Posted by: matheuscmss | February 17, 2012

## SPCS 6

Today we follow J.-C. Yoccoz’s 6th lecture and we begin our discussions around the simplicity of Lyapunov exponents of the Kontsevich-Zorich cocycle. By the end of this series of posts, we will focus on the case of square-tiled surfaces (along the lines of a work in progress by C.M., Martin Möller and J.-C. Yoccoz). However, since our future discussions will be based on the works of A. Avila and M. Viana, we dedicate today’s post to review (some aspects of) these works.

The main result of this article here of A. Avila and M. Viana (settling the Kontsevich-Zorich conjecture) is:

Theorem. The Lyapunov spectrum of the (Kontsevich-Zorich) cocycle $G_{KZ}$ is simple (i.e., all Lyapunov exponents have multiplicity $1$) with respect to the natural (Masur-Veech) measure on each connected component of each stratum of the moduli space of (unit area) Abelian differentials.

Roughly speaking, this theorem has two parts:

• a simplicity criterion for a certain class of cocycles;
• verification of the simplicity criterion in the particular case of the cocycle $G_{KZ}$ equipped with the Masur-Veech measure.

Of course, our goal here is not to make a complete discuss of the theorem of Avila and Viana, but instead only to revisit and adapt it in order to allow applications to the dynamics of $G_{KZ}$ with respect to other measures (than the Masur-Veech one).

In particular, for today we will content ourselves with the discussion of Avila-Viana simplicity criterion (and a straightforward adaptation of it to slightly more general contexts).

From the “historical” point of view, Avila-Viana simplicity criterion is inspired by previous results of H. Furstenberg, I. Goldsheid and G. Margulis, A. Raugi and Y. Guivarch, F. Ledrappier, C. Bonatti, X. Gomez-Mont and M. Viana, and C. Bonatti and M. Viana. Also, there are at least two versions of Avila-Viana criterion (serving for different purposes), one in this article here (published at Acta Math.) and the other in this article here (published at Portugaliae Math.). For the sake of today’s post, we will follow the version described in the paper published at Acta Math.

-Simplicity criterion-

Let $\Lambda$ be a finite or countable alphabet. Define $\Sigma = \Lambda^{\mathbb{N}}$, and $\hat{\Sigma} := \Lambda^{\mathbb{Z}}:=\Sigma_-\times\Sigma$. Denote by $f:\Sigma\to\Sigma$ and $\hat{f}:\hat{\Sigma}\to\hat{\Sigma}$ the natural (left) shift maps on $\Sigma$ and $\hat{\Sigma}$ resp. Also, let $p^+: \hat{\Sigma}\to\Sigma$ and $p^-:\hat{\Sigma}\to\Sigma_-$ be the natural projections

We denote by $\Omega = \bigcup\limits_{n\geq 0}\Lambda^n$ the set of words of the alphabet $\Lambda$. Given $\underline{\ell}\in\Omega$, let

$\Sigma(\underline{\ell}):=\{x\in\Sigma: x \textrm{ starts by }\underline{\ell}\}$

and

$\Sigma_-(\underline{\ell}):=\{x\in\Sigma: x \textrm{ ends by }\underline{\ell}\}$

Given a $f$-invariant probability measure $\mu$ on $\Sigma$, we denote by $\hat{\mu}$ the unique $\hat{f}$-invariant probability measure with $p^+_*(\hat{\mu})=\mu$, and we define $\mu_-:=p^-_*(\hat{\mu})$.

In the sequel, we will make the following bounded distortion assumption on $\mu$:

Hypothesis (Bounded distortion). There exists a constant $C(\mu)>0$ such that

$\frac{1}{C(\mu)}\mu(\Sigma(\underline{\ell}_1))\mu(\Sigma(\underline{\ell}_2)) \leq \mu(\Sigma(\underline{\ell}_1\underline{\ell}_2)) \leq C(\mu) \mu(\Sigma(\underline{\ell}_1)) \mu(\Sigma(\underline{\ell}_2))$

for any $\underline{\ell}_1, \underline{\ell}_2\in\Omega$.

This bounded distortion assumption says that, in some sense, $\mu$ is “not very far” from a Bernoulli measure. As an exercise, the reader can check that bounded distortion implies that $\mu$ is $f$-ergodic.

In any case, such $(f,\mu)$ and $(\hat{f},\hat{\mu})$ are the bases dynamics. Now, our next assumptions concerns the class of cocycles we want to investigate over these bases.

