Global rigidity of higher rank abelian Anosov algebraic actions

Abstract

We show that all $C^\infty$ Anosov $\mathbb {Z}^r$-actions on tori and nilmanifolds without rank-one factor actions are, up to $C^\infty$ conjugacy, actions by automorphisms.

Appendix A: A regularity theorem along Hölder foliations

Let $M$ be a compact Riemannian manifold with a continuous foliation $\mathcal {F}$ with smooth leaves. Write points in $\mathbb {R}^{\dim M}$ as $(x,y)\in \mathbb {R}^{\dim M-\dim \mathcal {F}}\times \mathbb {R}^{\dim \mathcal {F}}$. Suppose there are Hölder continuous local chart maps $\Gamma$ from open sets $O\subset \mathbb {R}^{\dim M}$ to $M$, such that $\Gamma$ sends subspaces parallel to the $y$-hyperplane to leaves of $\mathcal {F}$ by $C^\infty$ local immersions, and pushes the Lebesgue measure on $\mathbb {R}^{\dim M}$ to an absolutely continuous measure $J\mathrm {d}\mathrm {Vol}$ on $\Gamma (O)$. Moreover, assume that $J$ is also $C^\infty$ along the $\mathcal {F}$ leaves, and that all partial derivatives of $\Gamma$ and $J$ of arbitrary orders along the $y$ direction are Hölder continuous on $O$.

Theorem A.1

Suppose $M$ and $\mathcal {F}$ are as above. If a Hölder continuous function $\phi$ on $M$ satisfies that for all $\theta >0$, all partial derivatives of $\phi$ along $\mathcal {F}$ of all orders belong to $(C^\theta )^*$, then all these partial derivatives are Hölder continuous.

This theorem is motivated by [28, Theorem 1.1].

Let $B$ be the closed unit ball in $\mathbb {R}^m$. Given $\alpha \in [0,1)$ let us denote with $C_0^{\alpha }(B\times \mathbb {T}^l)$ the closure of the $C^\infty$ functions of compact support inside the interior of the unit ball times $\mathbb {T}^l$ w.r.t. the $C^\alpha$-norm $\Vert \phi \Vert _{C^\alpha }$ given in (3.14). When the space $B\times \mathbb {T}^l$ is understood we shall denote it directly with $C_0^{\alpha }$. Denote points in $B\times \mathbb {T}^l$ by $(x,y)$. Take $\alpha \in [0,1)$. Given $\phi \in C_0^\alpha$, $k\in \mathbb {N}$ and $\gamma \in [0,1)$, we say that $\phi$ has $k$-th derivative in the $y$ direction belonging to $(C^{\gamma })^*$ if for any multiindex $\mathbf{r}=(r_1,\dots r_l)$ with $|\mathbf{r}|=k$ we have that

$$\begin{aligned} \partial ^\mathbf{r}_y \phi \in (C^{\gamma })^* \end{aligned}$$

in the sense that, there is a constant $C=C(\gamma ,\mathbf{r})$ such that for any $u\in C^{\infty }_c(\mathbb {R}^m\times \mathbb {T}^l)$, we have that

$$\begin{aligned} \left| \,\,\int \limits _{\mathbb {R}^m\times \mathbb {T}^l}\phi \partial ^\mathbf{r}_y u\mathrm {d}x\mathrm {d}y\right| \le C\Vert u\Vert _{C^{\gamma }}. \end{aligned}$$

Here $|\mathbf {r}|=r_1+r_2+\cdots +r_l$, and $\partial _y^\mathbf {r}=\partial _{y_1}^{r_1}\cdots \partial _{y_l}^{r_l}$.

Proposition A.2

Let $\alpha \in (0,1)$ and let $\phi \in C_0^\alpha$. Let us assume that there is $\gamma \in [0,1)$ such that partial derivatives of all orders in the $y$ direction of $\phi$ belong to $(C^{\gamma })^*$. Then for some $\beta \in (0,\alpha )$ and for any multiindex $\mathbf{r}=(r_1,\dots r_l)$ we have that

$$\begin{aligned} \partial ^\mathbf{r}_y \phi \in C_0^{\beta }. \end{aligned}$$

That is, if $\phi$ is Hölder and has weak derivatives of all orders in the $y$ direction, then it has Hölder continuous derivatives in the $y$ direction.

Theorem A.1 is an almost immediate corollary to Proposition A.2.

