## Archive for May, 2010

### On C^1 closing lemma

May 25, 2010

Let $f: M \rightarrow M$ be a diffeomorphism. A point $p$ is non-wandering if for all neighborhood $U$ of $p$, there is increasing sequence $(n_k) \subseteq \mathbb{N}$ where $U \cap f^{n_k}(U) \neq \phi$. We write $p \in \mathcal{NW}(f)$.

Closing lemma: For any diffeomorphism $f: M \rightarrow M$, for any $p \in \mathcal{NW}(f)$. For all $\varepsilon>0$ there exists diffeomorphism $g$ s.t. $||f-g||_{C^1} < \varepsilon$ and $g^N(p) = p$ for some $N \in \mathbb{N}$.

Suppose $p \in \mathcal{NW}(f)$, $\overline{\mathcal{O}(p)}$ is compact, then for any $\varepsilon>0$, there exists $x_0 \in B(p, \varepsilon)$, $k \in \mathbb{N}$ s.t. $f^k(x) \in B(p, \varepsilon)$.

First we apply a selection process to pick an appropriate almost-orbit for the closing. Set $x_i = f^i(x_0), \ 0 \leq i \leq k$.

If there exists $0 < j < k$ where

$\min \{ d(x_0, x_j), d(x_j, x_k) \} < \sqrt{\frac{2}{3}}d(x_0, x_k)$

then we replace the origional finite sequence by $(x_0, x_1, \cdots, x_j)$ or $(x_j, \cdots, x_k)$. Iterate the above process. since the sequence is at least one term shorter after each shortening, the process stops in finite time. We obtain final sequence $(p_0, \cdots, p_n)$ s.t. for all $0 < i < n$,

$\min \{ d(p_0, p_i), d(p_i, p_n) \} \geq \sqrt{\frac{2}{3}}d(p_0, p_n)$.

Since the process is applied at most $k$ times, $x_0, x_k \in B(p, \varepsilon)$, after the first shortening, $d(p, x_{i_1}) \leq \max \{d(p, x_0), d(p, x_k) \} + \sqrt{\frac{2}{3}}d(x_0, x_k)$ $\leq \varepsilon + 2 \sqrt{\frac{2}{3}} \varepsilon$.

i.e. both initial and final term of the sequence is at most $(\frac{1}{2}+ \sqrt{\frac{2}{3}}) 2 \varepsilon$. Along the same line, we have, at the $i$-th shortening, the distance between the initial and final sequence and $p$ is at most $(\frac{1}{2} + \sqrt{\frac{2}{3}} + (\sqrt{\frac{2}{3}})^2 + \cdots (\sqrt{\frac{2}{3}})^i) 2 \varepsilon$. Hence for the final sequence $p_0, p_n \in B(p, 1+2 \sqrt{\frac{2}{3}}/(1-\sqrt{\frac{2}{3}}) \varepsilon) \subseteq B(p, 10 \varepsilon)$.

There is a rectangle $R \subseteq M$ where $p_0, p_n \in \sqrt{\frac{3}{4}}R$
(i.e. shrunk $R$ by a factor of $\sqrt{\frac{3}{4}}$ w.r.t. the center) and for all $0 < i < n, \ p_i \notin R$.

Next, we perturb $f$ in $R$ i.e. find $h: M \rightarrow M$ with $||h||_{C^1} < \delta$ and $h|_{M \backslash R} =$ id. Hence $||h \circ f - f ||_{C^1} < \delta$.

Suppose $R = I_1 \times I_2; L_1, L_2$ are the lengths of $I_1, I_2$, $L_1 < L_2$.
By main value theorem, for all $x \in M, \ d(x, h(x)) < \delta L_1$.
On the other hand, since $p_0 \in \sqrt{\frac{3}{4}}R$, it's at least $\frac{1}{2}(1-\sqrt{\frac{3}{4}})L_1$ away from the boundary of $R$. i.e. there exists bump function $h$ satisfying the above condition and $d(p_0, h(p_0)) > \frac{\delta}{8}(1-\sqrt{\frac{3}{4}})L_1$.

Hence in order to move a point by a distance $L_1$, we need about $1/ \delta$ such bump functions, to move a distance $L_2$, we need about $\frac{L_2}{\delta L_1}$ bumps.

