# Stationary probability measures on projective spaces 2: the critical case

Richard Aoun  University Gustave Eiffel, Champs-sur-Marne, 5 boulevard
Descartes Champs-sur-Marne 77420 Marne-la-Vallée Cedex 2, France
<EMAIL_ADDRESS>and  Cagri Sert  Institut für Mathematik,
Universität Zürich, 190, Winterthurerstrasse, 8057 Zürich, Switzerland
<EMAIL_ADDRESS>
###### Abstract.

In a previous article, given a finite-dimensional real vector space $V$ and a
probability measure $\mu$ on $\operatorname{PGL}(V)$ with finite first moment,
we gave a description of all $\mu$-stationary probability measures on the
projective space $\operatorname{P}(V)$ in the non-critical (or Lyapunov
dominated) case. In the current article, we complete the analysis by providing
a full description of the more subtle critical case. Our results demonstrate
an algebraic rigidity in this situation. Combining our results with those of
Furstenberg–Kifer (’83), Guivarch–Raugi (’07) $\&$ Benoist–Quint (’14), we
deduce a classification of all stationary probability measures on the
projective space for i.i.d random matrix products with finite first moment
without any algebraic assumption.

###### Key words and phrases:

Random walks, stationary measures, projective space, random matrix products

###### 2010 Mathematics Subject Classification:

Primary 37H15; Secondary 60J05, 60B15, 37A20

C.S. is supported by SNF Ambizione grant 193481

##  1\. Introduction

Let $V$ be a finite-dimensional real vector space and $\mu$ a probability
measure on $\operatorname{GL}(V)$. Let $(X_{n})_{n\in\mathbb{N}}$ denote a
$\operatorname{GL}(V)$-valued sequence of iid random variables with
distribution $\mu$ and write $L_{n}=X_{n}\ldots X_{1}$ for the associated
random matrix product. Via the action of $\operatorname{GL}(V)$ on the
projective space $\operatorname{P}(V)$ the random product $L_{n}$ induces a
Markov chain on $\operatorname{P}(V)$. The goal of the paper is to give a
classification of all stationary probability measures of this Markov chain. In
the sequel, we will refer to these as $\mu$-stationary, or simply, stationary
measures on $\operatorname{P}(V)$.

Let us start with a brief history. The study of stationary measures on
projective spaces was initiated by Furstenberg–Kesten [11] and Furstenberg
[10] who realized early on that these encode to a great extent the asymptotic
behaviour of random matrix products. Indeed, using stationary measures,
Furstenberg proved the key result of positivity of top Lyapunov exponent and
he gave a formula for the Lyapunov exponents (Furstenberg formula). He also
showed the uniqueness of stationary measure in the irreducible and proximal
case. Under the algebraic semisimplicity assumption, extending the work of
Furstenberg, the full description of stationary measures was obtained by
Guivarc’h–Raugi [14] and Benoist–Quint [5]. Their result demonstrates an
algebraic rigidity phenomenon in the semisimple situation and it shows that in
the semisimple case every $\mu$-stationary and ergodic probability measure on
$\operatorname{P}(V)$ is $\mu$-homogeneous, a notion which will make precise
below.

The study of stationary measures without the algebraic semisimplicity
assumption – a setting which encompasses many familiar examples such as
Bernoulli convolutions or more generally self-affine measures – was tackled by
Furstenberg–Kifer [12] and Hennion [15] who gave a first description of
stationary measures without additional algebraic assumptions. Crucially, they
used this classification to prove continuity of Lyapunov exponents in the
critical case, the setting which corresponds precisely to the setting of our
Theorem 1.1 below (see also Peres [20] for further applications). Broadly
speaking, without semisimplicity assumptions, for a general classification of
stationary measures in this context, there are two dynamically distinguished
cases that one needs to deal with. These are the so-called non-critical (or
Lyapunov-dominated) and critical cases. Non-criticality refers to a relative
contracting or expanding dynamics whereas the more subtle critical case is
characterized by absence of difference in dynamical behaviour, or expansion
rates, in different invariant subspaces.