Proof of implication Proposition A.2 $\Rightarrow$Theorem A.1] By hypothesis on the foliation $\mathcal {F}$ and making partition of unity, we may assume $M=\mathbb {R}^m\times \mathbb {R}^l$, $\mathcal {F}$ is the foliation into $\mathbb {R}^l$ hyperplanes in $y$ direction, and $\phi$ is compactly supported. We can even assume $\phi$ is supported on $B\times (-\frac{1}{4},\frac{1}{4})^l$. This new case follows from Proposition A.2. by identify $(-\frac{1}{4},\frac{1}{4})^l$ with an open subset of $\mathbb {T}^l$.

We now aim to prove Proposition A.2. Consider $\phi$ as in Proposition A.2. Let us write

$$\begin{aligned} \phi (x,y)=\sum _{\mathbf{n}\in \mathbb {Z}^l}\phi _\mathbf{n}(x)e^{2\pi i \mathbf{n}\cdot y} \end{aligned}$$

where

$$\begin{aligned} \phi _\mathbf{n}(x)=\int \limits _{\mathbb {T}^l}\phi (x,y)e^{-2\pi i \mathbf{n}\cdot y}\mathrm {d}y. \end{aligned}$$

We have that for any $\mathbf{n}\in \mathbb {Z}^l$,

$$\begin{aligned} \Vert \phi _\mathbf{n}\Vert _{C^\alpha (\mathbb {R}^m)}\le \Vert \phi \Vert _{C^\alpha (\mathbb {R}^m\times \mathbb {T}^l)}. \end{aligned}$$

(A.1)

On the other hand, since $\phi$ has partial derivatives of all orders in the $y$ direction in the weak $C^{\gamma }$-sense, for any $\mathbf {r}$ there is a constant $C_0(\mathbf {r})$ such that for any $u\in C^{\infty }_c(B\times \mathbb {T}^l)$,

$$\begin{aligned} \left| \,\,\int \limits _{B\times \mathbb {T}^l}\phi \partial ^\mathbf {r}_yu\mathrm {d}x\mathrm {d}y\right| \le C_0(\mathbf {r})\Vert u\Vert _{C^{\gamma }_0(B\times \mathbb {T}^l)}. \end{aligned}$$

(A.2)

In the sequel we shall use several standard embeddings of Sobolev’s spaces and interpolation. Given $s\in \mathbb {R}$ and $1\le p<\infty$ let $H^{s,p}=H^{s,p}(\mathbb {R}^m)$ be the Sobolev space of order $s$ over $L^p$, i.e. $H^{s,p}(\mathbb {R}^m)$ is the closure of $C_c^\infty (\mathbb {R}^m)$ w.r.t. the norm

$$\begin{aligned} \Vert u\Vert _{s,p}=\left\| \left( (1+|\xi |^2)^{s/2}|\hat{u}|\right) ^{\vee }\right\| _{L^p}. \end{aligned}$$

Here $\hat{u}$ stands for the Fourier transform and $u^\vee$ is the inverse transform. The following can be found in any book on Sobolev spaces (see for instance [31, 33]).

Lemma A.3

Let $s,t\in \mathbb {R}$, for $1<p<\infty$, let $p^*$ be such that $1=\frac{1}{p}+\frac{1}{p^*}$.

• (1) If $s<t$ then

  $$\begin{aligned} H^{s,p}\subset H^{t,p} \end{aligned}$$

  and the embedding is continuous.

• (2)
  $$\begin{aligned} \left( H^{s,p}\right) ^*=H^{-s,p^*}. \end{aligned}$$

• (3) If $\frac{1}{q}=\frac{1}{p}-\frac{s-t}{m}$ then

  $$\begin{aligned} H^{s,p}(\mathbb {R}^m)\subset H^{t,q}(\mathbb {R}^m) \end{aligned}$$

  where the embedding is continuous.

• (4) If $\alpha \in (0,1)$, $r\in \mathbb {N}$ and $\frac{s-r-\alpha }{m} = \frac{1}{p}$ then

  $$\begin{aligned} H^{s,p}(\mathbb {R}^m)\subset C^{r,\alpha }(\mathbb {R}^m) \end{aligned}$$

  and the embedding is continuous.

• (5) For any $\alpha \in (0,1)$ and for any $p\ge 1$, and for any $s<\alpha$,

  $$\begin{aligned} C^{\alpha }(\mathbb {R}^m)\cap L^p(\mathbb {R}^m)\subset H^{s,p}(\mathbb {R}^m) \end{aligned}$$

  and the embedding is continuous.