For simplicity, we now suppose $M$ is a surface. By starting with an $\varepsilon$ (and hence $R$) very small, we have for all $0 \leq i \leq N+M, \ f^i(R)$ is contained in a small neighbourhood of $p_i$. Hence on $f^i(B), f^i$ is $C^1$ close to the linear map $p_i + Df^i(p_0)(x-p_0)$. Hence mod some details we may reduce to the case where $f$ is linear in a neighborhood of $\mathcal{O}(p_0)$.

By choosing appropiate coordinate system in $R$, we can have $f$ preserving the horizontal and vertical foliations and the horizontal vectors eventually grow more rapidly than the vertical vectors.

It turns out to be possible to choose $R$ to be long and thin such that for all $i \leq 40 / \delta$, $f^i(R)$ has height greater than width. (note that $M = \lfloor 40/ \delta \rfloor$ bumps will be able to move the point by a distance equal to the width of the original rectangle $R$. Since horizontal vectors eventually grow more rapidly than the vertical vectors, there exists $N$ s.t. for all $N \leq i \leq N+M$, $f^i(R)$ has width greater than its height.
For small enough $\epsilon$, the boxes $f^i(R)$ are disjoint for $0 \leq i \leq N+40/ \delta$. Construct $h$ to be identity outside of

$\displaystyle \bigsqcup_{i=0}^M f^i(R) \sqcup \bigsqcup_{i=N}^{N + M} f^i(R)$

For the first $M$ boxes, we let $h$ preserve the horizontal foliation and move along the width so that $g = h \circ f$ has the property that $g^M(p_n)$ lies on the same vertical fiber as $f^M(p_0)$.

On the boxes $f^{N+i}(R), \ 0 \leq i \leq M$, we let $h$ pushes along the vertical direction so that

$g^{N+M}(p_n) = f^{N+M}(p_0)$

Since iterates of the rectangle are disjoint, for $N+M \leq i \leq n, \ h(p_i) = p_i$, $g(p_i) = f(p_i)$.

Hence $g^n(p_n) = g^{n-(N+M)} \circ g^{N+M}(p_n)$ $= g^{n-(N+M)} f^{N+M}(p_0) = g^{n-(N+M)} (p_{N+M}) = p_n$.

Therefore we have obtained a periodic point $p_n$ of $g$.

Since $p_n \in B(p, 10 \varepsilon)$, we may further perturb $g$ to move $p_n$ to $p$. This takes care of the linear case on surfaces.

### On compact extensions

May 10, 2010

This is again a note on my talk in the Szemerédi’s theorem seminar, going through Furstenberg’s book. In this round, my part is to introduce compact extension.
Let $\Gamma$ be an abelian group of measure preserving transformations on $(X, \mathcal{B}, \mu)$, $\alpha: (X, \mathcal{B}, \mu, \Gamma) \rightarrow ( Y, \mathcal{D}, \nu, \Gamma')$ be an extension map.
i.e. $\alpha: X \rightarrow Y$ s.t. $\alpha^{-1}$ sends $\nu-0$ sets to $\mu-0$ sets;

$\gamma'\circ \alpha (x) = \alpha \circ \gamma (x)$

Definition: A sequence of subsets $(I_k)$ of $\Gamma$ is a Folner sequence if $|I_k| \rightarrow \infty$ and for any $\gamma \in \Gamma$,

$\frac{| \gamma I_k \Delta I_k|}{|I_k|} \rightarrow 0$

Proposition: For any Folner sequence $I = (I_k)$ of $\Gamma$, for any $f \in L^1(X)$, $\displaystyle \frac{1}{|I_k|} \sum_{\gamma \in I_k} \gamma f$ converges weakly to the orthogonal projection of $f$ onto the subspace of $\Gamma$-invariant functions. (Denoted $P(f)$ where $P: L^2(X) \rightarrow L^2_{inv}(X)$.

Proof: Let $\mathcal{H}_0 = P^{-1}(\bar{0}) = (L^2_{inv}(X))^\bot$
For all $\gamma \in \Gamma$,

$\gamma (L^2_{inv}(X)) \subseteq L^2_{inv}(X)$

Since $\Gamma$ is $\mu$-preserving, $\gamma$ is unitary on $L^2(X)$. Therefore we also have $\gamma( \mathcal{H}_0) \subseteq \mathcal{H}_0$.

