Firstly, I would like to apologize for the long hiatus between the first and the second post on Asaoka’s theorem. Basically, my blog schedule was somewhat affected by the World Chess Championship! By the way, the third and fifth matches between V. Anand and V. Kramnik were really interesting: Anand won the third match with a variant of the semi-slav defense and he managed to use the SAME move (ok not exactly the same because he interposed the rook before the bishop) three days after (in the fifth match) but Kramnik (and his mates) were still unable to find an adequate defense! đź™‚

Anyway, since I’m not an expert in chess, let me return to the main purpose of this post. Following the plan of the previous post, today we’ll discuss the proof of Asaoka’s theorem:

**Theorem** (Asaoka)**.** Any transitive codimension-one Anosov flow is topologically conjugated to a smooth volume-preserving Anosov flow.

**Remark. **It is not hard to show that volume-preserving Anosov systems are transitive (exercise). Thus, Asaoka’s theorem is a converse of this fact *in the case of codimension-one Anosov flows*. In fact, it is a good open problem to know whether a similar statement is true in general.

Roughly speaking, the proof of this theorem has two steps:

- 1st step: a generalization of Elise Cawley’s work in order to show that one can
*deform*any codimension-one Anosov flow*inside*its topological conjugacy class so that the derivative along the stable and unstable fits any ‘coherently’ prescribed dynamical behaviour; in particular, one can deform the flow in order to get a topologically equivalent flow preserving a HĂ¶lder continuous volume form; - 2nd step: by a -small perturbation of a flow preserving a HĂ¶lder continuous volume form, one can get a flow preserving a smooth volume form (via the so-called ‘pasting lemma‘ technique developed by Alexander Arbieto and myself);

Since it is well-known that Anosov flows are structurally stable (i.e., they are topologically conjugated to any -nearby flow), these two steps complete the proof of Asaoka’s theorem.

**Remark 1. **Concerning the 1st step, it turns out that the resulting flow can’t be expected to preserve a smooth volume form in general. Indeed, this happens due to the lack of regularity of invariant foliations of Anosov flows: typically, they are only even when the codimension-one Anosov flow is .

Now let us turn to the details.

–**Generalization of Cawley’s deformation arguments**–

Consider a codimension-one Anosov flow with a hyperbolic splitting . For sake of concreteness, we assume that the unstable direction is one-dimensional (of course this can always be achieved by a time change ). Denote by the associated *strong-unstable* foliation and let the holonomy map of the *weak stable* foliation (i.e., the foliation tangent to the subbundle ).

**Definition 1****. **We denote by the set of families of measures such that

- is a Borel, non-atomic, positive on open sets, locally finite measure on for all ;
- implies ;
- the Radon-Nikodym derivative at is well-defined for any and close to ; morever is HĂ¶lder continuous on both variables and .

The next result is a precise statement of the generalization of Cawley’s work (quoted in the 1st step of the outline) about the realization of any dynamical behaviour along the unstable direction:

**Theorem 1 **(Radon-Nikodym realization theorem)**. **Given any positive HĂ¶lder continuous function , there are and such that

**Remark 2. **In fact, the reader should notice that we are *not* deforming the flow in order to obtaining the desired Radon-Nikodym derivative (although we promised to do so in the 1st step); of course, we could do it, but this would lead to *technical problems* which we are going to avoid in the following way: instead of deforming *directly* the flow (in order to get a desired dynamical behaviour), we keep the same flow at the cost of using the family of measures to deform the *differentiable structure* of the manifold. In other words, we make a little change of point of view: a deformed flow on a given manifold corresponds to a fixed flow on a deformed manifold (with the advantage that the deformation of the differentiable structure is easier to control than the deformation of the flow).

