stable submanifolds
We show that a partially hyperbolic system can have at most a finite number of compact center-stable submanifolds. We also give sufficient conditions for these submanifolds to exist and consider the question of whether they can intersect each other.
Key words and phrases:partial hyperbolicity, attractors, center-stable submanifolds
1991 Mathematics Subject Classification:37D30, 37C70
Much of the early theory of partially hyperbolic dynamics was developed by Hirsch, Pugh, and Shub in their foundational text, Invariant Manifolds [HPS77]. The book first considers the case of an invariant compact submanifold of the phase space where the dynamics normal to the submanifold is hyperbolic. Later chapters deal with systems where a partially hyperbolic splitting holds on the entire phase space. These two cases may overlap. For instance, if the system has a global splitting of the form , it may also have a compact submanifold tangent to and such a submanifold is therefore normally hyperbolic.
Recent discoveries show that a slightly different possibility exists. Rodriguez Hertz, Rodriguez Hertz, and Ures constructed an example of a partially hyperbolic system on the 3-torus with a compact submanifold, a 2-torus, tangent to the center and stable directions, [RHRHU16]. This center-stable submanifold is transverse to the expanding unstable direction and is therefore a normally repelling submanifold. Based on this, the author constructed further examples of compact center-stable submanifolds, both in dimension 3 and higher [Ham16].
The paper establishes general properties for these types of dynamical objects. In particular, we show that any partially hyperbolic system may have at most finitely many compact center-stable submanifolds and we give sufficient conditions under which these objects exist. Finally, we consider the consider the question of whether two of these submanifolds can have non-empty intersection.
2. Statement of results
A diffeomorphism of a closed connected manifold is (strongly) partially hyperbolic if there is a splitting of the tangent bundle
such that each subbundle is non-zero and invariant under the derivative and
hold for all and unit vectors , , and . There exist unqiue foliations and tangent to and . An immersed submanifold is a center-stable submanifold if it is tangent to .
A partially hyperbolic diffeomorphism has at most a finite number of compact center-stable submanifolds.
From this, the following result could be proved.
Every compact center-stable submanifold is periodic.
While being tangent to clearly requires the submanifold to be at least , it is equivalent to a condition which may be stated for submanifolds.
Suppose is partially hyperbolic and is a periodic compact submanifold. Then is a submanifold tangent to if and only if for all .
This also gives a way to find periodic submanifolds from non-periodic ones.
Suppose is partially hyperbolic, , and is a compact submanifold such that and for all . Then there exists a compact center-stable submanifold.
The next theorem assumes a one-dimensional unstable direction. It basically states that if a region has two boundary components and the ends of unstable curves inside this region tend towards the boundary in a uniform way, then there must be a compact center-stable submanifold inside the region.
Let be a partially hyperbolic diffeomorphism of a manifold , a compact connected submanifold of with boundary, a continuous function, and such that
if , then ,
if , , , and , then , and
if is a parameterized unstable leaf, then
Then, there is a compact center-stable submanifold in the interior of .
This result will be used in an upcoming paper as a critical step in giving a classification of all partially hyperbolic systems in dimension three which have center-stable tori. Section 5 gives the proof of creftype 2.5.
The proofs of the above results never use the sub-splitting of the center-stable bundle. Therefore, all of the above results also hold for weakly partially hyperbolic systems, where the diffeomorphism has an invariant splitting of the form . For further discussion of weak versus strong partial hyperbolicity, see sections 1 and 6 of [HP16] and the references therein.
Note that creftype 2.1 above only shows finiteness; it does not say anything about disjointedness.
Can two distinct compact center-stable submanifolds have non-empty intersection?
In the case of strongly partially hyperbolic systems in dimension 3, we have a number of special tools at our disposal including branching foliations and Anosov tori [BBI09, RHRHU11], and we can answer this question in the negative.
In a 3-dimensional strongly partially hyperbolic system, the compact center-stable submanifolds are pairwise disjoint.
To suggest why such intersections might be possible in higher dimensions, we give an example of an invariant partially hyperbolic subset of a 3-manifold which consists of two surfaces glued together, each of which is tangent to the center-stable direction of the splitting. Moreover, the partially hyperbolic splitting on the subset extends to a dominated splitting defined on the entire manifold.
There is a diffeomorphism with a sink and a global dominated splitting into three subbundles such that if denotes the basin of attraction of , then the boundary of is the union of two distinct intersecting tori tangent to , and the splitting is partially hyperbolic on all of .