Hypothesis (Locally constant integrable cocycles). We assume that our cocycle $A:\Sigma\to M_{d\times d}(\mathbb{R})$ satisfies

• (the cocycle is locally constant) $A(\underline{x})=A_{x_0}$, where $A_{\ell}\in GL(d,\mathbb{R})$ (“primitive case”) for $\ell\in\Lambda$, and $\underline{x}=(x_0,\dots)\in\Sigma$;
• (the cocycle is integrable) $\int_{\Sigma}\log\|A^{\pm1}(\underline{x})\|\,d\mu(\underline{x}) = \sum \mu(\Sigma(\ell))\log\|A^{\pm1}_{\ell}\|<\infty$.

In the work of Avila and Viana, they treat the primitive case of $A_\ell\in GL(d,\mathbb{R})$ together with the derived case of $A_\ell\in Sp(d,\mathbb{R})$, $d$ even, as they apply their criterion to the case of a symplectic cocycle closely related to the Kontsevich-Zorich cocycle. In our context, given our previous discussions, it is interesting to consider the following derived cases “together”:

• $Sp(d,\mathbb{R})$, $d$ even (and the base field is $\mathbb{K}=\mathbb{R}$);
• $U_{\mathbb{C}}(p,q)$, $p+q=d$ (and the base field is $\mathbb{K}=\mathbb{C}$);
• $U_{\mathbb{H}}(p,q)$, $p+q=d$ (and the base field is $\mathbb{K}=\mathbb{H}$).

Remark. Actually, concerning the results stated below, it seems that we can also include $O(p,q)$, $p+q=d$ (and the base field is $\mathbb{K}=\mathbb{R}$) in the list of derived cases, but we will not do this as currently we do not see interesting “applications” of this case.

As a matter of fact, in the cases $U(p,q)$, up to changing the sign of the preserved Hermitian form, we may assume that $p\geq q$.

Given $\underline{\ell}=(\ell_0,\dots,\ell_{n-1})\in\Omega$, we write

$A^{\underline{\ell}}:=A_{\ell_{n-1}}\dots A_{\ell_0}$

By definition, it satisfies $(f,A)^n(x)=(f^n(x),A^{\underline{\ell}})$ for any $x\in\Sigma(\underline{\ell})$.

The ergodicity of $\mu$ (coming from the bounded distortion property) and the integrability of the cocycle $A$ allow us to apply the Oseledets theorem to deduce the existence of Lyapunov exponents

$\theta_1\geq \dots\geq \theta_d$

where $d$ is the dimension over $\mathbb{K}$. Here, we count Lyapunov exponents with multiplicities.

Remark. In the case $\mathbb{K} = \mathbb{C}$ or $\mathbb{H}$, one can think of Lyapunov exponents by seeing our spaces over $\mathbb{R}$, but instead of the list above, we must repeat $2$ or $4$ times each $\theta_i$. So, when saying that we count the dimension $d$ over $\mathbb{K}$, we’re avoiding unnecessary repetitions of Lyapunov exponents, and keeping only “interesting” multiplicities.

In the derived cases, as we saw in previous posts, one has the following a priori constraints for the Lyapunov exponents:

• Symplectic ($Sp(d,\mathbb{R})$, $d$ even): $\theta_i=-\theta_{d+1-i}$;
• Unitary ($U_{\mathbb{K}}(p,q)$, $q\leq p$, $p+q=d$, $\mathbb{K}=\mathbb{C}, \mathbb{H}$): $\theta_i=-\theta_{d+1-i}$ and $\theta_i=0$ for $q.

Also, the unstable/stable Oseledets subspaces associated to positive/negative Lyapunov exponents are isotropic.

Definition. The Lyapunov spectrum of the cocycle $A$ is simple if

• $\theta_i>\theta_{i+1}$ for $1\leq i in the primitive case $GL(d,\mathbb{R})$ and the derived case $Sp(d,\mathbb{R})$;
• $\theta_i>\theta_{i+1}$ for $1\leq i\leq q$ in the derived cases $U_{\mathbb{K}}(p,q)$, $\mathbb{K}=\mathbb{C},\mathbb{H}$.

In other words, we say that a cocycle is simple when its Lyapunov spectrum is as simple as possible given the constraints presented above.