**Proof of theorem 1. **The basic idea here is the following: Anosov flows can be accurately modeled by the suspension flow over a subshift of finite type: more precisely, one can consider a *Markov partition * so that the dynamics of any point (outside the boundaries of ) can be ‘codified’ by its *itinerary* with respect to the elements of this partition (i.e., at each time we look at the number indexing the rectangle containing our point). For more details on the suspension flow (over an arbitrary discrete time dynamical system) the reader can consult my previous post and for more explanations about Markov partitions for hyperbolic systems the reader can see the excelent classical book of R. Bowen. It follows that the original Anosov flow and the suspension of the subshift are HĂ¶lder conjugated, so that the replacement of the Anosov system by this ‘toy model’ is harmless for our purposes.

Next, one look at the toy model provided by the suspension of the subshift with a mixing transition matrix and it turns out that the similar statement is true by the theory of equilibrium states (essentially the measure we are searching is the Gibbs measure obtained by maximizing the pressure of a certain potential). I.e., given a HĂ¶lder continuous function , take the *reference* measure for the *Gibbs (**-invariant) measure* attaining the topological pressure of among all -invariant measures (that is, ). It holds . Moreover, if verifies , we can find a constat such that the *Bowen’s equation* holds. On the other hand, after doing the translation of the Radon-Nikodym problem for to the Radon-Nikodym problem for the subshift with transition matrix A that it suffices to find a measure such that

where is a HĂ¶lder continuous function with . As we saw above, it suffices to use the theory of equilibrium states with the potential to obtain the desired measure.

Unfortunately, since the theory of equilibrium states for subshifts of finite type takes an entire post by itself, I’m unable to give further details of this argument here. I refer the reader to the sections 2 and 3 of Asaoka’s paper for further details.

**Remark 3. **Although it was not explicitly stated in the ‘proof’ of theorem 1, we are strongly using the codimension-one hypothesis here (as Cawley did when dealing with Anosov diffeomorphisms of ).

Now we proceed to investigate the deformation of the Anosov flow versus the deformation of the differentiable structure of the manifold. Recall that the weak stable foliation is . So, if we fix a (HĂ¶lder) Riemannian metric , it makes sense to consider an orthogonal splitting and the HĂ¶lder continuous flows and where (resp. ) denotes the orthogonal projection onto (resp. ). Also, we introduce the *cocycles* and over : the name cocycle comes from the property

and

We apply the Radon-Nikodym realization theorem with , so that we get a constant and a family of measures verifying

.

**Lemma 1. **.

**Proof.** Take a one-dimensional foliation transverse to . Note that this is possible since is a codimension-one Anosov flow. Fix an atlas on and a parametrization such that

- and ;
- and (where );
- and .

Using this atlas , the parametrizations and the family of measures provided by theorem 1, we can change the differentiable structure of in the following way: we define and we declare that the atlas given by

is our new differentiable structure. In order to avoid some confusion, we denote by the manifold equipped with this new atlas. It is not hard to check that is a manifold (although the ‘identity’ map is only a bi-HĂ¶lder homeomorphism). See lemmas 4.2 and 4.3 of Asaoka’s paper. In any case, it turns out that the flow is (despite the fact that it is generate by a HĂ¶lder continuous vector field). Furthermore, one can directly check that

and ,

where and (observe that is a HĂ¶lder continuous metric).

Assume *momentarily* that is generated by a (Anosov) vector field (recall that it is only generated by a HĂ¶lder vector field at this point). Then, we have , so that

.

In particular, there exists such that , i.e., . On the other hand, we know that . Hence, by computing this last expression at , we get

.

Since is a Anosov flow, it follows that (because contracts any direction tangent to which is transverse to the flow direction). Hence, .

Finally, it remains to get rid of the assumption that the vector field generating is . This can be accomplish by perturbation of the vector field (in the local coordinates provided by ). It follows that this nearby (deformed) vector field is Anosov (and topologically conjugated to the initial system) so that the previous argument applies: indeed, although the initial vector field associated to is only HĂ¶lder (and it does not make sense to say that it is Anosov), it ‘remembers’ that it was constructed from the Anosov smooth system . In particular, after a little bit of technical work (see lemma 4.4 of Asaoka’s paper), we can show that the initial HĂ¶lder vector field behaves as an Anosov system (in the sense that nearby smooth systems are Anosov and topologically conjugated to ). This completes the proof.

