Hölder forms and integrability of invariant distributions
We prove an inequality for Hölder continuous differential forms on compact manifolds in which the integral of the form over the boundary of a sufficiently small, smoothly immersed disk is bounded by a certain multiplicative convex combination of the volume of the disk and the area of its boundary. This inequality has natural applications in dynamical systems, where Hölder continuity is ubiquitous. We give two such applications. In the first one, we prove a criterion for the existence of global cross sections to Anosov flows in terms of their expansion-contraction rates. The second application provides an analogous criterion for non-accessibility of partially hyperbolic diffeomorphisms.
Key words and phrases:
2000 Mathematics Subject Classification:49Q15, 37D20, 37D30
Hölder continuity is ubiquitous in dynamical systems. Hölder continuous differential (though not differentiable) forms consequently play an important role there, especially in hyperbolic and partially hyperbolic dynamics. For instance, integrability of various invariant distributions (by which we mean bundles or plane fields) and the holonomy of the corresponding foliations can be expressed in terms of differential forms. Anosov used differential forms extensively for this purpose in his seminal paper .
Assume, for example, that is a Hölder continuous invariant splitting for a diffeomorphism . It is often important to know whether is an integrable distribution. One can locally define independent Hölder 1-forms such that the intersection of their kernels equals . If admits a global frame, these forms can be defined globally. Let . Then , for every vector tangent to (where denotes the inner multiplication by ), so we can write . The Frobenius integrability condition (see, e.g., ) requires that be divisible by , i.e., that , for some 1-form . Recall that this is equivalent to , for all . Since is only Hölder, this condition clearly does not apply. Hartman  (see also Plante ) proved an analogous integrability condition for continuous forms using the notion of the Stokes differential. Namely, is said to be Stokes differentiable if there exists a locally integrable -form such that
for every smoothly immersed -disk with piecewise smooth boundary . The form is then called the Stokes differential of . Hartman showed that is integrable if and only if divides in the above sense. The utility of this result is limited since there are no good criteria for the Stokes differentiability of continuous Hölder forms.
In certain dynamical situations, however, in order to show integrability of an invariant distribution one needs less than the Stokes or Frobenius theorem, as we will demonstrated in this paper. The crucial inequality is given in Theorem A: for any compact manifold , there exist numbers depending on , , and , such that for every Hölder -form () on and every immersed disk with piecewise boundary satisfying ,
The idea behind the proof is simple: we approximate locally by smooth forms , such that and , where is a universal constant. This is done by using the standard technique of regularization. By subtracting and adding from and to in , it is easy to show that
where is a constant depending only on . The trick is to allow to vary over a sufficiently large interval and then find the best by minimizing the right-hand side of (1.1). For this, needs to be sufficiently small. To eliminate the smallness requirement on , we use a special case of the isoperimetric inequality on Riemannian manifolds, supplied by Gromov .
We give two applications of this inequality in dynamical systems. First, we prove a criterion for the existence of a global cross section to an Anosov flow in terms of its expansion and contraction rates (Theorem B). We then translate this result into the language of partially hyperbolic diffeomorphisms and give a criterion for non-accessibility, also in terms of expansion-contraction rates (Theorem C). Both applications have strong limitations in that they apply only to a “small” set of systems. This is not surprising, since “most” distributions are not integrable. However, Theorem C suggests that there is a certain trade-off between the size of the Hölder exponent of the invariant splitting and accessibility: if is better than the standard lower estimate, accessibility is lost. This is illustrated by an example, due to an anonymous referee.
If is a -dimensional immersed submanifold of a Riemannian manifold , will always denote its Riemannian -volume. If is a smooth map between smooth manifolds, will denote its derivative (or tangent map) . For non-negative functions , we will write if there exists a uniform constant such that . If and , we write .
If is a metric space and , recall that a function is called Hölder (or just for short) if
The -norm of is defined by
We thank the anonymous referees for constructive criticism and the example at the end of the paper. While the paper was being revised for publication, Jenny Harrison pointed out that there are connections between Theorem A and some of her work in .
We start with a short overview of regularization of functions and the isoperimetric inequality.
2-A. Regularization in
We briefly review a well-known method of approximating locally integrable functions by smooth ones, which will be used in the proof of the main inequality.
Suppose is locally integrable and define its regularization (or mollification) as the convolution
with chosen so that . Note that the support of is contained in the ball of radius centered at and .
Let be locally integrable. Then:
If , then .
If , then .
If , then
Proof of (a) and (b) can be found in . For (c), we have
If , then the same estimates hold with replaced by .
Observe that since has compact support,
for . Note also that
Assuming , we obtain (d):
If is a -form on an open set , then can be regularized component-wise. Write
where , , , and . By definition, is of class () if and only if each is of class . Define the -regularization of by
where for each , is the -regularization of .