Due to the generality of projective dynamics without algebraic assumptions,
stationary measures on projective spaces were studied in various special cases
of interest: For the critical case, after the pioneering work of Furstenberg
[10], Kesten–Spitzer [17] treated the case of non-negative matrices and
Bougerol [7] gave a full description of stationary measures on vector spaces.
The case of affine random walks (also called affine stochastic recursion) was
brought to conclusion by Bougerol–Picard [9]. These works constitute the first
important situations where a complete description in the critical case were
achieved. More recently, Benoist–Bruère [3] classified stationary probability
measures on affine Grassmannians under a Zariski-density assumption.

The goal of the current paper is to give a classification of stationary
measures in the critical case, and combine this with our previous work [1] on
the non-critical case to obtain a full description without any assumptions on
the random matrix products (except finite first moment assumption). We will
prove a rigidity result, showing that in the critical case all ergodic
stationary measures are supported on semisimple subspaces (Theorem 1.1).
Combined with the aforementioned work of Guivarc’h–Raugi $\&$ Benoist–Quint,
this implies that in the critical case all $\mu$-stationary ergodic measures
are $\mu$-homogeneous (Corollary 1.2). Finally, combining this with the
results of Furstenberg–Kifer and our previous work, we obtain a description of
$\mu$-stationary measures (Theorem 1.3): given any probability measure $\mu$
with finite first moment, any $\mu$-stationary ergodic probability measure on
$\operatorname{P}(V)$ is the unique lift of a $\mu$-homogeneous stationary
measure.

We now proceed to make all these statements precise.

A probability measure $\mu$ on $\operatorname{GL}(V)$ is said to have finite
first moment if

$\int\log\max\\{\|g\|,\|g^{-1}\|\\}\,d\mu(g)<\infty,$

where $\|\cdot\|$ is any norm on $\operatorname{End}(V)$. Given a probability
measure $\mu$ with finite first moment, the top Lyapunov exponent of $\mu$ is
the constant, denoted $\lambda_{1}(\mu)$, which is the almost-sure limit of
$\lim_{n\to+\infty}\frac{1}{n}\log\|X_{n}\cdots X_{1}\|$. The existence of the
limit is guaranteed by Kingman’s subadditive ergodic theorem and was first
proved in this setting by Furstenberg–Kesten [11]. We denote by $\Gamma_{\mu}$
the closed semigroup of $\operatorname{GL}(V)$ generated by the support of
$\mu$. If $W$ is a $\Gamma_{\mu}$-invariant subspace of $V$, we will also
denote by $\lambda_{1}(W)$ the top Lyapunov exponent of the $\mu$-random
matrix products induced on $W$. In particular, with this notation,
$\lambda_{1}(\mu)=\lambda_{1}(V)$. Our first main result reads as follows:

######  Theorem 1.1.

(Critical case) Let $\mu$ be a probability measure on $\operatorname{GL}(V)$
with finite first moment. Assume that the top Lyapunov of any non-zero
$\Gamma_{\mu}$-invariant subspace of $V$ is equal to $\lambda_{1}(\mu)$. Then
the action of $\Gamma_{\mu}$ on the subspace generated by the support of any
$\mu$-stationary probability measure on $\operatorname{P}(V)$ is semisimple.

When the $\Gamma_{\mu}$-action on a vector space $W$ is semisimple
(equivalently when the Zariski-closure of $\Gamma_{\mu}$ in
$\operatorname{GL}(W)$ is reductive), Guivarc’h–Raugi [14] and Benoist–Quint
[5] proved that any $\mu$-stationary ergodic probability measure $\nu$ is
$\mu$-homogeneous in the sense that the Zariski-closure of the support of
$\nu$ is a closed orbit of $H_{\mu}$ in $\operatorname{P}(W)$, where $H_{\mu}$
denotes the Zariski closure of $\Gamma_{\mu}$ in $\operatorname{GL}(W)$.
Moreover they showed that this orbit supports a unique $\mu$-stationary
probability measure. This implies that the irreducible components of the
subspace $V_{\nu}$ generated by the support of $\nu$ have the same weight (in
particular $V_{\nu}$ is critical) so that the hypothesis of Theorem 1.1 is in
some sense sharp. On the other hand, their results combined with Theorem 1.1
imply the following.

