Posted by: matheuscmss | October 22, 2008

## Asaoka’s lecture on Verjovsky conjecture – part II

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 $C^1$-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 $C^1$-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 $C^{1+}$ even when the codimension-one Anosov flow is $C^{\infty}$.

Now let us turn to the details.

Generalization of Cawley’s deformation arguments

Consider a codimension-one Anosov flow $\phi^{t}$ with a hyperbolic splitting $TM = E^s\oplus T\phi\oplus E^u$. For sake of concreteness, we assume that the unstable direction $E^u$ is one-dimensional (of course this can always be achieved by a time change $t\mapsto -t$). Denote by $\mathcal{F}^{uu}$ the associated strong-unstable foliation and let $h_{pq}:\mathcal{F}^{uu}(p)\to\mathcal{F}^{uu}(q)$ the holonomy map of the weak stable foliation $\mathcal{F}^s$ (i.e., the foliation tangent to the subbundle $E^s\oplus T\phi$).

Definition 1. We denote by $\mathcal{M}(\phi)$ the set of families of measures $\nu= \{\nu\}_{p\in M}$ such that

• $\nu_p$ is a Borel, non-atomic, positive on open sets, locally finite measure on $\mathcal{F}^{uu}(p)$ for all $p\in M$;
• $\mathcal{F}^{uu}(p)=\mathcal{F}^{uu}(q)$ implies $\nu_p=\nu_q$;
• the Radon-Nikodym derivative $\frac{d(\nu_p\circ h_{pq})}{d\nu_q}(q)$ at $q$ is well-defined for any $p\in M$ and $q$ close to $p$; morever $\frac{d(\nu_p\circ h_{pq})}{d\nu_q}(q)$ is HĂ¶lder continuous on both variables $p$ and $q$.

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 $f:M\to\mathbb{R}$, there are $\nu\in\mathcal{M}(\phi)$ and $\rho>0$ such that

$\log\left(\frac{d(\nu_{\phi^t(p)}\circ \phi^t)}{d\nu_p}\right)(p) = \rho\cdot\int_0^t f\circ\phi^{\tau}(p)d\tau.$

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 $\nu$ 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 $\{R_1,\dots, R_n\}$ so that the dynamics of any point (outside the boundaries of $R_j$) can be ‘codified’ by its itinerary with respect to the elements of this partition (i.e., at each time we look at the number $j$ indexing the rectangle $R_j$ 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 $\sigma_A$ with a mixing transition matrix $A$ 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 $\ell$, take the reference measure $\mu_{\ell}$ for the Gibbs ($\sigma_A$-invariant) measure $\nu_{\ell}\ll\mu_{\ell}$ attaining the topological pressure $P_{\sigma_A}(\ell)=\sup\{h_{\nu}(\sigma_A)+\int k_{\ell}d\nu\}$ of $\ell$ among all $\sigma_A$-invariant measures (that is, $P_{\sigma_A}(\ell) = h_{\nu_{\ell}}+\int k_{\ell}d\nu_{\ell}$). It holds $\frac{d(\mu_{\ell}\circ \sigma_A)}{d\mu_{\ell}}(x)=-\ell(x)+P_{\sigma_A}(\ell)$. Moreover, if $\ell$ verifies $\inf\limits_{x\in\Sigma_A}\sum\limits_{j=0}^{m-1}\ell\circ\sigma^j(x)>0$, we can find a constat $\rho_{\ell}$ such that the Bowen’s equation $P_{\sigma_A}(-\rho_{\ell}\cdot \ell)=0$ holds. On the other hand, after doing the translation of the Radon-Nikodym problem for $\phi$ to the Radon-Nikodym problem for the subshift $\sigma_A$ with transition matrix A that it suffices to find a measure $\mu$ such that

$\log\frac{d(\mu\circ\sigma_A)}{d\mu} = \rho\cdot f_A$

where $f_A$ is a HĂ¶lder continuous function with $\inf\limits_{x\in\Sigma_A}\sum\limits_{j=0}^{m-1}f_A\circ\sigma^j(x)>0$. As we saw above, it suffices to use the theory of equilibrium states with the potential $f_A$ 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. $\square$

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 $T^2$).