This shows in particular that there is no local obstruction to having an intersection. We first construct the example which demonstrates creftype 2.8 in section 6 and then prove creftype 2.7 in section 7.
The above results are stated for center-stable submanifolds. By replacing with its inverse, one may state analogous results for compact center-unstable submanifolds. It is easier, in some cases, to prove a result in this alternate setting and so we switch back and forth between the two viewpoints in the proofs below.
In related work, theorems 2.1 and 2.2 generalize results given in [RHRHU15] and their proofs are based on the techniques given there. creftypecap 2.3 is closely related to the main result of [BC16], which considers an arbitrary compact invariant set where for all . creftypecap 2.3 could be proved as a consequence of this result. However, the fact that has the structure of a submanifold means that we can give a direct, intuitive, and comparatively simple proof of creftype 2.3 in the space of a few pages. For this reason, we give a full self-contained proof in this paper.
Let denote Hausdorff distance. Equipped with , the space of compact subsets of is a compact metric space. If and are points on the same unstable leaf, let denote the distance between them as measured along the leaf. If and are on distinct unstable leaves, then . Similar to the definition of Hausdorff distance, for subsets define
In what follows, we write -submanifold as shorthand for a center-stable submanifold. Using the transversality of and , one may prove the following
There is such that if and are compact -submanifolds and , then .∎
In this section, call a compact subset “well positioned” if for all distinct .
Let be a compact -submanifold. Then, there is an integer such that is well positioned for all .
As is transverse to , there is such that for all distinct and . Then take such that and use the above assumption on . ∎
If is a well-positioned compact -submanifold and for some , then there is a unique well-positioned periodic submanifold such that .
For , define as the unique point in such that
Existence follows from and uniqueness from the fact that is well positioned. By the same reasoning, there an inverse map and so is a homeomorphism. For , define One can show that for all . In other words, restricted to a neighbourhood of is a fiber contraction of a fiber bundle. By the fiber contraction theorem [HPS77, Theorem 3.1], there is an invariant submanifold in this neighbourhood. Applying , one sees that .
Suppose is a well-positioned periodic submanifold with . Then and tends to zero as . This shows that . ∎
Let be a compact -submanifold. For any , there is a well-positioned periodic submanifold such that .
Proof of creftype 2.2.
4. Regularity of submanifolds
Proof of creftype 2.4.
The above proof further shows that the submanifolds in creftype 2.4 satisfy , .
One direction of creftype 2.3 readily follows from results in the last section. To prove the other direction, it will be easier to exchange the roles of and . Therefore, we assume is partially hyperbolic and is a periodic submanifold such that for all . Our goal is then to show that is a submanifold tangent to . To prove this, we may freely replace by an iterate. In particular, assume . Also assume that associated to the partially hyperbolic splitting is a continuous function such that for all and unit vectors and .
Let be a cone family associated to the dominated splitting. That is, for every , is a closed convex subset of such that , , and depends continuously on . Define the dual cone family as the closure of . The properties of the splitting imply that
Replacing by some , by a large iterate , and the function by
assume for any and non-zero vector that
if , then ;
if , then ; and
if , then
Let be the exponential map. Up to rescaling the Riemannian metric on , assume that if , then there is a unique vector with such that . Define a continuous map
by requiring that for all and with .
There is such that for any and with
if , then ;
if , then ; and
if , then
The properties of the exponential map imply that tends uniformly to zero as . The lemma may then be proved from the corresponding properties of . ∎
Later on, we will also need the following fact.
For any , there is such that if and , then .
There is a lower bound on the angle between any non-zero vectors and . Take small enough that if , then the angle between and is smaller than this bound. ∎
If is such that and for all , then lies on the stable leaf through .
This lemma is more or less one of the steps in establishing the existence of the stable foliation. See for instance [HPS77, Section 5]. For completeness, we give a proof which assumes that the stable foliation exists.
Since is transverse to , there is such that the (incomplete) submanifold
is transverse to . There is also such if , then intersects in a point which satisfies for all .
Write and . Then , so that for all . As for large , there is and a vector such that and lie on the same stable leaf and for all . Without loss of generality, assume and write . If , the result is proved. Therefore, we assume . Then implies and therefore for all . However, one can show that for all large , and this gives a contradiction. ∎
For the rest of the section, if and are distinct points on define and for all . For those indices where , define such that and .