######  Corollary 1.2.

Keep the assumptions of Theorem 1.1. Then any $\mu$-stationary ergodic
probability measure is $\mu$-homogeneous, and there is a natural bijection

$\\{\mu\text{-stationary and ergodic measures on
$\operatorname{P}(V)$}\\}\simeq\\{\text{Compact}\;H_{\mu}\text{-orbits
in}\;\operatorname{P}(V)\\}.$

Corollary 1.2 combined with the works of Furstenberg–Kifer [12] and Hennion
[15] and our previous work [1] yields the following classification of
stationary probability measures on $\operatorname{P}(V)$.

######  Theorem 1.3 (General Classification).

Let $\mu$ be a probability measure on $\operatorname{GL}(V)$ with finite first
moment. Then there exists a filtration $V=F_{1}\supsetneq
F_{2}\cdots\supsetneq F_{k}\supsetneq\\{0\\}$ of $\Gamma_{\mu}$-invariant
subspaces such that for each $\mu$-stationary ergodic probability $\nu$, there
exists $i\in\\{1,\cdots,k\\}$ such that
$\nu(\operatorname{P}(F_{i})\setminus\operatorname{P}(F_{i+1}))=1$ and $\nu$
is the unique lift in
$\operatorname{P}(F_{i})\setminus\operatorname{P}(F_{i+1})$ of a
$\mu$-homogeneous stationary measure in $\operatorname{P}(F_{i}/F_{i+1})$.

Here and elsewhere, when $W<V$ is an invariant subspace, we employ the term
lift to mean that $\nu$ is a $\mu$-stationary probability measure on
$\operatorname{P}(V)\setminus\operatorname{P}(W)$ whose pushforward on
$\operatorname{P}(V/W)$ under the map induced by the natural projection $V\to
V/W$ is $\overline{\nu}$.

In the above statement, the fact that each ergodic stationary probability
measure $\nu$ lives in some
$\operatorname{P}(F_{i})\setminus\operatorname{P}(F_{i+1})$ is contained in
Furstenberg–Kifer and Hennion’s work. The new information given by Theorem 1.3
is in the description of these measures: the projection of each such measure
$\nu$ on $\operatorname{P}(F_{i}/F_{i+1})$ is one of the stationary measures
described by Guivarc’h–Raugi and Benoist–Quint (i.e. $\mu$-homogeneous) and
$\nu$ is the unique lift thereof on the open subset
$\operatorname{P}(F_{i})\setminus\operatorname{P}(F_{i+1})$. In particular,
combined with the unique ergodicity results of [14, 5] for $\mu$-homogeneous
measures, this yields a parametrization of $\mu$-stationary and ergodic
probability measures by compact $H_{\mu}$-orbits on the quotients of
Furstenberg–Kifer–Hennion spaces:

######  Corollary 1.4.

$\\{\mu\text{-stationary and ergodic probabilities on
$\operatorname{P}(V)$}\\}\simeq\bigcup_{i=1}^{k}\,\\{\text{Compact
$H_{\mu}$-orbits in}\,\operatorname{P}(F_{i}/F_{i+1})\\}.$

###### Proof of Theorem 1.3.

Let $\nu$ be a $\mu$-stationary and ergodic probability measure. By
Furstenberg–Kifer [12], there exists a filtration $V=F_{1}\supsetneq
F_{2}\cdots\supsetneq F_{k}\supsetneq\\{0\\}$ of $\Gamma_{\mu}$-invariant
subspaces and $i\in\\{1,\ldots,k\\}$ such that
$\nu(\operatorname{P}(F_{i}))=1$ and $\nu(\operatorname{P}(F_{i+1}))=0$.
Therefore, $\nu$ projects down to a stationary ergodic measure
$\overline{\nu}$ on $\operatorname{P}(F_{i}/F_{i+1})$. By [12, Lemma 3.7], the
quotient $F_{i}/F_{i+1}$ satisfies the assumption of Theorem 1.1 (i.e.
$F_{2}(F_{i}/F_{i+1})=\\{0\\}$) and therefore Theorem 1.1 applies and it
yields that $\overline{\nu}$ is supported on a $\Gamma_{\mu}$-completely
reducible subspace. Now Guivarc’h–Raugi [14] $\&$ Benoist–Quint [5] applies
and shows that $\overline{\nu}$ is a $\mu$-homogeneous stationary measure.
Since $\lambda_{1}(F_{i+1})<\lambda_{1}(F_{i}/F_{i+1})$, [1, Theorem 1.1]
implies that $\nu$ is the unique lift of the $\mu$-homogeneous stationary
measure $\overline{\nu}$, concluding the proof. ∎

