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# Partially Hyperbolic Dynamics and 3-Manifold Topology

## 1. Introduction

The hairy ball theorem implies that the 2-dimensional sphere cannot admit a vector field without singularities. This is just an example of a restriction imposed by the topology of a phase space on the possible dynamics it can support. In this note we would like to present and relate two results in this direction. These are about restrictions imposed by the topology of certain 3-manifolds on the dynamics it can support.

The first result we will present corresponds to the theory of Anosov flows and was proved by Margulis Mar when he was still an undergraduate student. It appears in an appendix to a paper of Anosov and Sinai AnS. The result was later revisited by Plante and Thurston PT who extended its scope and proposed a different approach that used some finer properties of foliations.

A flow generated by a (smooth) vector field on a closed manifold is said to be an *Anosov flow* if there is a continuous splitting of the tangent bundle -invariant satisfying that there is so that for every ( a unit vector we have that ) This immediately implies that stable vectors (i.e., those in . are contracted exponentially by ) while unstable vectors (i.e., those in are expanded exponentially fast by ) .

Examples of Anosov flows include geodesic flows in negative curvature Ano as well as suspensions of certain toral automorphisms. Their definition goes back at least to the paper of Anosov and Sinai AnS where they extracted the properties from geodesic flows in negative curvature needed to obtain ergodicity. We point out that in 3-manifolds we know that Anosov flows contain the space of robustly transitive flows (i.e., flows so that every perturbation has some dense orbit); see DoBDV. The result by Margulis and Plante-Thurston says that if a 3-manifold admits an Anosov flow, then its fundamental group has exponential growth (see Theorem 3.1) and implies in particular that 3-manifolds such as the sphere or the 3-torus do not admit such flows.

The second result is more recent and essentially due to Burago and Ivanov BI. This result gives some obstructions for some mapping classes of certain 3-manifolds to admit partially hyperbolic diffeomorphisms. A diffeomorphism is said to be *partially hyperbolic* if the tangent space splits as a direct sum of non-trivial continuous subbundles which are and satisfy that there is some -invariant so that for every if , ( are unit vectors, then )

Naturally, time one maps of Anosov flows (i.e., the diffeomorphism

In dimension 3, we will explain how the result of Burago-Ivanov implies that if a diffeomorphism

The connection between these two results will allow us to briefly comment on the classification of partially hyperbolic diffeomorphisms in 3-manifolds, referring the interested reader to recent surveys such as CHHUHPPotBFFPBFP for a more complete presentation.

## 2. Anosov Flows and Foliations

Consider an Anosov flow *invariant cone-fields*, which we will not define but just point out that these are objects that are robust (i.e., if a system has invariant cone-fields, then this is true in a

The importance of this infinitesimal condition is that it can be pushed into the manifold in a way that one obtains objects whose dynamics mimic the dynamics of the differential map.

We need to say a few words to explain what we mean. Let *uniquely integrable* if through every point

The same is true for

The proof of this result when

The curves tangent to *strong stable* and *weak stable* foliations, which together with their dual strong unstable and weak unstable foliations are one of the main tools to understand the dynamics and geometry of Anosov systems. Let us just state an easy fact about these that we will use later:

This is a direct consequence of the fact that the time

Recall that a *foliation by surfaces* of a 3-manifold *leaves*) that locally look like horizontal planes

A *transversal* to a foliation is an embedded circle which is everywhere transverse to the leaves of

We will not prove this beautiful result which has several expositions. In fact, Novikov’s result is much stronger and implies the existence of what are known as *Reeb components*. One should think that in 3-manifolds compact leaves (or Reeb components) of foliations play the role that singularities play in vector fields in surfaces, and therefore Novikov’s theorem acts as the Poincaré-Bendixon’s theorem in this setting.Footnote^{1} Even if much deeper, there is a part of the proof of Novikov’s theorem (which is indeed enough to rule out homotopically trivial transversal loops for Anosov flows) that is very much modelled in the proof of Poincaré-Bendixon’s theorem. It is known as *Haefliger’s* argument: using the transverse loop, one constructs a disk whose boundary is transverse to the foliation and which is in general position; studying the induced flow on the disk is enough to find a configuration which is not compatible with Anosov flows, and such that with much more work produces a Reeb component. We note that for the partially hyperbolic case to be treated later, the full version of Novikov’s theorem is important.^{✖} We refer the reader to Ca, Chapter 4 for a friendly account of foliations in 3-manifolds.