For $f \in \mathcal{H}_0$, suppose there is subsequence $(n_k)$ where $\displaystyle \frac{1}{|I_{n_k}|} \sum_{\gamma \in I_{n_k}} \gamma (f)$ converges weakly to some $g \in L^2(X)$.

By the property that $\frac{| \gamma I_k \Delta I_k|}{|I_k|} \rightarrow 0$, we have for each $\gamma \in \Gamma$, $\gamma(g) = g, \ g$ is $\Gamma$-invariant. i.e. $g \in (\mathcal{H}_0)^\bot$

However, since $f \in \mathcal{H}_0$ hence all $\gamma(f)$ are in $\mathcal{H}_0$ hence $g \in \mathcal{H}_0$. Therefore $g \in \mathcal{H}_0 \cap (\mathcal{H}_0)^\bot$, $g=\bar{0}$

Recall: 1)$X \times_Y X := \{ (x_1, x_2) \ | \ \alpha(x_1) = \alpha(x_2) \}$.

i.e. fibred product w.r.t. the extension map $\alpha: X \rightarrow Y$.

2)For $H \in L^2(X \times_Y X), \ f \in L^2(X)$,

$(H \ast f)(x) = \int H(x_1, x_2) f(x_2) d \mu_{\alpha(x_1)}(x_2)$

Definition: A function $f \in L^2(X)$ is said to be almost periodic if for all $\varepsilon > 0$, there exists $g_1, \cdots g_k \in L^2(X)$ s.t. for all $\gamma \in \Gamma$ and almost every $y \in Y$,

$\displaystyle \min_{1 \leq i \leq k} || \gamma (f) - g_i||_y < \varepsilon$

Proposition: Linear combination of almost periodic functions are almost periodic.

Proof: Immediate by taking all possible tuples of $g_i$ for each almost periodic function in the linear combination corresponding to smaller $\varepsilon$l.

Definition: $\alpha: (X, \mathcal{B}, \mu, \Gamma) \rightarrow ( Y, \mathcal{D}, \nu, \Gamma')$ is a compact extension if:

C1: $\{ H \ast f \ | \ H \in L^\infty (X \times_Y X) \cap \Gamma_{inv} (X \times_Y X)$, $f \in L^2(X) \}$ contains a basis of $L^2(X)$.

C2: The set of almost periodic functions is dense in $L^2(X)$

C3: For all $f \in L^2(X), \ \varepsilon, \delta > 0$, there exists $D \subseteq Y, \ \nu(D) > 1- \delta, \ g_1, \cdots, g_k \in L^2(X)$ s.t. for any $\gamma \in \Gamma$ and almost every $y \in Y$, we have

$\displaystyle \min_{1 \leq i \leq k} || \gamma (f)|_{f^{-1}(D)} - g_i||_y < \varepsilon$

C4: For all $f \in L^2(X), \ \varepsilon, \delta > 0$, there exists $g_1, \cdots, g_k \in L^2(X)$ s.t. for any $\gamma \in \Gamma$, there is a set $D \subseteq Y, \ \nu(D) > 1- \delta$, for all $y \in D$

$\displaystyle \min_{1 \leq i \leq k} || \gamma (f) - g_i||_y < \varepsilon$

C5: For all $f \in L^2(X)$, let $\bar{f} \in L^1(X \times_Y X)$ where

$\bar{f}: (x_1, x_2) \mapsto f(x_1) \cdot f(x_2)$

Let $I=(I_k)$ be a Folner sequence, then $\bar{f}=\bar{0}$ iff $P \bar{f} = \bar{0}$.

Theorem: All five definitions are equivalent.

Proof: “C1 $\Rightarrow$ C2″

Since almost periodic functions are closed under linear combination, it suffice to show any element in a set of basis is approximated arbitrarily well by almost periodic functions.

Let our basis be as given in C1.

For all $H \in L^\infty (X \times_Y X) \cap \Gamma_{inv} (X \times_Y X)$, the associated linear operator $\varphi_H: L^2(X) \rightarrow L^2(X)$ where $\varphi_H: f \mapsto H \ast f$ is bounded. Hence it suffice to check $H \ast f$ for a dense set of $f \in L^2(X)$. We consider the set of all fiberwise bounded $f$ i.e. for all $y \in Y$, $||f||_y \leq M_y$.