Definition. We say that $1\leq k is admissible if

• without further restrictions in the primitive case $GL(d,\mathbb{R})$ or the derived case $Sp(d,\mathbb{R})$;
• $1\leq k\leq q$ or $p\leq k in the derived cases $U_{\mathbb{K}}(p,q)$, $\mathbb{K}=\mathbb{C},\mathbb{H}$.

Given $k$ admissible, we denote by $G(k)$:

• the Grassmanian of $k$-planes of $\mathbb{R}^d$ in the primitive case $GL$;
• the Grassmanian of $k$-planes which are isotropic if $1\leq k\leq d/2$, and coisotropic if $d/2\leq k< d$ in the derived case $Sp$;
• the Grassmanian of $k$-planes over $\mathbb{K}$ which are isotropic if $1\leq k\leq q$, and coisotropic if $p\leq k< d$ in the derived case $U_{\mathbb{K}}(p,q)$.

At this point, we are ready to make our main assumptions on our cocycle $A$:

1. Pinching. There exists $\underline{\ell}^*\in\Omega$ such that the spectrum of $A^{\underline{\ell}^*}$ is simple.
2. Twisting (“strong form”). For all $m\geq 1$, $k_1,\dots,k_m$ admissibles, $F_i\in G(k_i)$, $F_i'\in G(d-k_i)$, $1\leq i\leq m$, there exists $\underline{\ell}\in\Omega$ such that  $A^{\underline{\ell}}(F_i)\cap F_i'=\{0\}$.

In this language, the simplicity criterion of Avila-Viana (compare with Theorem 7.1 in Avila-Viana’s article) can be stated as follows:

Theorem 1 (A. Avila and M. Viana). Under these assumptions, the Lyapunov spectrum of $A$ is simple.

Remark. In their article, A. Avila and M. Viana considered exclusively the derived case $Sp$, but, as we’re going to sketch below, their arguments can be adapted (in a straightforward way) to give the statement above.

We begin by noticing that one can consider the cocycle $A$ over the invertible dynamics $\hat{f}:\hat{\Sigma}\to\hat{\Sigma}$ because the Lyapunov spectrum is not affected by this procedure.

The main result towards the proof of Theorem 1 above is:

Theorem 2 (A. Avila and M. Viana). For every $k$ admissible, there exists a map $\Sigma_-\to G(k)$, $x\mapsto \xi(x)$ verifying the properties:

1. Invariance: the map $\hat{\xi} = \xi\circ p^-$ satisfies $A(x)\hat{\xi}(x) = \hat{\xi}(\hat{f}(x))$;
2. for $\mu_-$-almost every $x\in\Sigma_-$, $\frac{\sigma_k(A^{\underline{\ell}(x,n)})}{\sigma_{k+1}(A^{\underline{\ell}(x,n)})}\to+\infty$ and $\xi_{\underline{\ell}(x,n)}\to\xi(x)$ as $n\to +\infty$;
3. for all $F'\in G(d-k)$, we have $\xi(x)\cap F'=\{0\}$ for a set of positive $\mu_-$-measure.

Here,

• $\underline{\ell}(x,n)$ is the terminal word of $x\in \Sigma_-$ of length $n$;
• $A^{\underline{\ell}}(\{v:\|v\|=1\})$ is an ellipsoid of semi-major axis of lengths $\sigma_1(A^{\underline{\ell}})\geq \dots\geq \sigma_d(A^{\underline{\ell}})$;
• $\xi_{\underline{\ell}}$ is the subspace generated by the $k$ largest semi-major axis.

This result corresponds to Theorem A.1 in Avila-Viana’s article. As the reader can check (from the [one-page] subsection A.6 [Proof of Theorem 7.1] of this article), it is not hard to deduce Theorem 1 from Theorem 2.

Just to say a few words on why is this so, we observe that Theorem 2 provides us with $\xi(x)\in G(k)$ (verifying the properties above) for each $x\in \Sigma_-$. By using the same theorem with the time “reversed”, one gets $\xi_*(y)$ (verifying analogs of the properties of $\xi(x)$) for $y\in\Sigma_+$. By the third property above, one can show the “transversality property”: $\xi(x)\cap\xi_*(y)=\{0\}$ for almost every $(x,y)\in\hat{\Sigma}$. Also, the second property above implies that $\xi(x)$ is associated to the $k$ largest exponents and $\xi_*(y)$ is associated to the $d-k$ smallest exponents, so that the transversality property permits to “separate” the $k$th exponent $\theta_k$ from the $(k+1)$th exponent $\theta_{k+1}$ (in the sense that $\theta_k>\theta_{k+1}$) for any $k$ admissible, and the simplicity of the Lyapunov spectrum of $A$ (in the sense we defined above) follows.

