Posted by: matheuscmss | November 28, 2016

## “Mesures stationnaires absolument continues”

About 3+1/2 weeks ago, Jean-François Quint gave a very nice talk (with same title as this post) during Paris 6 and 7 “Journées de dynamique” about his joint work with Yves Benoist on the regularity properties of stationary measures.

In what follows, I’m reproducing my notes for Jean-François Quint’s lecture. (As usual, all errors/mistakes in the sequel are my responsibility.)

1. Introduction

1.1. Limit sets of semigroups of matrices

Let ${\Gamma\subset GL_d(\mathbb{R})}$ be a semigroup of invertible ${d\times d}$ real matrices.

Recall that:

• ${\Gamma}$ is irreducible if there are no non-trivial ${\Gamma}$-invariant subspaces, i.e., ${V\subset\mathbb{R}^d}$ and ${\Gamma(V)=V}$ imply ${V=\{0\}}$ or ${\mathbb{R}^d}$;
• ${\Gamma}$ is proximal if it contains a proximal element ${g\in\Gamma}$, i.e., ${g}$ has an unique eigenvalue with maximal modulus which has multiplicity one in the characteristic polynomial of ${g}$; equivalently, ${\mathbb{R}^d = \mathbb{R} x_g^+ \oplus V_g^{<}}$, ${g(x_g^+)=\lambda x_g^+}$, ${g(V_g^{<})=V_g^{<}}$ and ${g|_{V_g^{<}}}$ has spectral radius ${<|\lambda|}$ or, in other terms, the action of ${g}$ on the projective space ${\mathbb{P}^{d-1}}$ has an attracting fixed point.

Proposition 1 Let ${\Gamma\subset GL_d(\mathbb{R})}$ be a irreducible and proximal semigroup. Then, the action of ${\Gamma}$ on ${\mathbb{P}^{d-1}}$ admits a smallest non-empty invariant closed subset ${\Lambda_{\Gamma}}$ called the limit set of ${\Gamma}$.

Proof: Let ${\Lambda_{\Gamma}:=\overline{\{\mathbb{R}x_g^+: g\in\Gamma \textrm{ proximal}\}}}$. It is clear that ${\Lambda_{\Gamma}}$ is non-empty, closed and invariant. Moreover, ${\Lambda_{\Gamma}}$ is the smallest subset with these properties thanks to the following argument. Let ${g\in\Gamma}$ be a proximal element. If ${x\notin\mathbb{P}(V_g^{<})}$, then ${g^n(x)}$ converges to ${\mathbb{R}x_g^+}$ as ${n\rightarrow\infty}$. If ${x\in\mathbb{P}(V_g^{<})}$, we use the irreducibility of ${\Gamma}$ to find an element ${\gamma\in\Gamma}$ such that ${\gamma(x)\notin\mathbb{P}(V_g^{<})}$ and, a fortiori, ${g^n(\gamma(x))}$ converges to ${\mathbb{R}x_g^+}$ as ${n\rightarrow\infty}$. $\Box$

1.2. Stationary measures

Suppose that ${\mu}$ is a probability measure on a semigroup ${G}$ acting on a space ${X}$. We say that a probability measure ${\nu}$ on ${X}$ is ${\mu}$stationary if it is ${G}$-invariant on average, i.e.,

$\displaystyle \mu\ast\nu:=\int_G g_{\ast}(\nu) d\mu(g)$

is equal to ${\nu}$.

In the case of irreducible and proximal semigroups of matrices, the following theorem of Furstenberg and Kesten ensures the existence and uniqueness of stationary measures for the corresponding projective actions:

Theorem 2 (Furstenberg-Kesten) Let ${\mu}$ be a Borel probability measure on ${GL_d(\mathbb{R})}$ and denote by ${\Gamma_{\mu}}$ the subsemigroup generated by the elements in the support ${\textrm{supp}(\mu)}$ of ${\mu}$. Suppose that ${\Gamma_{\mu}}$ is irreducible and proximal. Then, ${\mu}$ has an unique ${\mu}$-stationary measure on ${\mathbb{P}^{d-1}}$ and ${\nu(\Lambda_{\Gamma_{\mu}})=1}$.

In what follows, we shall also assume that ${\Gamma_{\mu}}$ is strongly irreducible, i.e., ${\nu(\mathbb{P}V)=0}$ for all non-trivial proper subspaces ${V\subset \mathbb{R}^d}$, and we will be interested in the nature of ${\nu}$ in Furstenberg-Kesten theorem.

It is possible to show that if ${\mu}$ is absolutely continuous with respect to the Lebesgue (Haar) measure (on ${GL_d(\mathbb{R})}$), then ${\nu}$ is absolutely continuous with respect to the Lebesgue measure (on ${\mathbb{P}^{d-1}}$).

For this reason, we shall focus in the sequel on the following question:

Can ${\nu}$ be absolutely continuous when ${\mu}$ is finitely supported?