## 3. Margulis/Plante-Thurston’s Result

In the late 60s Margulis showed the following beautiful result:

A finitely generated group *exponential growth* if for some finite generating set

In a closed manifold

where

It is an easy exercise to show that, up to changing the constants *exponential growth of fundamental group*.

The proof by Margulis Mar is direct and independent of any deep result in foliation theory (even if the foliations are used crucially). Later, Plante and Thurston PT gave a more conceptual proof that works for general codimension 1 Anosov flowsFootnote^{2} i.e., those whose stable or unstable bundle is 1-dimensional^{✖} and uses some deeper results in foliation theory. The proof we shall present here has ingredients from both organized in a way that will lead us naturally to the generalization of these arguments to the classification problem of partially hyperbolic diffeomorphisms in dimension 3.

We emphasize the following fact. In dimension 2, the hairy ball theorem, or the Poincaré-Hopf index theorem, implies that admitting a continuous subbundle is already enough to get some topological obstruction (i.e., only the 2-torus and the Klein bottle admit a continuous splitting of the tangent bundle). However, this is not the case in dimension 3; up to double cover, every closed 3-manifold has trivial tangent bundle. That is,

## 4. The Proof

We provide here a quick proof of Theorem 3.1 based on the original arguments, but probably with a more modern viewpoint. The goal is motivating tools that provide an understanding of the interaction between topology and dynamics.

An easy consequence of Theorem 3.1 is the non-existence of Anosov flows in the sphere

With these elements in hand, we are ready to give the proof. The reader not comfortable with the basics of algebraic topology can use as a model the 3-torus

Margulis’s proof is more elementary since it does not use any deep results about foliations; however, it depends crucially on the fact that the weak stable/unstable foliation is *complete* in the sense that a weak stable/unstable leaf is the union of the strong stable/unstable manifolds through points of a given orbit. This fact fails when one goes to the partially hyperbolic setting. This property is used by Margulis to construct by hand the universal cover of

The proof of Plante and Thurston is similar to the one we present here; however, instead of computing volume they construct many loops that they show are pairwise non-homotopic. For this, they use Haefliger’s argument (cf. footnote Footnote^{1}). In particular, as in the proof presented here, in contrast with Margulis’s proof, it only needs one of the two foliations and hence it extends to codimension 1 Anosov flows. But what is important here is that this line of reasoning does not depend on understanding the internal structure of the codimension 1 foliation, and so it is well suited to be extended in other contexts.

## 5. Classification of Partially Hyperbolic Systems

We will now come to the problem of understanding the structure of general partially hyperbolic systems in 3-dimensional manifolds by modelling the questions and ideas of the work done in the previous section. Here we shall concentrate on the following questions of current research interest which can be considered as continuations of the problem discussed above for Anosov flows:

Which

We refer the reader to CP for a general exposition of the basic facts about partially hyperbolic systems as well as a long list of examples. Here we will concentrate on a few relevant aspects specific to 3-dimensions.

A main difference which makes studying partially hyperbolic diffeomorphisms much harder than Anosov flows is that even if the strong bundles *center stable* and *center unstable* bundles *dynamical coherence* since all the known examples had them. We say that a partially hyperbolic diffeomorphism is *dynamically coherent* if there are

A recent breakthrough result by Burago and Ivanov BI provided a tool for avoiding such an undesirable hypothesis.Footnote^{4} The reason it is undesirable is that it is not easy to check, and several examples have appeared where it is known not to hold.^{✖}

Up to finite cover, there is a Reebless foliation

This implies by iterating backwards that one can choose the foliation to be as close to tangent to *branching foliation*. This is an incredibly useful tool for the study of partially hyperbolic diffeomorphisms, but we do not discuss it here. We note here that the proof of Theorem 5.2 depends very strongly on the fact that

In fact, to show the result it is enough to show that there exists a foliation transverse to

Theorem 5.2 has the following consequence which is the first known topological obstruction for the existence of partially hyperbolic diffeomorphisms:

The sphere

The proof of Theorem 3.1 in §4 has as a moral that to expand a 1-dimensional foliation transverse to a 2-dimensional foliation in a 3-manifold one needs space. This moral extends to the diffeomorphism case, only diffeomorphisms can wrap the manifold onto itself and then obtain expansions without much space.