This “reduces” our considerations to the discussion of Theorem 2.

-Proof of Theorem 2-

Definition. A u-state (“u” standing for “unstable”) is a probability measure $\hat{m}$ on $\hat{\Sigma}\times G(k)$ such that $p'_*(\hat{m})=\hat{\mu}$, where $p':\hat{\Sigma}\times G(k)\to\hat{\Sigma}$ is the natural projection, and there exists a constant $C(\hat{m})$ with

$\frac{\hat{m}(\Sigma_-(\underline{\ell}^0)\times\Sigma(\underline{\ell})\times X)}{\mu(\Sigma(\underline{\ell}))} \leq C(\hat{m})\frac{\hat{m}(\Sigma_-(\underline{\ell}^0)\times\Sigma(\underline{\ell}')\times X)}{\mu(\Sigma(\underline{\ell}'))}$

for any Borelian $X\subset G(k)$, $\underline{\ell}^0,\underline{\ell},\underline{\ell}'\in\Omega$.

Roughly speaking, the previous condition says that u-states are almost product measures.

Example. Given any probability measure $\nu$ on $G(k)$, $\hat{m}:=\hat{\mu}\times\nu$ is a u-state with $C(\hat{m})=C(\mu)^2$.

Proposition 1. There exists a u-state invariant under $(\hat{f},A)$.

Proof. The argument is very classical and we will only sketch its main steps.

Even though the space $\hat{\Sigma}$ may not be compact (in the case of an alphabet $\Lambda$ with countably many symbols), the space of probability measures on $\hat{\Sigma}\times G(k)$ projecting to $\hat{\mu}$ is compact in the weak-* topology. In particular, for each $C>0$, it follows that the space of u-states $\hat{m}$ with $C(\hat{\mu})\leq C$ is a convex compact set.

Now we observe that:

Exercise. If $\hat{m}_0$ be a u-state, then $\hat{m}(n):=(\hat{f},A)_*^n\hat{m}_0$ is also a u-state with $C(\hat{m}(n))\leq C(\hat{m})C(\mu)^2$.

This exercise is a direct (3 lines) computation (whose solution can be found in Lemma A.2 of Avila-Viana’s article).

Using this exercise, one can complete the proof of the proposition by the standard Krylov-Bogolyubov argument: by letting $\hat{m}_0$ be an arbitrary u-state, and by putting $C:=C(\hat{m}_0)C(\mu)^2$, we have that, by convexity, compactness and the statement of the exercise, the Cesaro averages of $\hat{m}(n)$ accumulate on $(\hat{f},A)$-invariant u-states $\hat{m}$ (with $C(\hat{m})\leq C(\hat{m}_0)C(\mu)^2$). $\square$

The following result is a simple application of the martingale convergence theorem (for a direct [half-page] proof of it, see Lemma A.4 of Avila-Viana’s article):

Proposition 2. Let $\hat{m}$ be a probability measure on $\hat{\Sigma}\times G(k)$ with $p'_*\hat{m} = \hat{\mu}$. For $x\in\Sigma_-$, and $X\subset G(k)$ Borelian, let

$\hat{m}_n(x)(X):=\frac{\hat{m}(\Sigma_-(\underline{\ell}(x,n))\times\Sigma\times X)}{\hat{m}(\Sigma_-(\underline{\ell}(x,n))\times\Sigma\times G(k))}$

Then, for $\mu_-$-almost every $x\in\Sigma_-$, $\hat{m}_n(x)$ converges to some $\hat{m}(x)$. Here, we recall that $\underline{\ell}(x,n)$ is the terminal word of $x\in\Sigma_-$ of length $n$.