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 $\mathcal{F}$ is $C^{1+}$. So, if we fix a (HĂ¶lder) Riemannian metric $g$, it makes sense to consider an orthogonal splitting $TM=T\phi\oplus E_{\phi}\oplus E_{\phi}^{\perp}$ and the HĂ¶lder continuous flows $N_{\mathcal{F}}\phi^t=\pi_{\phi}\circ D_{\mathcal{F}}\phi^t|_{E_{\phi}}$ and $N_{\mathcal{F}}^{\perp}\phi^t=\pi_{\phi}^{\perp}\circ D_{\mathcal{F}}\phi^t|_{E_{\phi}}$ where $\pi_{\phi}$ (resp. $\pi_{\phi}^{\perp}$) denotes the orthogonal projection onto $E_{\phi}$ (resp. $E_{\phi}^{\perp}$). Also, we introduce the cocycles $\alpha_{\phi}(p,t,\mathcal{F},g) = \log\det_g(N_{\mathcal{F}\phi^t})_p$ and $\alpha_{\phi}^{\perp}(p,t,\mathcal{F},g) = \log\det_g(N_{\mathcal{F}^{\perp}\phi^t})_p$ over $\phi^t$: the name cocycle comes from the property

$\alpha_{\phi}(p,t+s,\mathcal{F},g)=\alpha_{\phi}(p,t,\mathcal{F},g)+ \alpha_{\phi}(\phi^t(p),s,\mathcal{F},g)$

and

$\alpha_{\phi}^{\perp}(p,t+s,\mathcal{F},g)=\alpha_{\phi}^{\perp}(p,t,\mathcal{F},g)+ \alpha_{\phi}^{\perp}(\phi^t(p),s,\mathcal{F},g)$

We apply the Radon-Nikodym realization theorem with $f(p)=-\frac{\partial\alpha_{\phi}}{\partial t}(p,0,\mathcal{F},g)$, so that we get a constant $\rho_{\phi}>0$ and a family of measures $\nu\in \mathcal{M}(\phi)$ verifying

$\log\left(\frac{d(\nu_{\phi^t(p)}\circ \phi^t)}{d\nu_p}\right)(p) = \rho_{\phi}\cdot\int_0^t f\circ\phi^{\tau}(p)d\tau=-\rho_{\phi}\cdot\alpha_{\phi}(p,t,\mathcal{F},g)$.

Lemma 1. $\rho_{\phi}=1$.

Proof. Take $\mathcal{L}$ a $C^{2+}$ one-dimensional foliation transverse to $\mathcal{F}$. Note that this is possible since $\phi$ is a codimension-one Anosov flow. Fix an atlas $a_p:U_p\to\mathbb{R}^n$ on $M$ and a $C^{1+}$ parametrization $L_p:(-1,1)\to \mathcal{F}^{uu}(p)$ such that

• $a_p(p)=0$ and $a_p(U_p)=(-1,1)^{n}$;
• $a_p^{-1}((-1,1)^{n-1}\times \{y\})\subset\mathcal{F}(q)$ and $a_p^{-1}(\{x\}\times (-1,1))\subset\mathcal{L}(q)$ (where $a_p(q)=(x,y)$);
• $L_p(0)=p$ and $L_p(y)\in\mathcal{F}^{uu}(p)\cap a_p^{-1}((-1,1)^{n-1}\times \{y\})$.

Using this atlas $a_p$, the parametrizations $L_p$ and the family of measures $\nu$ provided by theorem 1, we can change the differentiable structure of $M$ in the following way: we define $\eta_p(y)=\int_0^y d(\nu_p\circ L_p)$ and we declare that the atlas $\widehat{a}_p$ given by

$\widehat{a}_p\circ a_p^{-1}(x,y):=(x,\eta_p(y))$

is our new differentiable structure. In order to avoid some confusion, we denote by $\widehat{M}$ the manifold $M$ equipped with this new atlas. It is not hard to check that $\widehat{M}$ is a $C^{1+}$ manifold (although the ‘identity’ map $i:M\to\widehat{M}$ 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 $\widehat{\phi}^t=i\circ \phi^t\circ i^{-1}$ is $C^{1+}$ (despite the fact that it is generate by a HĂ¶lder continuous vector field). Furthermore, one can directly check that