We note in passing that unlike many results for limit theorems in random
matrix products theory, Theorem 1.1 is specific to vector spaces over
archimedean local fields (so $\mathbb{R}$ or $\mathbb{C}$111Indeed, Theorem
1.1 holds for a complex vector space; this follows easily from the real
case.); it fails for non-archimedean local fields (finite extensions of
$\mathbb{Q}_{p}$ or $\mathbb{F}_{p}((T))$). Indeed, this result is ultimately
a generalization of the fact that a random walk on $\mathbb{R}$ has a
stationary probability measure if and only if it is trivial (i.e. the law of
the random walk is the Dirac measure on $0$) and already this statement
obviously fails for non-archimedean local fields.

In specific models, such as the ones coming from homogeneous dynamics as
random walks on affine spaces [9] or affine Grassmanians [3], it is desirable
to have a formulation in terms of lifts with respect to a specific invariant
subspace. More precisely, given a $\Gamma_{\mu}$-invariant subspace $W$ of
$V$, one is interested in the existence of $\mu$-stationary probability
measures on $\operatorname{P}(V)\setminus\operatorname{P}(W)$. For instance, a
direct consequence of Theorem 1.1 is that if $\lambda_{1}(V)=\lambda_{1}(W)$
and $W$ and $V/W$ are irreducible, then there is no invariant stationary
probability measure on $\operatorname{P}(V)\setminus\operatorname{P}(W)$
unless $W$ has an invariant complement. Treating the general case (i.e.
without irreducibility assumptions) requires some more care and is done in the
following result by combining Theorem 1.1, together with [12] and the
contracting case [1].

######  Theorem 1.5 (Lift with respect to a given subspace).

Let $\mu$ be a probability measure on $\operatorname{GL}(V)$ with finite first
moment and $W$ a $\Gamma_{\mu}$-invariant subspace. Let $\overline{\nu}$ be a
$\mu$-stationary and ergodic probability measure on $\operatorname{P}(V/W)$
such that $\lambda_{1}(V_{\overline{\nu}})\leqslant\lambda_{1}(W)$. Then, the
following are equivalent:

  * (i)

There exists a $\mu$-stationary lift $\nu$ of $\overline{\nu}$ on
$\operatorname{P}(V)\setminus\operatorname{P}(W)$.

  * (ii)

There exists a $\Gamma_{\mu}$-invariant subspace $W^{\prime}<V$ such that
$\lambda_{1}(W^{\prime}\cap W)<\lambda_{1}(V_{\overline{\nu}})$ and
$\operatorname{P}(V_{\overline{\nu}})=\operatorname{P}(W^{\prime}/W^{\prime}\cap
W)$.

In this case, we have $\lambda_{1}(V_{\overline{\nu}})=\lambda_{1}(V_{\nu})$.

This theorem generalizes Bougerol’s result [7, Theorem 5.1] and in particular
Bougerol–Picard [9] in the invertible case. It also recovers Benoist–Bruère’s
results [3, Theorem 1.6]. We note that, unlike the expanding case
$\lambda_{1}(V_{\overline{\nu}})<\lambda_{1}(W)$ (see [1, Theorem 1.5]), the
existence of $\mu$-stationary lift does not imply its uniqueness. We refer to
Remark 4.4 for a more detailed explanation.

Finally, it is well-known, thanks to a classical result of Chevalley, that for
any real algebraic group $G$ and algebraic subgroup $R$, the algebraic
homogeneous space $G/R$ can be realized as a $G$-orbit in
$\operatorname{P}(V)$ via some representation $G\to\operatorname{GL}(V)$. On
the other hand, since $H_{\mu}$-orbits are locally closed in
$\operatorname{P}(V)$, any $\mu$-stationary ergodic probability measure is
supported by a single $H_{\mu}$-orbit $\mathcal{O}$. Therefore, given a
probability measure $\mu$ on $G<\operatorname{GL}(V)$, it is important to be
able to describe $H_{\mu}$-orbits $\mathcal{O}\subseteq\operatorname{P}(V)$
that support $\mu$-stationary measures. The following corollary that we obtain
by combining Theorem 1.1, [1, Theorem 1.1], and [5] gives such a description.