By Proposition 1, we can fix $\hat{m}$ a $(\hat{f},A)$-invariant u-state. Denote by $\nu=p^{''}_*\hat{m}$, where $p^{''}:\hat{\Sigma}\times G(k)\to G(k)$ is the natural projection. Of course, $\nu$ is a probability measure on $G(k)$. Given $x\in\Sigma_-$, we define the following sequence of probabilities on $G(k)$:

$\nu_n(x):=A^{\underline{\ell}(x,n)}_*\nu$

By the $(\hat{f},A)$-invariance of $\hat{m}$, we have

$\hat{m}_n(x)(X)=\frac{\hat{m}(\Sigma_-(\underline{\ell}(x,n))\times\Sigma\times X)}{\hat{m}(\Sigma_-(\underline{\ell}(x,n))\times\Sigma\times G(k))} = \frac{\hat{m}(\Sigma_-\times\Sigma(\underline{\ell}(x,n))\times A^{-\underline{\ell}(x,n)}(X))}{\hat{m}(\Sigma_-\times\Sigma(\underline{\ell}(x,n))\times G(k))}$

Since $\hat{m}$ is a u-state, we obtain

$\hat{m}_n(x)(X)\leq C(\hat{m})^2\frac{\hat{m}(\Sigma_-\times\Sigma\times A^{-\underline{\ell}(x,n)}(X))}{\hat{m}(\Sigma_-\times\Sigma\times G(k))} = C(\hat{m})^2 \hat{m}(\hat{\Sigma}\times A^{-\underline{\ell}(x,n)}(X))$

On the other hand, by definition,

$\nu_n(x)(X) = \nu(A^{-\underline{\ell}(x,n)}(X)) = \hat{m}(\hat{\Sigma}\times A^{-\underline{\ell}(x,n)}(X))$

so that we get

$\hat{m}_n(x)(X)\leq C(\hat{m})^2 \nu_n(x)(X)$

Of course, since the definition of u-state is “symmetric”, we can “reverse” the role of $\hat{m}_n(x)$ and $\nu_n(x)$ in the argument above. The conclusion is

$C(\hat{m})^{-2}\leq \hat{m}_n(x)(X)/\nu_n(x)(X)\leq C(\hat{m})^2$

for any Borelian $X\subset G(k)$.

In particular,

Corollary. The probability measure $\hat{m}(x)=\lim\hat{m}_n(x)$ is equivalent to any accumulation point of the sequence $\nu_n(x)$.

As it is explained in the end of subsection A.4 of Avila-Viana’s article, the proof of Theorem 2 (i.e., Theorem A.1 in their article) is complete once we can prove the following fact:

Proposition 3. For $\mu_-$-almost every $x\in\Sigma_-$, there exists a subsequence $\nu_{n_k}(x)$, $n_k=n_k(x)\to\infty$, converging to a Dirac mass.

Indeed, this proposition says that $\nu_n(x)$ accumulates a Dirac mass. Since, by the corollary above, $\hat{m}(x)$ is equivalent to any accumulation point of $\nu_n(x)$, it follows that $\hat{m}(x)$ itself is a Dirac mass, say $\hat{m}(x) = \delta_{\xi(x)}$. Then, it is possible to check (cf. end of subsection A.4 of Avila-Viana’s article) that $x\mapsto\xi(x)$ has the desired properties in the statement of Theorem 2: for instance, the fact that $\xi(x)$ has the desired invariance properties (first item of Theorem 2) follows from the $(\hat{f},A)$-invariance of $\hat{m}$, while the other two remaining items need (of course) the pinching and twisting assumptions.

In any event, this “reduces” our task to the proof of Proposition 3. We start by noticing that the pinching and twisting assumptions on $A$ imply that

• a) there exists $\underline{\ell}^0\in\Omega$ such that, for each $k$ admissible and for every pair of $A^{\underline{\ell}^*}$-invariant subspaces $F\in G(k), F'\in G(d-k)$, one has $A^{\underline{\ell}^0}(F)\cap F'=\{0\}$;
• b) there exists $m\geq 1$, $\underline{\ell}_1,\dots,\underline{\ell}_m\in\Omega$ and $\delta>0$ such that, for each $k$ admissible and for every $F'\in G(d-k)$, there exists $\underline{\ell}_i$ with $A^{\underline{\ell}_i}(F_+(A^{\underline{\ell}^*}))\cap F'=\{0\}$ and the angle between $A^{\underline{\ell}_i}(F_+(A^{\underline{\ell}^*}))$ and $F'$ is $\geq\delta$. Here, $F_+(A^{\underline{\ell}^*})$ is the subspace associated to $k$ largest exponents of $A^{\underline{\ell}^*}$.