It was shown by Kaimanovich and Le Prince that the answer to this question is not always positive:

Theorem 3 (Kaimanovich-Le Prince) There exists ${S\subset SL_2(\mathbb{R})}$ finite (actually, ${\# S=2}$) such that ${S}$ spans a Zariski dense subsemigroup of ${SL_2(\mathbb{R})}$, but ${S}$ is the support of a probability measure ${\mu}$ such that the associated stationary measure ${\nu}$ on ${\mathbb{P}^1}$ is singular with respect to the Lebesgue measure.

On the other hand, Bárány-Pollicott-Simon and Bourgain showed that the answer to this question is sometimes positive:

Theorem 4 (Bárány-Pollicott-Simon, Bourgain) There exists ${S\subset SL_2(\mathbb{R})}$ finite supporting a probability measure ${\mu}$ such that the corresponding stationary measure ${\nu}$ is absolutely continuous with respect to Lebesgue.

Remark 1 As it was pointed out by Quint, the examples produced by Bourgain are explicit, but it would be desirable to get simpler explicit examples of sets ${S\subset SL_2(\mathbb{R})}$ satisfying the previous theorem. In this direction, he asked the following question. Denote by ${R_{\theta} = \left(\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right)}$, ${0<\theta<\pi/2}$ and ${g_t = \left(\begin{array}{cc} e^t & 0 \\ 0 & e^{-t} \end{array}\right)}$, ${t\in\mathbb{R}}$, and consider the probability measures

$\displaystyle \mu_{t,\theta}=\frac{1}{2}\left(\delta_{g_t} + \delta_{R_{\theta} g_t R_{\theta}^{-1}}\right)$

Is it true that, for each fixed ${\theta}$, if ${t}$ is small enough (and typical?), then the stationary measure ${\nu_{t,\theta}}$ associated to ${\mu_{t,\theta}}$ is absolutely continuous with respect to the Lebesgue measure? (Note that if ${t}$ is very large, then we are in the regime described by Kaimanovich-Le Prince theorem 3.)

1.3. Statement of the main result

In a recent paper, Benoist and Quint extended Theorem 4 to higher dimensions:

Theorem 5 (Benoist-Quint) For any ${d\geq 3}$, there exists ${S\subset GL_d(\mathbb{R})}$ finite and a probability measure ${\mu}$ with ${\textrm{supp}(\mu)=S}$ and ${\Gamma_{\mu}=\Gamma_S}$ proximal and strongly irreducible such that the corresponding stationary measure ${\nu}$ on ${\mathbb{P}^{d-1}}$ is absolutely continuous with respect to Lebesgue.

The remainder of this post is dedicated to the proof of this result.

2. Proof of the main theorem

2.1. Spectral theory of quasi-compact operators

Let ${E}$ be a Banach space and denote by ${\mathcal{B}(E)}$ the space of bounded linear operators on ${E}$.

Given ${T\in\mathcal{B}(E)}$, recall that the compact non-empty set

$\displaystyle \sigma(T):=\{\lambda\in\mathbb{C}: \lambda\cdot \textrm{Id} - T \textrm{ is not invertible}\}\subset\mathbb{C}$

is the spectrum of ${T}$, and the quantity

$\displaystyle \rho(T):=\lim\limits_{n\rightarrow\infty} \|T^n\|^{1/n} = \max\{|\lambda|: \lambda\in\sigma(T)\}$

is the spectral radius of ${T}$.

The space ${\mathcal{K}(E)\subset \mathcal{B}(E)}$ of compact operators is an ideal and the quotient ${\mathcal{C}(E) = \mathcal{B}(E)/\mathcal{K}(E)}$ comes equipped with a natural norm ${\|.\|_e}$.

Recall that the essential spectrum of ${T\in\mathcal{B}(E)}$ is

$\displaystyle \sigma_e(T):=\{\lambda\in\mathbb{C}: \lambda\cdot\textrm{Id}-T \textrm{ is not Fredholm}\}$

and the essential spectral radius of ${T\in\mathcal{B}(E)}$ is

$\displaystyle \rho_e(T):=\max\{|\lambda|: \lambda\in\sigma_e(T)\} = \lim\limits_{n\rightarrow\infty} \|T^n\|_e^{1/n}$

Note that ${\sigma_e(T)\subset\sigma(T)}$ and ${\rho_e(T)\leq\rho(T)}$. Moreover, these objects are the same for ${T}$ and its adjoint ${T^*\in\mathcal{B}(E^*)}$:

Proposition 6 One has the following identities: ${\sigma(T) = \sigma(T^*)}$, ${\rho(T)=\rho(T^*)}$, ${\sigma_e(T)=\sigma_e(T^*)}$ and ${\rho_e(T) = \rho_e(T^*)}$.