**Theorem 2. ** is topologically conjugated to an Anosov flow preserving a HĂ¶lder continuous volume form.

**Proof.** We have that is topologically conjugated to an Anosov system such that . In particular, it follows that preserves the volume form associated to the HĂ¶lder metric . Sending back this structure (from to ) via the HĂ¶lder homeomorphism gives the desired conclusion.

At this point, the proof of Asaoka’s theorem is essentially complete: up to the somewhat boring fact that the volume form is HĂ¶lder continuous (although one would like to get smooth volume form for the application to Verjovsky conjecture because S. Simic’s method needs some regularity). We overcome this technical problem in the next section.

–**‘Pasting lemma’ technique and the regularization of the volume form**–

A basic problem in order to get a smooth volume form is the following: since the differentiable structure on is only , our volume forms can’t be more regular than . However, one can bypass this problem with the following result of Hart:

**Theorem 3 **(Hart)**. **For any foliation on a manifold , there exists a diffeomorphism on such that is a subbundle of . Moreover, can be taken arbitrarily ** **close to the identity.

The proof of this result is not very difficult (for a short proof [of 1 page] of it see the appendix A of Asaoka’s paper), but we will skip it here. An interesting direct corollary of this theorem is the following fact:

**Corollary 1. **Given a manifold and an oriented foliation , we can find a differentiable structure on which is compatible with the initial structure such that is generated by a vector field.

We apply this corollary in the context of the theorem 2. We consider the orbit foliation of on the manifold . Since is , the corollary 1 gives a smooth structure on compatible with the initial structure and a nearby Anosov vector field generating such that preserves a HĂ¶lder volume form.

In order to conclude the proof of Asaoka’s theorem, it suffices to construct a vector field preserving a smooth volume form such that is nearby to (in the topology). To do so, we fix a Riemannian metric on and we consider a HĂ¶lder function such that for all (where is the flow generated by ). Note that this implies

.

Thus, although is only HĂ¶lder, we have that is differentiable along and

.

In particular, is a HĂ¶lder function. Now, we consider a function $h$ such that $h$ is close to and is close to , and a vector field $Y$ nearby in the topology.Â Of course, this can be achieved by a standard use of mollifiers. It follows that is close to .

Next, we recall the formula

for any vector field , any (continuous) Riemannian metric and any function differentiable along the direction . Using this formula and the fact , we obtain

.

Therefore, is close to 0. In other words, we are *almost* done since the condition of the type means that the vector field preserves the volume form associated to .

At this point, we ‘adjust’ the vector field by a technique called ‘*Pasting Lemma*‘ (due to my coauthor A. Arbieto and myself in this paper). Initially, we developed the Pasting Lemma to deal with questions related to volume preserving diffeomorphisms (where the corresponding equation is ): the basic idea going back to the seminal paper of Dacorogna and Moser is the simple and powerful remark that one can adjust the diffeomorphism (or vector field) via the resolution of a very well-know PDE, namely, (perhaps with Dirichlet or Neumann boundary condition), but it turns out that this pasting lemma has some applications by other authors, e.g., Bochi, Fayad and Pujals, Araujo and Bessa (besides Asaoka himself as we are going to see).

In the present case, the application is quite direct because the PDE is linear on (while the case of diffeomorphisms is technically more subtle since is highly non-linear). In any case, we fix a reference point and we take the unique solution of the PDE with . Note that exists because is (and, *a fortiori*, ) and (since is compact and boundaryless). Moreover, the Schauder estimates say that is close to 0 because is close to 0. In particular, the vector field is a vector field $C^{1+}$ close to such that

,

i.e., preserves the (smooth) volume form of . This ends the proof of Asaoka’s theorem.

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