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.

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