There is a uniform constant such that for any distinct either or for some .
First note that since by assumption, lemma 4.3 implies that such an exists for each pair considered on its own. The goal is to find a uniform constant which works for all pairs. Let be such that implies . The set is compact. One may then use an open cover to show that there is a uniform constant such that if then either or for some .
Now suppose . Let be the smallest integer such that either or . Such an must exist as uniformly expands vectors in . If , then and the minimality of implies that . If , then by the choice of and so there is such that . Since , this implies that . ∎
There is a sequence of positive numbers such that if then .
Since this shows that is a submanifold tangent to .
5. Cross sections
Let be a compact connected manifold with boundary, a flow on , a continuous function, and a constant such that
if , , and , then .
Then, there is a compact codimension one submanifold in the interior of which intersects any orbit in at most one point.
Assume now that the hypotheses in the creftype 5.1 hold. Note that is a global flow defined for all time. For each , is a homeomorphism, and so is invariant under the flow. For , define
Since , at least one boundary component is contained in and at least one is contained in . The second item in the theorem implies that .
The subsets and are closed.
Suppose is a sequence in , converging to . Then there is such that . As is continuous, for all large . Then for all large and all with . By continuity, for all with . ∎
At least one of or is non-empty.
Otherwise, and disconnect into two clopen subsets. ∎
Without loss of generality, assume is non-empty.
Also define as
and define similarly.
The subsets , , , and are open.
If , then there is such that
This inequality also holds for all points in a neighbourhood of and implies that . If , then there is and such that
This also holds on a neighbourhood of and shows that is open. The cases of and are analogous. ∎
For the remainder of the proof, we assume . This can always be achieved by rescaling time for the flow, and makes the definitions simpler in what follows. We now adapt the averaging method of Fuller [Ful65] to this setting. For each integer , define by
Let denote the derivative along the flow. That is, for a function , define
The Fundamental Theorem of Calculus implies that
The assumption implies that if , then at least one of or holds. Hence, for all and for a fixed and large . Define by
If , one can show that , , and
Hence, any orbit in intersects in exactly one point. As in [Ful65], one can then show that locally has the structure of a codimension one submanifold.
For , the subset is open in the topology of .
The cases of and follow immediately from lemma 5.4. If , then implies that cannot be zero for all . Thus, and is therefore open. Similarly for . ∎
The set is a finite disjoint union of compact connected codimension one submanifolds.
As noted above, locally has the structure of a submanifold. By lemma 5.5 and the fact that splits into the disjoint union , the subset is clopen in the topology of . In particular, this subset is compact and therefore consists of a finite number of compact connected submanifolds. ∎
To complete the proof of creftype 5.1, take to be one of the components of .
We now look at how these components interact with a diffeomorphism which preserves the orbits of the flow. In what follows, let be the connected components of .
Suppose is a homeomorphism which preserves the orbits of and such that
for all . Then there is such that an orbit of intersects a component if and only if the orbit intersects .
The hypotheses imply that . For each point , there is a unique point on the orbit of such that . Moreover, depends continuously on . The image is a compact manifold in and is therefore equal to one of the . This shows that, up to flowing along the orbits of , the homeomorphism permutes the components . Up to replacing again by an iterate, we may assume this is the identity permutation. ∎
Proof of creftype 2.5.
First, consider the case where is orientable. Define a flow such that the orbits of are exactly the unstable leaves of . This flow satisfies the hypotheses of creftype 5.1 (with in place of ). Consequently, there is a compact submanifold in the interior of which intersects each unstable leaf in at most one point. By creftype 5.7, there is an iterate such that an unstable leaf intersects if and only if it intersects . Then, and creftype 2.4 implies that there is a compact periodic center-stable submanifold as desired. This concludes the orientable case.
Instead of handling the non-orientable case directly, we assume now that there is an involution which commutes with , preserves the unstable foliation, and reverses the orientation of . If and ) are disjoint, then and are disjoint. If, instead, intersects ), then the argument in proof of creftype 5.7 shows that and the fact that is -periodic implies that . In either case, quotients down to a compact submanifold embedded in . ∎
6. Making a calzone
In this section, we construct the example in creftype 2.8. As in section 4, it is slightly easier from the notational viewpoint to switch the roles of and in the construction. Therefore, we will actually build a system with a normally repelling fixed point and two intersecting center-unstable tori.