2-B. Regularization on Riemannian manifolds
Suppose now that is a compact Riemannian manifold and fix a finite atlas of . Let be a -form on . This means that is of class in every local coordinate system on , i.e., is on , for each chart . Define the -norm of by
We regularize in each coordinate chart as follows. For each , choose an open set in such that the closure of is contained in and the collection still covers . Since is compact, without loss of generality we can assume that – hence – is bounded. Define
If the representation of in the -coordinates is
then we define the -regularization of on by , where is the -regularization of defined as in (2.2). If , then is defined on , for every .
This produces a family of -forms, with approximating on in the sense of Proposition 2.1. Using partitions of unity, this family can be patched together into a globally defined smooth form; however, for our purposes local regularization will be sufficient.
2-C. Remarks on the isoperimetric inequality
We will also need a special case of the isoperimetric inequality on Riemannian manifolds, which we now briefly review.
and denotes the volume of the unit ball in .
On Riemannian manifolds the situation is more complicated, so we will only discuss a special case we need in this paper.
Given a cycle () in a Riemannian manifold , recall that the isoperimetric problem asks whether there is a volume minimizing chain in , such that . For our purposes it suffices to consider this problem for small . The solution is given by the following result.
2.2 Lemma (, Sublemma 3.4.B’).
For every compact manifold , there exists a small positive constant such that every -dimensional cycle in of volume less than bounds a chain in , which is small in the following sense:
, for some constant depending only on ;
The chain is contained in the -neighborhood of , where .
The following corollary is immediate.
If is a -immersed -dimensional disk with piecewise boundary in a compact manifold with , then there exists a -disk such that ,
and is contained in the -neighborhood of , where .
3. The Main Inequality
We now have a necessary set-up for proving our main inequality.
Let be a compact manifold and let be a -form on , for some and . There exist constants , depending only on , , and , such that for every -immersed -disk in with piecewise boundary satisfying , we have
The proof is divided into three steps. First, we show that the inequality holds for small, sufficiently flat disks in . By sufficiently flat, we mean that the ratio is small enough. Second, we extend this result to compact manifolds. Finally, we use the isoperimetric inequality to remove the smallness assumption on and complete the proof of the theorem.
We now prove the inequality for small, sufficiently flat disks in . Let be bounded (open) domains in such that the closure of is contained in . Define
Observe that . Let be a -form defined on . Then we have:
For every -immersed -disk in with piecewise boundary, satisfying
Here is as in §2-A.
Let be the -regularization of as above. If , then is defined on . Furthermore, by Proposition 2.1
Let be a -disk in satisfying . Subtracting and adding , and using the Stokes theorem, we obtain:
The estimate is valid for all for which is defined on , that is, for . The minimum of
is achieved at , which lies in the permissible range . This minimum equals
If is , then it is , for all , and it is not hard to check that as ,
Let be a compact Riemannian manifold. We fix an atlas such that each is bounded. For each chart , choose an open set so that:
the closure of is contained in ;
the collection covers .
Let be defined as in (2.4) and denote the Lebesgue number of the covering by . This means that for every set , if , then , for some chart .
Since is finite and the sets are relatively compact, and are finite and positive.
If is a -form on , then for every -immersed -disk with piecewise boundary in satisfying and
where is the same as above, was defined in (2.3), and is a constant depending only on , , and .
Let be a disk satisfying the above assumptions. Since , there exists a chart such that . Observe that
Therefore, we can use the change of variables formula and apply Proposition 3.1 to on . We obtain:
The completes the proof of the proposition with . ∎
To extend the inequality to all small disks, we proceed as follows. Let
and is contained in the -neighborhood of , with . If , we can simply take , so without loss we assume .
The above assumptions imply
so we can apply Proposition 3.2 to on . This yields
If , then , so the assumption is superfluous.
The estimate also holds for “long, thin” disks , namely, those that can be decomposed into finitely many small disks such that , and Theorem A applies to each . For then and
Theorem A is also valid for immersed submanifolds with piecewise smooth boundary. The proof goes through word for word.
4. Global cross sections to Anosov flows
Recall that a non-singular smooth flow on a closed (compact and without boundary) Riemannian manifold is called Anosov if there exists a -invariant continuous splitting of the tangent bundle,
and constants , , and such that for all ,
The center bundle is one dimensional and generated by the vector field tangent to the flow. The distributions , and are called the strong unstable, strong stable, center unstable, and center stable bundles, respectively. Typically they are only Hölder continuous [12, 11], yet uniquely integrable , giving rise to continuous foliations denoted by , and , respectively. Recall that a distribution is called uniquely integrable if it is tangent to a foliation and every differentiable curve everywhere tangent to is wholly contained in a leaf of the foliation.
The idea of studying the dynamics of a flow by introducing a (local or global) cross section dates back to Poincaré. Recall that a smooth compact codimension one submanifold of is called a global cross section for a flow if it intersects every orbit transversely. If this is the case, then every point returns to , defining the Poincaré or first-return map . The flow can then be reconstructed by suspending under the roof function equal to the first-return time [6, 13, 19].