$\alpha_{\widehat{\phi}}(i(p),t,\widehat{\mathcal{F}},\widehat{g})=\alpha_{\phi}(p,t,\mathcal{F},g)$ and $\alpha_{\widehat{\phi}}^{\perp}(i(p),t,\widehat{\mathcal{F}},\widehat{g})=-\rho_{\phi}\cdot\alpha_{\phi}(p,t,\mathcal{F},g)$,

where $\widehat{\mathcal{F}}=i(\mathcal{F})$ and $\widehat{g}=i^*g$ (observe that $\widehat{g}$ is a HĂ¶lder continuous metric).

Assume momentarily that $\widehat{\phi}$ is generated by a $C^{1+}$ (Anosov) vector field (recall that it is only generated by a HĂ¶lder vector field at this point). Then, we have $\widehat{\phi}^t(\widehat{M})=\widehat{M}$, so that

$vol_{\widehat{g}}(\widehat{\phi}^t(M))=\int_{\widehat{M}} |\det_{\widehat{g}}D\widehat{\phi}^t|=\int_{\widehat{M}} 1 = vol_{\widehat{g}}(\widehat{M})$.

In particular, there exists $p_0\in\widehat{M}$ such that $|\det_{\widehat{g}}D\widehat{\phi}^t(p_0)|=1$, i.e., $\log|\det_{\widehat{g}}D\widehat{\phi}^t(p_0)|=0$. On the other hand, we know that $\log|\det_{\widehat{g}}D\widehat{\phi}^t|=\alpha_{\widehat{\phi}} + \alpha_{\widehat{\phi}}^{\perp}=(1-\rho_{\phi})\alpha_{\widehat{\phi}}$. Hence, by computing this last expression at $p_0$, we get

$0=(1-\rho_{\phi})\alpha_{\widehat{\phi}}$.

Since $\widehat{\phi}$ is a $C^{1+}$ Anosov flow, it follows that $\sup\limits_{p\in M} \alpha_{\widehat{M}}(p)<0$ (because $\widehat{\phi}$ contracts any direction tangent to $\widehat{\mathcal{F}}$ which is transverse to the flow direction). Hence, $\rho_{\phi}=1$.

Finally, it remains to get rid of the assumption that the vector field generating $\widehat{\phi}$ is $C^{1+}$. This can be accomplish by perturbation of the vector field (in the $C^{1+}$ local coordinates provided by $a_p$). 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 $\widehat{\phi}$ 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 $\phi$. 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 $\widehat{\phi}$). This completes the proof. $\square$

Theorem 2. $\phi$ is topologically conjugated to an Anosov flow $\psi$ preserving a HĂ¶lder continuous volume form.

Proof. We have that $\phi$ is topologically conjugated to an Anosov system $\psi$ such that $\alpha_{\widehat{\psi}} + \alpha_{\widehat{\psi}}^{\perp} = 0$. In particular, it follows that $\widehat{\psi}$ preserves the volume form associated to the HĂ¶lder metric $\widehat{g}$. Sending back this structure (from $\widehat{M}$ to $M$) via the HĂ¶lder homeomorphism $i:M\to \widehat{M}$ gives the desired conclusion. $\square$

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 $\widehat{M}$ is only $C^{1+}$, our volume forms can’t be more regular than $C^{1+}$. However, one can bypass this problem with the following result of Hart:

Theorem 3 (Hart). For any $C^{r+\alpha}$ foliation $\mathcal{G}$ on a $C^{\infty}$ manifold $M$, there exists a $C^{r+\alpha}$ diffeomorphism $H$ on $M$ such that $T H(\mathcal{G})=DH(T\mathcal{G})$ is a $C^{r+\alpha}$ subbundle of $TM$. Moreover, $H$ can be taken arbitrarily $C^r$ 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 $C^{1+}$ manifold $\widehat{M}$ and an oriented $C^{1+}$ foliation $\widehat{O}$, we can find a $C^{\infty}$ differentiable structure on $\widehat{M}$ which is compatible with the initial $C^{1+}$ structure such that $\widehat{O}$ is generated by a $C^{1+}$ vector field.