First, we build a partially hyperbolic subset of which is the union of two non-disjoint -tori. Then, we explain how this partially hyperbolic subset can be glued into the 3-torus in such a way to produce a global dominated splitting.
The two -tori each have the same derived-from-Anosov dynamics with a repelling fixed point. They are glued together on the complement of the basin of repulsion of this fixed point. The -tori are, of course, tangent along this intersection and the construction vaguely resembles the type of food called a calzone, where two pieces of dough are pressed together to enclose a region which is full of other ingredients. A depiction of this construction is given in figure 1.
Let be a weakly partially hyperbolic diffeomorphism with a splitting of the form . That is,
hold for all and unit vectors , and . Further suppose that is a repelling fixed point for . Let
be the basin of repulsion of and define . Define a constant small enough that for all unit vectors . Define a smooth function with the following properties:
is an odd function with fixed points exactly at -1, 0, and +1;
the fixed point at zero is expanding with for any unit vector ;
the fixed points -1 and +1 are contracting with ; and
there is a constant such that for all with .
See figure 2. Define a smooth function which equals on a neighbourhood of and equals on a neighbourhood of . Then define by
We now look at the behaviour of tangent vectors under the action of the derivative. If , a tangent vector may be decomposed as with horizontal component and vertical component .
For any point and any tangent vector
define and let and be its horizontal and vertical components respectively.
If is non-zero, then the ratio tends to 0 as .
If , then the angle between and tends to 0 as .
The non-wandering set of is
At the condition on implies that vectors in the horizontal direction are expanded more strongly than vectors in the vertical direction. At all other points in , the condition on implies that a vector in the vertical direction is contracted more strongly than any vector in the horizontal direction. Hence, there is a neighbourhood of and a constant such that if , then
Since for all large , this implies item (1).
From the definition of , note that for all , and item (2) follows directly from fact that is weakly partially hyperbolic. ∎
From lemma 6.1, one may show that on the invariant subset
there is a dominated splitting with three one-dimensional subbundles. We will use to denote this splitting, even though the direction is not uniformly contracting.
The fixed point is hyperbolic with a two-dimensional unstable direction. Let denote the two-dimensional unstable manifold though this point. This manifold may be expressed as the graph of a function from to . Let be the closure of . Then may be expressed as the graph of a continuous function from to which is zero on all points in . One may show, either directly or by a variant of creftype 2.3, that is a submanifold tangent to . Since is uniformly attracting on , this implies that the tangent bundle restricted to has a strongly partially hyperbolic splitting. By symmetry, the closure of the unstable manifold through the point is also a surface, denoted , with similar properties. Thus, the union is a partially hyperbolic set and the intersecton is non-empty.
We now describe how this example may be embedded into . The constant was defined so that the equality holds for all with . By rescaling the vertical direction of , one may, for any given , define a similar example such that this equality holds for all with . Then, take the construction of given in the proof of [Ham16, Theorem 1.2] and replace the dynamics on defined there with that of the defined here. Using lemma 6.1 and the techniques in [Ham16] one may show that this new system has a global dominated splitting and that, outside a basin of repulsion, this dominated splitting is partially hyperbolic. This establishes all of the properties listed in creftype 2.8.
As a final note, it is possible to define a variation on this example by composing with the reflection . This new system has two compact center-unstable tori which intersect and which are the images of each other.
7. No calzones
The last section constructed an example which was only partially hyperbolic on a subset of . Here we prove creftype 2.7, showing that the example cannot be improved to a global partially hyperbolic splitting. The basic idea of the proof is that the region between the two tori must have finite volume, even after lifting to the universal cover. This region also has unstable curves of infinite length. The “length-versus-volume” argument of [BBI09] then gives a contradiction.
Assume is a partially hyperbolic diffeomorphism of a 3-manifold , and that and are two intersecting compact -submanifolds. Up to replacing by an iterate, assume each is -invariant. Up to replacing by a double cover, assume is orientable. The results in [RHRHU11] then imply that is either
the suspension of “minus the identity” on , or
the suspension of a hyperbolic toral automorphism on .
We only consider the case The other two cases have analogous proofs. Further, after applying a change of coordinates to the system, we assume without loss of generality that .
The lifted partially hyperbolic map on the universal cover is a finite distance from a map of the form where is linear and hyperbolic. The subset covers and is invariant under the lifted dynamics. By a slight abuse of notation, if