######  Corollary 1.6.

Let $\mu$ be a probability measure on $\operatorname{GL}(V)$ with a finite
first moment. Let $\mathcal{O}$ be a $H_{\mu}$-orbit in $\operatorname{P}(V)$.
Then the following hold.

  1. (1)

The orbit $\mathcal{O}$ supports at most one $\mu$-stationary probability
measure.

  2. (2)

Denote by $V_{\mathcal{O}}$ the linear space generated by the orbit
$\mathcal{O}$ and by $F_{\mu}:=F_{2}(V_{\mathcal{O}})$ the maximal
$H_{\mu}$-invariant subspace of slower expansion. Then the following are
equivalent.

    1. 2.1.

The orbit $\mathcal{O}$ supports a $\mu$-stationary probability measure.

    2. 2.2.

The image of $\mathcal{O}$ under the natural projection
$\operatorname{P}(V_{\mathcal{O}})\setminus\operatorname{P}(F_{\mu})\to\operatorname{P}(V_{\mathcal{O}}/F_{\mu})$
is compact and $\overline{O}\setminus O\subseteq\operatorname{P}(F_{\mu})$.

We note that since $\operatorname{P}(F_{\mu})\cap\mathcal{O}=\emptyset$, the
projection map above is well-defined. Finally, the subspace
$F_{\mu}<V_{\mathcal{O}}$ can be trivial (equivalently $V_{\mathcal{O}}$ is
critical) in which case (2) above boils down to Corollary 1.2.

###  1.1. Outline of the proof

We give an outline of the proof of Theorem 1.1. After certain simplifying
preliminary reductions (such as restricting, via return times, to a Zariski-
connected group and a non-degenerate stationary measure) carried out in §2,
the proof hinges on proving Theorem 3.1 which says that the existence of non-
degenerate stationary measure implies that every $\Gamma_{\mu}$-invariant and
irreducible subspace $W<V$ with $\lambda_{1}(W)=\lambda_{1}(V)$ has a
$\Gamma_{\mu}$-invariant complement. To prove this, we start by decomposing a
given stationary measure $\nu$ into random pieces $\nu_{b}$ living on a
(projective) subspace $W_{b}$ of the most contracting (unstable) Oseledets
space for the inverse random walk. Using the criticality assumption, the
invariant space $W$ is shown to intersect the space $W_{b}$, and $W\cap W_{b}$
has co-dimension at least one in $W_{b}$. We then go on to construct a
(random) idempotent quasi-projective transformation $\pi$ whose image is
$W_{b}$. This key construction uses recurrence of Birkhoff sums (Lemma 3.6)
and some observations on random matrix products theory (Lemma 3.3 $\&$ Lemma
3.5). Using this transformation $\pi$, we construct a new (random) semigroup
$\mathcal{S}_{b}$ acting on the space $W_{b}$ which, on the one hand, reads
information from (a Zariski-dense subset of the) original semigroup
$\Gamma_{\mu}$, and on the other hand, preserves the random probability
measure $\nu_{b}$. By a classical result of Furstenberg, the latter property
implies that the semigroup $\mathcal{S}_{b}$ acts on $W_{b}$ relatively
compactly (all this is expressed in Proposition 3.7 which provides a crucial
handle to construct a complement to $W$). Now, by relative compactness, we get
an $\mathcal{S}_{b}$-invariant complement to $W\cap W_{b}$ inside $W_{b}$. By
the aforementioned relation between $\mathcal{S}_{b}$ and $\Gamma_{\mu}$, we
can then transfer this data to $\Gamma_{\mu}$. We finally finish the proof of
Theorem 3.1 in an inductive way.

### Acknowledgements

The authors are thankful to Alex Eskin for helpful remarks.

##  2\. Preliminary reduction

As mentioned in the introduction, the proof of Theorem 1.1 will involve
arguments making use of the Zariski topology on $\Gamma_{\mu}$. It will be
more convenient at several places to work with a semigroup $\Gamma_{\mu}$
having a connected Zariski-closure, and also with stationary measures which
are non-degenerate. Accordingly, our goal in this section is to show that the
following Theorem 2.1 implies Theorem 1.1.

######  Theorem 2.1.

The conclusion of Theorem 1.1 holds if we additionally assume that
$\Gamma_{\mu}$ is Zariski-connected and $\nu$ is ergodic and non-degenerate in
$\operatorname{P}(V)$.