For all $\delta > 0$, we perturb $H \ast f$ by multiplying it by the characteristic function of a set of measure at least $1- \delta$ to get an almost periodic function.

“C2 $\Rightarrow$ C3″:

For any $f \in L^2(X)$, there exists $f'$ almost periodic, with $||f-f'||< \frac{\epsilon \sqrt{\delta}}{2}$ . Let $\{ g_1, g_2, \cdots, g_{k-1} \}$ be the functions obtained from the almost periodicity of $f'$ with constant $\varepsilon/2$, $g_k = \bar{0}$.

Let $D = \{ y \ | \ ||f-f'||_y < \varepsilon/2 \}$, since

$|| f - f'||^2 = \int ||f-f'||_y^2 d \nu(y)$

Hence $||f-f'||< \frac{\varepsilon \sqrt{\delta}}{2} \ \Rightarrow \ ||f-f'||^2 < \frac{\varepsilon^2 \delta}{4}$, $\{ y \ | \ ||f-f'||_y > \varepsilon/2 \}$ has measure at most $\delta/2$, therefore $\nu(D) > 1- \delta$.

For all $\gamma \in \Gamma$, if$y \in \gamma^{-1}(D)$ then

$|| \gamma f|_{\alpha^{-1}(D)} - \gamma f'||_y = ||f|_{\alpha^{-1}(D)} - f'||_{\gamma(y)} < \varepsilon /2$

Hence $\displaystyle \min_{1 \leq i \leq k-1} ||\gamma f|_{\alpha^{-1}(D)} - g_i||_y < \varepsilon /2 + \varepsilon /2 = \varepsilon$

If $y \notin \gamma^{-1}(D)$ then $f|_{\alpha^{-1}(D)}$ vanishes on $\alpha^{-1}(\gamma y)$ so that $|| \gamma f|_{\alpha^{-1}(D)} - g_i||_y = 0 < \varepsilon$.

Hence $\alpha$ satisfies C3.

“C3 $\Rightarrow$ C4″:

This is immediate since for all $y \in \gamma^{-1}(D)$, we have $\gamma f = \gamma f|_{\alpha^{-1}(D)}$ on $\alpha^{-1}(y)$ hence

$\displaystyle \min_{1 \leq i \leq k} ||\gamma f - g_i||_y < \min_{1 \leq i \leq k-1} ||\gamma f_{\alpha^{-1}(D)} - g_i||_y < \varepsilon$

$\nu(\gamma^{-1}(D)) = \nu(D) > 1-\delta$. Hence $\alpha$ satisfies C4.

“C4 $\Rightarrow$ C5″:

For all $f \in L^2(X), \ \varepsilon, \delta > 0$, by C4, there exists $g_1, \cdots, g_k \in L^2(X)$ s.t. for any $\gamma \in \Gamma$, there is a set $D \subseteq Y, \ \nu(D) > 1- \delta$, for all $y \in D$

$\displaystyle \min_{1 \leq i \leq k} || \gamma (f) - g_i||_y < \varepsilon$

W.L.O.G. we may suppose all $g_i$ are bounded since by making $\delta$ slighter larger we can modify the unbounded parts to be bounded.

$\bar{g_j} \otimes g_j \in L^\infty(X \times_Y X)$, suppose $P(\bar{f}) = 0$.

Recall in C5 we have $\bar{f}: (x_1, x_2) \mapsto f(x_1) \cdot f(x_2)$, and $\displaystyle P_I \bar{f}(x_1, x_2) = \lim_{k \rightarrow \infty} \frac{1}{|I_k|} \sum_{\gamma \in I+k} f(\gamma x_1) \bar{ f(\gamma x_2)}$.