The next proposition explains that the spectrum and the essential spectrum morally differ only by eigenvalues of finite multiplicity:

Proposition 7 If ${\lambda\in\sigma(T)-\{z\in\mathbb{C}:|z|\leq\rho_e(T)\}}$, then there exists a decomposition into closed subspaces such that ${F}$ is finite-dimensional, ${T(F)\subset F}$, ${T(G)\subset G}$, ${\sigma(T|_F)=\{\lambda\}}$ and ${\sigma(T|_G)=\sigma(T)-\{\lambda\}}$.

2.2. Spectral criterion for absolute continuity

Let ${\mu}$ be a Borel probability measure on ${GL_d(\mathbb{R})}$ and consider the natural action of ${GL_d(\mathbb{R})}$ on ${\mathbb{P}^{d-1}}$.

Given a function ${\varphi:\mathbb{P}^{d-1}\rightarrow\mathbb{C}}$, let ${P_{\mu}\varphi(x):=\int_{GL_d(\mathbb{R})} \varphi(gx) d\mu(g)}$.

Remark 2 ${P_{\mu}\varphi(x)=\sum\limits_{g\in\textrm{supp}(\mu)}\varphi(gx)}$ when ${\mu}$ is finitely supported.

We equip ${\mathbb{P}^{d-1}\simeq \mathbb{S}^{d-1}/\{\pm 1\}}$ with the round measure ${\rho}$ induced from the natural Lebesgue measure on the sphere ${\mathbb{S}^{d-1}}$.

If ${\mu}$ has compact support, then ${P_{\mu}}$ is a bounded operator on ${L^2(\mathbb{P}^{d-1},\rho)}$.

One can infer the absolute continuity of the stationary measure of ${\mu}$ from the spectral properties of ${P_{\mu}}$ thanks to the following proposition:

Proposition 8 If ${\Gamma_{\mu}}$ is proximal and strongly irreducible, and ${\|P_{\mu}\|_e < 1}$, then the ${\mu}$-stationary measure ${\nu}$ on ${\mathbb{P}^{d-1}}$ is absolutely continuous with respect to ${\rho}$, i.e., ${\nu\ll\rho}$.

Proof: Note that ${P_{\mu}1=1}$ (where ${1}$ is the constant function with value one), so that ${1\in\sigma(P_{\mu})}$.

By hypothesis, ${1\in\sigma(P_{\mu}) - \{z\in\mathbb{C}:|z|\leq \rho_e(P_{\mu})\} = \sigma(P_{\mu}^*) - \{z\in\mathbb{C}:|z|\leq \rho_e(P_{\mu}^*)\}}$. Thus, there exists ${\psi\in L^2(\mathbb{P}^{d-1},\rho)}$ with ${P_{\mu}^*\psi=\psi}$. By definition, this means that the absolutely continuous measure ${\nu=|\psi|\rho}$ is the ${\mu}$-stationary measure. $\Box$

2.3. Application of the spectral criterion

The result in Theorem 5 (i.e., the case ${d\geq 3}$) is easier to derive than Theorem 4 (i.e., the case ${d=2}$) because ${SO(3)}$ has elements equidistributing very quickly. Here, the word equidistribution means the following: if ${\mu}$ is a probability measure on ${SO(3)}$, ${\Gamma_{\mu}}$ is Zariski dense on ${SO(3)}$, then we say that the elements in the support of ${\mu}$ equidistribute whenever ${P_{\mu}^n\varphi\rightarrow\int\varphi(g) dg}$ for all ${\varphi\in L^2(SO(3))}$ (with ${dg}$ standing for the Haar measure).

This equidistribution property holds in presence of spectral gap, i.e., ${\rho(P_{\mu}|_{L^2_0(SO(3))}) < 1=\rho(P_{\mu}|_{L^2(SO(3))})}$ (where ${L^2_0(SO(3))}$ is the subspace of ${L^2}$-functions with zero average [for Haar measure]). In particular, the works of Drinfeld and Margulis provide examples of elements of ${SO(3)}$ equidistributing very quickly:

Theorem 9 (Margulis and Drinfeld) There exists ${S\subset SO(3)}$ finite and a probability measure ${\mu}$ with ${\textrm{supp}(\mu)=S}$ and ${\rho(P_{\mu}|_{L^2_0(SO(3))})<1}$.

At this point, the proof of Theorem 5 is almost over. Indeed, if we take ${g\in GL_3(\mathbb{R})}$ a proximal element and we denote by ${\mu}$ the probability measure provided by Margulis and Drinfeld, then the sequence of measures

$\displaystyle \mu_n:=\underbrace{\mu\ast\dots\ast\mu}_{n \textrm{ times}}\ast\delta_g$

have the property that ${P_{\mu_n}}$ converges to a rank one operator. Therefore,

$\displaystyle \lim\limits_{n\rightarrow\infty} \|P_{\mu_n}\|_e = 0$

In particular, by Proposition 8, it follows that ${\mu_n}$ satisfies the conclusions of Benoist-Quint theorem 5 for any ${n}$ sufficiently large.