• (6) If $s_1,s_2\in \mathbb {R}$, $1< p_1,p_2<\infty$, $\theta \in [0,1]$ and $t=\theta s_1+(1-\theta )s_2$, $\frac{1}{p}=\frac{\theta }{p_1}+\frac{1-\theta }{p_2}$ then there is a constant $C=C(s_1,s_2,\theta ,p_1,p_2)$ such that

  $$\begin{aligned} |u|_{t,p}\le C |u|_{s_1,p_1}^{\theta }|u|_{s_2,p_2}^{1-\theta }. \end{aligned}$$

Lemma A.4

For all multiindex $\mathbf {r}$, there is a constant $C(\mathbf {r})$ such that, for any $\mathbf {n}\in \mathbb {Z}^l$, with $|\mathbf {n}^{\mathbf {r}}|\ne 0$,

$$\begin{aligned} \Vert \phi _{\mathbf {n}}\Vert _{(C^\gamma )^*}\le \frac{C(\mathbf {r})|\mathbf {n}|}{|\mathbf {n}^{\mathbf {r}}|}. \end{aligned}$$

(A.3)

Here and below, $|\mathbf {z}|=\sum _{i=1}^l|z_i|$ and $\mathbf {z}^\mathbf {r}=z_1^{r_1}z_2^{r_2}\cdots z_l^{r_l}$ for all $\mathbf {z}\in \mathbb {C}^l$.

Proof

Take $w\in C_c^\infty (B)$.

$$\begin{aligned}&\displaystyle \left| \int \limits _B\phi _\mathbf {n}(x) w(x)\mathrm {d}x\right| \\&\quad =\left| \,\,\int \limits _{B\times \mathbb {T}^l}\phi (x,y)\cdot e^{2\pi i\mathbf {n}\cdot y}w(x)\mathrm {d}x\mathrm {d}y\right| \\&\quad =\left| (2\pi i\mathbf {n})^{-\mathbf {r}}\int \limits _{B\times \mathbb {T}^l}\phi (x,y)\partial _y^\mathbf {r}\left( e^{2\pi i\mathbf {n}\cdot y}w(x)\right) \mathrm {d}x\mathrm {d}y\right| \end{aligned}$$

By (A.2),

$$\begin{aligned} \displaystyle \left| \int \limits _B\phi _\mathbf {n}(x) w(x)\mathrm {d}x\right|&\le \frac{1}{(2\pi )^{|\mathbf {r}|}|\mathbf {n}^\mathbf {r}|}\cdot \left\| e^{2\pi i\mathbf {n}\cdot y}w(x)\right\| _{C^\gamma }\\&\le \frac{C_0(\mathbf {r})}{(2\pi )^{|\mathbf {r}|}|\mathbf {n}^\mathbf {r}|}\cdot \left\| e^{2\pi i\mathbf {n}\cdot y}\right\| _{C^\gamma }\Vert w\Vert _{C^\gamma }\\&\le \frac{C_0(\mathbf {r})}{(2\pi )^{|\mathbf {r}|}|\mathbf {n}^\mathbf {r}|}\cdot \left\| e^{2\pi i\mathbf {n}\cdot y}\right\| _{C^1}\Vert w\Vert _{C^\gamma }\\&= \frac{2\pi |\mathbf {n}|C_0(\mathbf {r})}{(2\pi )^{|\mathbf {r}|}|\mathbf {n}^\mathbf {r}|}\cdot \Vert w\Vert _{C^\gamma }, \end{aligned}$$

which is the lemma. $\square$

Lemma A.5

There exist $\beta >0$ and $\theta >0$ such that for any multiindex $\mathbf {r}$, there is a constant $C(\mathbf {r},\beta )$ such that for any $\mathbf {n}\in \mathbb {Z}^l$, with $|\mathbf {n}^{\mathbf {r}}|\ne 0$,

$$\begin{aligned} \Vert \phi _{\mathbf {n}}\Vert _{C^\beta }\le \frac{C(\mathbf {r},\beta )|\mathbf {n}|^\theta }{|\mathbf {n}^{\mathbf {r}}|^{\theta }}. \end{aligned}$$

(A.4)

Proof

Fix a sufficiently large $p^*$ such that $\frac{m}{p^*}<\frac{\alpha }{4}$.