The verification of a) and b) is not hard: by the pinching assumption, there are only finitely many $A^{\underline{\ell}^*}$-invariant subspaces $F\in G(k), F'\in G(d-k)$ (for $k$ admissible), so that a) follows from the twisting assumption; also, for each $F'\in G(d-k)$, we can use the twisting assumption to find $\underline{\ell}(F')\in\Omega$, $\delta(F')>0$ and a neighborhood $U(F')$ of $F'$ in $G(d-k)$ such that $A^{\underline{\ell}(F')}(F_+(A^{\underline{\ell}^*}))\cap F''=\{0\}$ and the angle between $A^{\underline{\ell}(F')}(F_+(A^{\underline{\ell}^*}))$ and $F''$ is $>\delta(F')>0$ for every $F''\in U(F')$; then, by compactness of $G(d-k)$ (and Lebesgue covering lemma), b) follows.

Lemma 1 (cf. Lemma A.6 of Avila-Viana’s article). Let $\varepsilon>0$ and $\rho$ a probability measure on $G(k)$. There exists $n_0=n_0(\rho,\varepsilon)$ and, for each $\widetilde{\underline{\ell}}\in\Omega$, there exists $i=i(\widetilde{\underline{\ell}})\in\{1,\dots,m\}$ such that, for $n\geq n_0$, we have

$A^{\underline{\ell}}_*(\rho)(B)>1-\varepsilon$

where $\underline{\ell}:=(\underline{\ell}^*)^n\underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i\widetilde{\underline{\ell}}$ (here $(\underline{\ell}^*)^n := \underbrace{\underline{\ell}^*\dots \underline{\ell}^*}_n$) and $B$ is the ball of radius $\varepsilon>0$ centered at $\xi_{\underline{\ell}}$.

This lemma is harder to state than to explain: geometrically, it says that, independently of the (long) word $\widetilde{\underline{\ell}}$ one picks, we can choose an appropriate “start” ($(\underline{\ell}^*)^n\underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i$) so that the word $\underline{\ell}$ obtained by the concatenation of $(\underline{\ell}^*)^n\underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i$ and $\widetilde{\underline{\ell}}$ has the property that $A^{\underline{\ell}}$ concentrates the most of the mass of any probability measure $\rho$ on $G(k)$ on a tiny ball $B$.

For the proof of this lemma, we will need the following fact: the family $A^{\widetilde{\underline{\ell}}}$ is equicontinuous with respect to $\widetilde{\underline{\ell}}$ on to the complement of a neighborhood of $k$-planes non-transverse to $F_-(A^{\widetilde{\ell}})$, that is, on the set of $k$-planes making an angle $\geq\delta>0$ with the $d-k$-dimensional subspace $F_-(A^{\widetilde{\ell}})$ (for some fixed $\delta$). Below we exemplify this fact in the particular case of $3\times 3$ matrices:

Example. Any matrix $A$ can be decomposed as $A= O' D O$ where $O$ and $O'$ are orthogonal and $D$ is diagonal. Since the orthogonal matrices act on Grassmanians in an equicontinuous way, it suffices to understand the action of the diagonal part. In the case of a $3\times 3$ matrix $A$, we write

$A=\left(\begin{array}{ccc}\lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \end{array}\right)$

with $\lambda_1\geq\lambda_2\geq\lambda_3$. Suppose we are interested in studying the action on $2$-planes (as the case of lines is too simple). Then, the $F_-$-direction is $x_3$, and, in this case, a $2$-plane makes angle $\geq\delta$ with the $x_3 = F_-$ direction if and only if it contain a pair of vectors

$(1,0,x_3), (0,1,y_3)$

whose coordinates verify $|x_3|,|y_3|\leq 1/\delta$. Of course, we see that in this part of the Grassmanian of $2$-planes, we can use $x_3$ and $y_3$ as coordinates in order to study the action of $A$. Since

$A(1,0,x_3)=(\lambda_1,0,\lambda_3 x_3), A(0,1,y_3) = (0,\lambda_2,\lambda_3 y_3)$

we get that the action of $A$ (on this part of the Grassmanian of $2$-planes) in terms of the coordinates $x_3$ and $y_3$ is

$(x_3',y_3')=A(x_3,y_3)=\left(\frac{\lambda_3}{\lambda_1}x_3,\frac{\lambda_3}{\lambda_2}y_3\right)$

Thus, we see that the action of this family $A$ on the complement of a $\delta$ neighborhood of $F_-(A)$ is equicontinuous.