For each $1 \leq j \leq k$, we have $\int (\bar{g_j} \otimes g_j) \cdot P \bar{f} d(\mu \times_Y \mu) = 0$

Hence we have $\displaystyle \lim_{i \rightarrow \infty} \frac{1}{|I_i|} \sum_{\gamma \in I_i} \int (\bar{g_j(x_1)} g_j(x_2)) \cdot$ $\gamma f(x_1) \bar{\gamma f(x_2)} d\mu \times_Y \mu(x_1, x_2) = 0$

$\Rightarrow \displaystyle \lim_{i \rightarrow \infty} \frac{1}{|I_i|} \sum_{\gamma \in I_i} \int | \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 d \nu(y) = 0$

$\Rightarrow \displaystyle \lim_{i \rightarrow \infty} \frac{1}{|I_i|} \sum_{\gamma \in I_i} \{ \sum_{j=1}^k \int | \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 d \nu(y) \} = 0$

Hence for large enough $i$, there exists $\gamma \in I_i$ s.t. $\sum_{j=1}^k \int | \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 d \nu(y)$ is as small as we want.

We may find $D' \subseteq Y$ with $\nu(D) > 1-\delta$ s.t. for all $y \in D'$ and for all $j$, we have

$| \int \bar{g_j(x)} \gamma f(x) d\mu_y(x)|^2 < \varepsilon^2$

On the other hand, by construction there is $j$ with $|| \gamma f - g_j||^2_y < \varepsilon^2$ for all $y \in D$, with $\nu(D) > 1-\delta$.

Hence for $y \in D \cap D', \ ||f||_{\gamma'^{-1}(y)}^2 = || \gamma f||_y^2 < 3 \varepsilon^2$.

Let $\varepsilon \rightarrow 0, \ \delta \rightarrow 0$ we get $f = \bar{0}$. Hence C5 holds.

“C5 $\Rightarrow$ C1″

Let $f \in L^2(X)$ orthogonal to all of such functions. Let $(I_k)$ be a Folner sequence.

Define $\displaystyle H(x_1, x_2) := \lim_{i \rightarrow \infty} \frac{1}{|I_i|}\sum_{\gamma \in I_i} \gamma f(x_1) \cdot \gamma f(x_2) = P \bar{f}(x_1, x_2)$

Let $H_M(x_1, x_2)$ be equal to $H$ whenever $H(x_1, x_2) \leq M$ and $0$ o.w.

$H$ is $\Gamma$-invariant $\Rightarrow \ H_M$ is $\Gamma$-invariant and bounded.

Therefore $f \bot H_M \ast f$, i.e.

$\int \bar{f(x_1)} \{ \int H_M(x_1, x_2) d \mu_{\alpha(x_1)}(x_2) \} d \mu(x_1) = 0$ <\p>

Since $\mu = \int \mu_y d \nu(y)$, we get

$\int \bar{f} \otimes f \cdot H_M d \mu \times_Y \mu = 0$ <\p>

Hence $H_M \bot (\bar{f} \otimes f)$. For all $\gamma \in \Gamma, \ \gamma (\bar{f} \otimes f) \bot \gamma H_M = H_M$.

Since $H = P \bar{f}$ is an average of $\gamma (\bar{f} \otimes f), \ \Rightarrow \ H \bot H_M$.
$0 = \int \bar{H} \cdot H_M = \int |H_M|^2 \ \Rightarrow \ H_M = \bar{0}$ for all $M$

Hence $H = \bar{0}$. By C5, we obtain $f = \bar{0}$. Hence $\{ H \ast f \ | \ H \in L^\infty (X \times_Y X) \cap \Gamma_{inv} (X \times_Y X)$, $f \in L^2(X) \}$ contain a basis for $L^2(X)$.

Definition: Let $H$ be a subgroup of $\Gamma$, $\alpha: (X, \mathcal{B}, \mu, \Gamma) \rightarrow ( Y, \mathcal{D}, \nu, \Gamma')$ is said to be compact relative to $H$ if the extension $\alpha: (X, \mathcal{B}, \mu, H) \rightarrow ( Y, \mathcal{D}, \nu, H')$ is compact.

### On plaque expansiveness

May 5, 2010

This note is mostly based on parts of (RH)^2U (2006) and conversations with R. Ures while he was visiting Northwestern.

Let $\mathcal{F}$ be a foliation of the manifold $M$, for $p \in M$, a plaque in of $\mathcal{F}$ through $p$ is a small open neighborhood of $p$ in the leaf $\mathcal{F}_p$ that’s pre-image of a disc via a local foliation chart. (i.e. plaques stuck nicely to make open neighborhoods where the foliation chart is defined.) For $\varepsilon$ small enough, whenever the leaves of $\mathcal{F}$ are $C^1$, the path component of $B(p, \varepsilon)$ containing $p$ is automatically a plaque, we denote this by $\mathcal{F}_\varepsilon(p)$.