For instance, a matrix in

With essentially the same proof as for Theorem 3.1 by replacing the stable manifold theorem with Theorem 5.2, one can obtain the following result which provides obstructions for the mapping classes which admit partially hyperbolic diffeomorphisms.

If

Recall that the *mapping torus* of a map

But Theorem 5.2 is indeed stronger, since it can also provide further obstructions thanks to the well-developed theory of Reebless foliations. There are manifolds with exponential growth of fundamental group known not to admit foliations without compact leaves, including some hyperbolic 3-manifolds (see, e.g., Ca, Example 4.4.6). These also provide obstructions to the existence of partially hyperbolic diffeomorphisms. Up to recently, these were more or less all the known obstructions to the existence of partially hyperbolic diffeomorphisms. At the moment of this writing, we do not know any manifold with exponential growth of fundamental group which admits a partially hyperbolic diffeomorphism but does not admit an Anosov flow. But lots of developments have been made recently that give us hope that understanding partially hyperbolic diffeomorphisms is not far from understanding Anosov flows.

## 6. Further Discussion

As mentioned, the obstruction given by Theorem 5.4 is not sharp, so it makes sense to see to what extent one can characterize the homotopy classes of diffeomorphisms of

But somehow, all examples we know build in some way or the other on some Anosov system. The examples in BGHP are constructed by using the cone-field criterium to guarantee partial hyperbolicity together with a careful understanding of the global structure of the invariant bundles. This way, it is possible to construct diffeomorphisms of the manifold which respect transversalities between the bundles, and this allows us to create new partially hyperbolic diffeomorphisms in new isotopy classes. These kinds of constructions are still in their infancy, and it is likely that new examples can be created using these ideas. Nonetheless, there are some manifolds and isotopy classes of diffeomorphisms where the partially hyperbolic dynamics seem amenable to classification, notably hyperbolic and Seifert 3-manifolds BFFPBFP. A notion of *collapsed Anosov flow* has been proposed recently that may account for all new examples, and which needs to be tested against new potential constructions BFP.

In higher dimensions, Anosov systems are far from being classified, and new ways to construct partially hyperbolic examples have been devised GHO, which depend to some extent on Anosov systems, but seem likely to be more flexible and may be combinable with techniques in BGHP. Even the most basic questions in high dimensions remain quite open.

We refer the reader to BDV for a general overview of smooth dynamics and to Wil for a recent account of partial hyperbolicity. In CHHU the reader can find a survey of the dynamics of partially hyperbolic diffeomorphisms specialized to dimension 3, which also touches upon the classification problem.

If the reader wishes to know more about the classification problem of partially hyperbolic diffeomorphisms in dimension 3, then the following references could provide a useful introduction CHHUHPPotBFFPBFP.

## Acknowledgments

The author was partially supported by CSIC-618, FCE-135352, the Minerva Research Foundation Membership Fund, and the grant NSF DMS-1638352. This work was written while the author was a von Neumann Fellow at the Institute for Advanced Study, and he wants to acknowledge the excellent working conditions and environment. Comments of Silvia Ghinassi, Mariana Haim, Santiago Martinchich, Martín Reiris, and Jan Vonk were very helpful in the writing of the note. The author wishes to particularly thank the referees who provided a lot of helpful input to improve the paper.

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Rafael Potrie is an associate professor at Universidad de la República, Uruguay. His email address is rpotrie@cmat.edu.uy.

Article DOI: 10.1090/noti2286

## Credits

Opener photo is courtesy of Free art director via Getty.

Figures 1–4 are courtesy of Rafael Potrie.

Author photo is courtesy of Natalia de León.