Proof of Lemma 1. Given an arbitrary probability $\rho$, we apply $(A^{\underline{\ell}^*})^n$. In this way, we get a probability $\rho':= (A^{\underline{\ell}^*})^n_*\rho$ whose mass concentrates nearby some point of $G(k)$ invariant under $A^{\underline{\ell}^*}$. Now, we use $A^{\underline{\ell}^0}$ (given by item a) above) in order to get probability measures $\rho'':=A^{\underline{\ell}^0}_*\rho'$ concentrated nearby a point of of $G(k)$ in general position. Next, we re-apply $(A^{\underline{\ell}^*})^n$ to get probability measures $\rho'''=(A^{\underline{\ell}^*})^n_*\rho''$ concentrated nearby the point $F_+(A^{\underline{\ell}^*})$.

Here we see a subtle but crucial gain of information: in the first application of $(A^{\underline{\ell}^*})^n$, we get nearby some isotropic $k$-plane invariant under $A^{\underline{\ell}^*}$, but after adjusting with the “twisting” $A^{\underline{\ell}^0}$, in the second application of $(A^{\underline{\ell}^*})^n$, we get nearby a very specific isotropic $k$-plane, namely, $F_+(A^{\underline{\ell}^*})$ associated to the $k$ largest exponents of $A^{\underline{\ell}^*}$.

Continuing the argument, we take $F'=F_-(A^{\widetilde{\underline{\ell}}})$ and, by using item b) above, we select $i = i(\widetilde{\underline{\ell}})$ such that the angle between

$A^{\underline{\ell}_i}(F_+(A^{\underline{\ell}^*}))$

and

$F_-(A^{\widetilde{\underline{\ell}}})$

is $\geq\delta>0$. Then, by applying $A^{\underline{\ell}_i}$, we get probability measures $\rho''''=A^{\underline{\ell}_i}\rho'''$ concentrated nearby a point uniformly transverse to (i.e., making angle $\delta$ with) $F'=F_-(A^{\widetilde{\underline{\ell}}})$.

By the equicontinuity of the family $A^{\widetilde{\underline{\ell}}}$ on the complement of a $\delta$-neighborhood of $F_-(A^{\widetilde{\underline{\ell}}})$, we see that, by applying $A^{\widetilde{\underline{\ell}}}$, one ends up with a probability measure $\rho^(v)=A^{\widetilde{\underline{\ell}}}\rho''''$ concentrated nearby $\xi_{\underline{\ell}}$, where $\underline{\ell}=(\underline{\ell}^*)^n\underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i\widetilde{\underline{\ell}}$. $\square$

Coming back to the proof of Proposition 3, we recall that our task is to prove that for $\mu_-$-almost every $x$, there exists a subsequence $\nu_{n_k}(x)$ converging to a Dirac mass (with the aid of Lemma 1 above). Here, we take a slightly different route from Avila-Viana’s article and, instead of following the arguments of their Lemma A.7, we will conclude with a density point plus bounded distortion argument (in the spirit of the proof of the ergodicity of Bernoulli shifts).

We wish to apply Lemma 1 above with $\rho=\nu$ (as, by definition, $\nu_n(x)=A^{\underline{\ell}(x,n)}_*\nu$).

In order to do so, we fix $n\in\mathbb{N}$, and we claim that for $\mu_-$-almost every $x$, there are infinitely many integers $N\geq 1$ such that $\underline{\ell}(x,N)=(\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i\widetilde{\underline{\ell}}$ with $i=i(\widetilde{\ell})$.

The proof of this claim is by contradiction. Suppose that, for some fixed $n\in\mathbb{N}$, the set $C=C_n$ of points $x\in\Sigma_-$ for which the claim is false has positive $\mu_-$-measure. By definition, we can write $C=\bigcup\limits_{a\in\mathbb{N}} C(a)$, where $C(a)$ is the set of $x\in\Sigma_-$ such that $\underline{\ell}(x,N)$ has not the desired form (i.e., $\underline{\ell}(x,N)\neq (\underline{\ell}^*)^n \underline{\ell}^0 (\underline{\ell}^*)^n\underline{\ell}_i\widetilde{\underline{\ell}}$ with $i=i(\widetilde{\ell})$) for every $N\geq a$. Since $C(a)\subset C(a+1)$ for all $a\in\mathbb{N}$, and $C$ has positive $\mu_-$-measure, we can find $a_0\in\mathbb{N}$ such that $B:=C(a_0)$ still has positive $\mu_-$-measure.