We apply this corollary in the context of the theorem 2. We consider the orbit foliation $\widehat{O}$ of $\widehat{\psi}$ on the manifold $\widehat{M}$. Since $\widehat{O}$ is $C^{1+}$, the corollary 1 gives a smooth structure on $\widehat{M}$ compatible with the initial $C^{1+}$ structure and a $C^{1+}$ nearby Anosov vector field $\widehat{X_0}$ generating $\widehat{O}$ such that $\widehat{X_0}$ preserves a HĂ¶lder volume form.

In order to conclude the proof of Asaoka’s theorem, it suffices to construct a vector field $X$ preserving a smooth volume form such that $X$ is nearby to $X_0$ (in the $C^1$ topology). To do so, we fix a $C^{\infty}$ Riemannian metric $g_0$ on $M$ and we consider $h_0$ a HĂ¶lder function such that $\det_{e^{h_0}g_0}D\phi_0^t(p)=1$ for all $p\in M$ (where $\phi_0$ is the flow generated by $X_0$). Note that this implies

$0=\log\det_{e^{h_0}g_0}D\phi_0^t(p) = \log\det_{g_0}D\phi_0^t(p)+dim(M)\cdot(h_0(\phi_0^t(p)) - h_0(p))$.

Thus, although $h_0$ is only HĂ¶lder, we have that $h_0$ is differentiable along $X_0$ and

$dim(M)\cdot X_0 h_0 = -div_{g_0}X_0$.

In particular, $X_0 h_0$ is a HĂ¶lder function. Now, we consider a $C^{\infty}$ function $h$ such that $h$ is $C^{0+}$ close to $h_0$ and $X_0h$ is $C^{0+}$ close to $X_0h_0$, and a $C^{\infty}$ vector field $Y$ nearby $X_0$ in the $C^{1+\alpha}$ topology.Â  Of course, this can be achieved by a standard use of mollifiers. It follows that $div_{e^h g_0}Y$ is $C^{0+}$ close to $div_{e^h g_0}X_0$.

Next, we recall the formula

$div_{e^h g} X(p) = dim(M)\cdot Xh + div_g X$

for any $C^1$ vector field $X$, any (continuous) Riemannian metric $g$ and any function $h$ differentiable along the direction $X$. Using this formula and the fact $dim(M)\cdot X_0 h_0 = -div_{g_0}X_0$, we obtain

$div_{e^h g_0} Y = (div_{e^h g_0} Y-div_{e^h g_0} X_0)+dim(M)\cdot(X_0 h -X_0 h_0)$.

Therefore, $div_{e^h g_0} Y$ is $C^{0+}$ close to 0. In other words, we are almost done since the condition of the type $div_{g}X=0$ means that the vector field $X$ preserves the volume form associated to $g$.

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 $f$ (where the corresponding equation is $\det_{g} Df=1$): 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, $\Delta a = b$ (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 $div_g X=0$ is linear on $X$ (while the case of diffeomorphisms is technically more subtle since $\det_g Df$ is highly non-linear). In any case, we fix a reference point $p_0\in M$ and we take the unique solution $f$ of the PDE $\Delta f = -div_{e^h g_0}Y$ with $f(p_0)=0$. Note that $f$ exists because $div_{e^h g_0} Y$ is $C^{\infty}$ (and, a fortiori, $C^{0+}$) and $\int_M div_{e^h g_0} Y=0$ (since $M$ is compact and boundaryless). Moreover, the Schauder estimates say that $f$ is $C^{2+}$ close to 0 because $div_{e^h g_0}Y$ is $C^{0+}$ close to 0. In particular, the vector field $X=Y+grad_{e^h g_0} f$ is a vector field $C^{1+}$ close to $Y$ such that

$div_{e^h g_0}X = div_{e^h g_0}Y + div_{e^h g_0}grad_{e^h g_0}f=div_{e^h g_0}Y+\Delta_{e^hg_0} f=0$,

i.e., $X$ preserves the (smooth) volume form of $e^{h}g_0$. This ends the proof of Asaoka’s theorem.

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