The Poincaré map of a global cross section to an Anosov flow is automatically an Anosov diffeomorphism. Therefore, any classification of Anosov diffeomorphisms immediately translates into a classification of the corresponding class of suspension Anosov flows.
Geometric criteria for the existence of global cross sections to Anosov flows were obtained by Plante , who showed that the flow admits a smooth global cross section if the distribution is (uniquely) integrable. He also showed that is integrable if and only if the foliations and are jointly integrable. This means that in every joint foliation chart for and , the -holonomy takes -plaques to -plaques. Joint integrability of and (in that order) is defined analogously; by symmetry, is uniquely integrable if and only if and are jointly integrable.
We now present a criterion for the existence of a global cross section to an Anosov flow in terms of its expansion-contraction rates.
Let and be in the same local strong unstable manifold of an Anosov flow . Assume and are close enough so that they lie in a foliation chart for both and .
If is of class , a immersed 2-disk is called a -disk if:
Its boundary is the concatenation of four simple paths: , where , , for some as above; furthermore, and , where is the terminal point of .
is a union of -arcs, i.e., arcs contained in the strong stable plaques.
We will call the base of and its sides. See Fig. 1.
Define a 1-form on by requiring
Since is of class , so is . It is clear that is invariant under the flow: , for all .
If is of class , then the following statements are equivalent.
and are jointly integrable.
, for every -disk .
Follows directly from the definitions of and joint integrability of and . ∎
Suppose is a Anosov flow on a closed Riemannian manifold . Assume:
and , for all , and some constants , , and .
is of class .
, where is the Hölder exponent of .
Then admits a global cross section.
Let be an -disk with base and sides , as above. Then:
, for all .
There exists a vector field tangent to with flow such that can be parametrized by
for some continuous function . Since is a parametrization of , the area element of is
By the chain rule,
The vector decomposes into relative to the splitting . Since , as , and is constant, it follows that for ,
Clearly, . Therefore,
which is dominated by , as . This clearly implies the claim of the lemma. ∎
By Lemma 4.2, we need to show that
for every -disk . It is enough to prove this for small . The idea is to use the flow invariance of and change of variables,
and then apply Theorem A to show that the right-hand side converges to zero, as . Since is very “long”, we cannot use Theorem A directly. However, is also very “thin”, since the length of its -sides go to zero, as . More precisely,
where and . We therefore proceed by cutting into -disks , as in Figure 2. We decompose as , so that for each , gives rise to a -disk with . To determine how large has to be as a function of , recall that we need , for each , in order to apply Theorem A. Using the notation (with clear meaning), for each we have:
where is some constant depending only on the hyperbolicity of the flow and . So to ensure , we can take to be of the order . More precisely, assuming is so large that , choose so that , i.e.,
We also require that so that .
By Lemma 4.3, we have
Applying Theorem A to each , we obtain the following estimate:
since . Letting , we obtain , as desired. ∎
It is likely that Theorem B could be slightly improved by using the extra smoothness of along the leaves of the center unstable foliation. This extra smoothness comes from the fact that along the leaves of the center unstable foliation , the strong unstable distribution (assumed to be only Hölder) is actually as smooth as the flow, i.e., . The condition in Theorem B would then be replaced by a weaker one , where .
It needs to be pointed out that the assumptions (b) and (c) of Theorem B are quite restrictive and are satisfied only by a small set of Anosov flows. However, once a system does verify (b) and (c), by structural stability there exists a neighborhood of such that each flow in admits a global cross section.
In this section we prove a sufficient condition for a partially hyperbolic diffeomorphism to be non-accessible.
Recall that a diffeomorphism of a compact Riemannian manifold is called partially hyperbolic if the tangent bundle of splits continuously and invariantly into the stable, center, and unstable bundle, , such that exponentially contracts , exponentially expands and this hyperbolic action on dominates the action of on . The stable and unstable bundles are always uniquely integrable, giving rise to the stable and unstable foliations, . In contrast, the center bundle , the center stable , and the center unstable bundle are not always integrable. If they are, is called dynamically coherent (cf., [3, 17, 4]).
A partially hyperbolic diffeomorphism is called accessible if every two points of can be joined by an -path, that is, a continuous piecewise smooth path consisting of finitely many arcs lying in a single leaf of or a single leaf of .
If is dynamically coherent and the foliations and are jointly integrable (in the same sense as in Section §4), then it is clear that is not accessible. We can also speak of joint integrability of and (in that order), which is defined analogously; it also implies non-accessibility.
Let be a partially hyperbolic diffeomorphism with and integrable center-unstable bundle . Assume that both and have dimension . We now define objects that will play the role of -disks in this context.
Assume is and pick an arbitrary and . Let be an -dimensional -immersed surface (with piecewise boundary) contained in and define to be a -immersed -disk (or “cube”) by making the following requirements:
is foliated by arcs tangent to ;