Take $s$ and $p$, such that $\frac{s-\gamma }{m}=\frac{1}{p}$ and $1=\frac{1}{p}+\frac{1}{p^*}$. By Lemma A.3, we have a continuous embeddings $(C^\gamma )^*\subset (H^{s,p})^*=H^{-s,p^*}$. Therefore by Lemma A.4, there is $C_1(\mathbf {r})$ such that

$$\begin{aligned} \Vert \phi _\mathbf {n}\Vert _{H^{-s,p^*}}\le \frac{C_1(\mathbf {r})|\mathbf {n}|}{|\mathbf {n}^\mathbf {r}|}. \end{aligned}$$

(A.5)

On the other hand, $\phi _\mathbf {n}$ is clearly bounded by $\Vert \phi \Vert _{L^\infty }\le \Vert \phi \Vert _{C^\alpha }$. Hence $\Vert \phi _\mathbf {n}\Vert _{L^{p^*}}\le a_1\Vert \phi \Vert _{C^\alpha }$ where $a_1$ depends only on the volume of the unit ball $B$. Combining this with (A.1), we know by Lemma A.5(5) that there is a constant $C_2$ that depends only on the dimension, such that

$$\begin{aligned} \Vert \phi _\mathbf {n}\Vert _{H^{\frac{\alpha }{2},p^*}}\le C_2 \Vert \phi \Vert _{C^\alpha }. \end{aligned}$$

(A.6)

Choose $\theta \in (0,1)$ such that $\theta \cdot (-s)+(1-\theta )\cdot \frac{\alpha }{2}=\frac{\alpha }{4}$. The interpolation formula Lemma A.3(6) allows to merge (A.5) and (A.6) into:

$$\begin{aligned} \Vert \phi _\mathbf {n}\Vert _{H^{\frac{\alpha }{4},p^*}}\le C_1(\mathbf {r})^\theta C_2^{1-\theta }\Vert \phi \Vert _{C^\alpha }^{1-\theta }\left( \frac{|\mathbf {n}|}{|\mathbf {n}^\mathbf {r}|}\right) ^\theta . \end{aligned}$$

(A.7)

Take $\beta >0$ such that $\frac{\frac{\alpha }{4}-\beta }{m}=\frac{1}{p^*}$, which is possible thanks to the choice of $p^*$. (A.7) establishes the lemma since $H^{\frac{\alpha }{4},p^*}$ continuously embeds into $C^\beta$. $\square$

Corollary A.6

For any $T>0$ there is $C(T)>0$ such that

$$\begin{aligned} \Vert \phi _{\mathbf {n}}\Vert _{C^\beta }\le \frac{C(T)}{|\mathbf {n}|^T} \end{aligned}$$

for any $0\ne \mathbf {n}\in \mathbb {Z}^l$.

Proof

Given $\mathbf {n}\ne 0$, there is $1\le i\le l$ such that $|n_i|\ge \frac{|\mathbf {n}|}{l}$. Choose $\mathbf {r}$ by letting $r_i=\lceil \frac{T+1}{\theta }\rceil$ and $r_j=0$ if $j\ne i$. Then the corollary directly follows from Lemma A.5. $\square$

Proof of Proposition A.2

Let $A>0$ and $\mathbf {r}$ be a multiindex with $|\mathbf {r}|=r$, define

$$\begin{aligned} \phi _{\mathbf {r},A}(x,y)=\sum _{\mathbf {n}\in \mathbb {Z}^l, |\mathbf {n}|\le A}(2\pi i)^r\mathbf {n}^\mathbf {r}\phi _\mathbf {n}(x)e^{2\pi i\mathbf {n}\cdot y}. \end{aligned}$$

Then there are constants $C_1(\mathbf {r}), C_2(r)$ such that

$$\begin{aligned}&|\phi _{\mathbf {r},A}(x,y)-\phi _{\mathbf {r},A}(z,y)|\\&\quad \le C_1(\mathbf {r})\sum _{\mathbf {n}\in \mathbb {Z}^l, |\mathbf {n}|\le A}|\mathbf {n}|^r|\phi _\mathbf {n}(x)-\phi _\mathbf {n}(z)|\\&\quad \le C_1(\mathbf {r})\sum _{\mathbf {n}\in \mathbb {Z}^l, |\mathbf {n}|\le A}|\mathbf {n}|^r\Vert \phi _\mathbf {n}\Vert _{C^{\beta }_0}\mathrm{dist }(x,z)^\beta \\&\quad \le C_1(\mathbf {r})C(r+2l+1)\left( \sum _{\mathbf {n}\in \mathbb {Z}^l, |\mathbf {n}|\le A}\frac{1}{|\mathbf {n}|^{2l+1}}\right) \mathrm{dist }(x,z)^\beta \\&\quad \le C_2(r)\mathrm{dist }(x,z)^\beta , \end{aligned}$$

where $C(r+2l+1)$ is that from Corollary A.6.

This same computation gives that

$$\begin{aligned} \lim _{A\rightarrow \infty }\phi _{\mathbf {r},A}=\partial ^\mathbf {r}_y\phi \end{aligned}$$

in $C^{\beta }$-topology. Which implies that $\partial ^\mathbf {r}_y\phi$ is $C^{\beta }$. $\square$