Now, we take a density point $x_0\in B$. By definition, given $\varepsilon>0$, we can find $N(\varepsilon)=N(x_0,\varepsilon)\geq 1$ such that

$\mu_-(\Sigma_-(\underline{\ell}(x_0,N))\cap B)>(1-\varepsilon)\mu_-(\Sigma_-(\underline{\ell}(x,N)))$

for all $N\geq N(\varepsilon)$.

On the other hand, the bounded distortion property ensures the existence of a constant $c(n)>0$ such that

$\mu_-(\Sigma_-((\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i \widetilde{\underline{\ell}}))\geq c(n) \mu_-(\Sigma_-(\widetilde{\underline{\ell}}))$

for every $1\leq i\leq m$ and $\widetilde{\underline{\ell}}\in\Omega$. Of course, $c(n)$ may go to $0$ as $n\to\infty$, but since $n$ is fixed in this argument here (as we stressed out above), this is not a serious issue.

By taking $\varepsilon:=c(n)/2$ and $N_0:=N(c(n)/2)$, we have

$\mu_-(\Sigma_-(\underline{\ell}(x_0,N))\cap B)>(1-\varepsilon)\mu_-(\Sigma_-(\underline{\ell}(x_0,N)))$

for all $N\geq N_0$. In particular, by denoting by $B^c$ the complement of $B$, one has

$\mu_-(\Sigma_-((\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i \underline{\ell}(x_0,N))\cap B^c)\leq \mu_-(\Sigma_-(\underline{\ell}(x_0,N))\cap B^c)$

$\leq \varepsilon\mu_-(\Sigma_-(\underline{\ell}(x_0,N))):=(c(n)/2)\mu_-(\Sigma_-(\underline{\ell}(x_0,N)))$

for every $1\leq i\leq m$ and $N\geq N_0$. Combining this with the estimate coming from the bounded distortion property, we find that

$\mu_-(\Sigma_-((\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i \underline{\ell}(x_0,N))\cap B) + (c(n)/2)\mu_-(\Sigma_-(\underline{\ell}(x_0,N)))\geq$

$\mu_-(\Sigma_-((\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i \underline{\ell}(x_0,N))\cap B)+ \mu_-(\Sigma_-((\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i \underline{\ell}(x_0,N))\cap B^c)$

$= \mu_-(\Sigma_-((\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i \underline{\ell}(x_0,N)))\geq c(n) \mu_-(\Sigma_-(\underline{\ell}(x_0,N)))$

for every $1\leq i\leq m$ and $N\geq N_0$. In other words, we conclude that

$\mu_-(\Sigma_-((\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i \underline{\ell}(x_0,N))\cap B)\geq (c(n)/2)\mu_-(\Sigma_-(\underline{\ell}(x_0,N)))>0$

for all $1\leq i\leq m$ and $N\geq N_0$. Of course, this is a contradiction because, by the definition $B:=C(a_0)$, one has $\Sigma_-((\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i \underline{\ell}(x_0,N))\cap B=\emptyset$ for $N\geq \max\{a_0,N_0\}$ and $i=i(\underline{\ell}(x_0,N))$.

So we proved the claim that, for each fixed $n$, the set of $x$ such that there are infinitely many integers $N\geq 1$ such that $\underline{\ell}(x,N)=(\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i\widetilde{\underline{\ell}}$ with $i=i(\widetilde{\ell})$ has full $\mu_-$-measure.

Since the intersection of a countable family of sets of full measure still has full measure, the claim implies the slightly stronger property that for $\mu_-$-almost every $x\in\Sigma_-$, one has that, for every $n$, there are infinitely many integers $N\geq 1$ such that $\underline{\ell}(x,N)=(\underline{\ell}^*)^n \underline{\ell}^0(\underline{\ell}^*)^n\underline{\ell}_i\widetilde{\underline{\ell}}$ with $i=i(\widetilde{\ell})$.

As one can check from the statement of Lemma 1, this slightly stronger property is exactly what we need to be able to apply Lemma 1 to $\rho=\nu$ and $\nu_n(x):=A^{\underline{\ell}(x,n)}_*\nu$. The conclusion is: for $\mu_-$-almost every $x\in\Sigma_-$, we can extract a subsequence $\nu_{n_k}(x)$ converging to a Dirac mass, so that the proof of Proposition 3 (and today’s post) is complete.

Next time, we will start the discussion of some strategies towards the verification of the pinching and twisting assumptions in concrete examples.