Given a partially hyperbolic diffeomorphism $f: M \rightarrow M$, suppose the center integrates to foliation $\mathcal{F}^c$.

Definition: An $\varepsilon$-pseudo orbit w.r.t. $\mathcal{F}^c$ is a sequence $(p_n)$ where for any $n \in \mathbb{Z}$, $f(x_n) \in \mathcal{F}^c_\varepsilon(x_{n+1})$.

i.e. $p_{n+1}$ is the $f$-image of $p_n$ except we are allowed to move along the center plaque for a distance less than $\varepsilon$.

Definition: $f$ is plaque expansive at $\mathcal{F}^c$ if there exists $\varepsilon>0$ s.t. for all $\varepsilon$-pseudo orbits $(p_n), (q_n)$ w.r.t. $\mathcal{F}^c$, $d(p_i, q_i)<\varepsilon$ for all $i \in \mathbb{Z}$ then $p_0 \in \mathcal{F}^c_\varepsilon(q_0)$.

i.e. any two pseudo-orbits in different plagues will eventually (under forward or backward iterates) be separated by a distance $\varepsilon$.

In the book Invariant Manifolds (Hirsch-Pugh-Shub), it’s proven that

Theorem: If a partially hyperbolic system has plaque expansive center foliation, then the center being integrable and plaque expansiveness are stable under perturbation (in the space of diffeos). Furthermore, the center foliation of the perturbed system $g$ is conjugate to the center foliation of the origional system $f$ in the sense that there exists homeomorphism $h: M \rightarrow M$ where

1) $h$ sends leaves of $\mathcal{F}^c_f$ to leaves of $\mathcal{F}^c_g$ i.e. for all $p \in M$,

$h(\mathcal{F}^c_f(p)) = \mathcal{F}^c_g(p)$

2) $h$ conjugates the action of $f$ and $g$ on the set of center leaves i.e. for all $p \in M$,

$h \circ f \ (\mathcal{F}^c_f(p)) = g \circ h \ ( \mathcal{F}^c_f(p))$

(both sides produce a $\mathcal{F}^c_g$ leaf)

Morally this means plaque expansiveness implies structurally stable in terms of permuting the center leaves.

It’s open whether or not any partially hyperbolic diffeomorphism with integrable center is plaque expansive w.r.t. its center foliation.

Another problem, stated in HPS about plaque expansiveness is:

Question: If $f$ is partially hyperbolic and plaque expansive w.r.t. center foliation $\mathcal{F}_c$, then is $\mathcal{F}_c$ the
unique $f$−invariant foliation tangent to $E^c$?

(RH)^2U has recently gave a series of super cool examples where the 1-dimensional center bundles of a $C^1$ partially hyperbolic diffeomorphism 1) does not integrate OR 2) integrates to a foliation but leaves through a given point is not unique (there is other curves through the point that’s everywhere tangent to the bundle). I will say a few words about the examples without spoil the paper (which is still under construction).

Start with the cat map on the $2$-torus (matrix with entries $( 2, 1, 1, 1)$, take the direct product with the source-sink map on the circle, we obtain a diffeo on the $3$ torus. For the purpose of our map, we make the expansion in the source-sink map weaker than that of the cat map and the contraction stronger.

Then we perturb the map by adding appropriate small rotations to the system, the perturbation vanish on the $\mathbb{t}^2$ fibers corresponding to the two fixed points in the source-sink map. This will make our system partially hyperbolic, with center bundles as shown below:

To construct a non-integrable center, we make a perturbation that gives center boundle (inside the unstable direction of the cat map times the circle):

For intergrable but have non-unique center leaves, we simply rotate the upper and bottom half in opposite directions and obtain:

Note that in this case, all center leaves are merely copies of $S^1$. The example is plaque expansive due to to fact that all centers leaves are compact (and of uniformly bounded length). However, although the curve through any given point tangent to the bundle is non-unique, there is only one possible foliation of the center. Hence this does not give a counter example to the above mentioned question in HPS.

I think there are hopes to modify the example and make one that has similar compact leafs but non-unique center foliation, perhaps by making the unique integrability fail not only on a single line.