Let us first fix once for all our notation and recall standard definitions and
notions used above and in all the paper. All probability measures will be
understood as Borel probability measures and all vector spaces will be real or
complex. Given a probability measure $\mu$ on $\operatorname{GL}(V)$, denote
by $\Gamma_{\mu}$ the closed semigroup of $\operatorname{GL}(V)$ generated by
the support of $\mu$. Recall that a probability measure $\nu$ on the
projective space $\operatorname{P}(V)$ of $V$ is said to be $\mu$-_stationary_
if for every continuous function $f$ on $\operatorname{P}(V)$,
$\int{fdv}=\iint{f(g[x])d\mu(g)d\nu([x])}$ where $[x]$ denotes the projection
of a non-zero vector $x$ of $V$ on the projective space $\operatorname{P}(V)$.
It is said to be $\mu$-ergodic if it is extremal in the convex set of
$\mu$-stationary probability measures on $\operatorname{P}(V)$. We say that
$\nu$ is _non-degenerate in $PP(V)$_ if $\nu(\operatorname{P}(W))=0$ for any
proper projective subspace $\operatorname{P}(W)$ of $\operatorname{P}(V)$. We
denote by $V_{\nu}$ the linear subspace of $V$ generated by the support of
$\nu$.

We recall now standard facts about linear algebraic groups. A linear algebraic
group $G$ is a subgroup of $\operatorname{GL}(V)$ for some finite-dimensional
vector space $V$ which is closed for the Zariski topology, i.e. it is a
subgroup of $\operatorname{GL}(V)$ such that the matrix entries of all
elements of $\operatorname{GL}(V)$ satisfy a set of polynomial equations. A
standard fact is that the Zariski-closure of a semigroup
$\Gamma<\operatorname{GL}(V)$ is a linear algebraic group. We will denote by
$G^{o}$ the connected component of $G$ for the Zariski-topology; it is an
algebraic normal subgroup of $G$ of finite index. Recall finally that for
linear algebraic groups Zariski-connectedness is equivalent to
(Zariski-)topological irreducibility (i.e. the intersection of any two open
non-empty subsets is non-empty). Given a probability measure $\mu$ on
$\operatorname{GL}(V)$, we denote by $H_{\mu}$ the Zariski-closure of
$\Gamma_{\mu}$.

All the random variables we consider will be defined on the probability space
$(B,\mathcal{A},\beta)$ where $B=\operatorname{GL}(V)^{\mathbb{N}}$,
$\mathcal{A}$ the product Borel $\sigma$-algebra, and
$\beta=\mu^{\otimes\mathbb{N}}$. Given a probability measure $\mu$ on
$\operatorname{GL}(V)$ and $k\in\mathbb{N}$, we denote by $\mu^{k}$ the
$k$-fold convolution $\mu\ast\cdots\ast\mu$ which is the distribution of the
random variable $b=(b_{i})_{i\in\mathbb{N}}\mapsto b_{k}\cdots b_{1}$. For
convenience we put $\mu^{0}=\delta_{\textrm{id}}$. We denote by
$\tau:B\to\mathbb{N}$, the stopping time defined by
$b\mapsto\tau(b):=\inf\\{k\in\mathbb{N}:b_{k}\cdots b_{1}\in H_{\mu}^{o}\\}$
and by $\mu^{\tau}$ the distribution of the random variable $b\mapsto
b_{\tau}\cdots b_{1}$. Note that $\tau$ is the hitting time of a state for the
Markov chain given by the $\mu$-random walk on the finite group
$H_{\mu}/H_{\mu}^{o}$. This Markov chain is irreducible since, by Zariski-
density, $\Gamma_{\mu}$ surjects onto $H_{\mu}/H_{\mu}^{o}$, therefore $\tau$
is almost-surely finite.

In the proof of the implication Theorem 2.1 $\implies$ Theorem 1.1, we will
make use of the decomposition of $\nu$ given by the following result.

######  Lemma 2.2.

Let $\nu$ be a $\mu$-stationary ergodic probability measure on
$\operatorname{P}(V)$. Then there exist finitely many probability measures
$\nu_{1},\cdots,\nu_{k}$ on $\operatorname{P}(V)$, where
$k:=[H_{\mu}:H_{\mu}^{o}]$, such that

  * •

$\nu=\frac{1}{k}\sum_{i=1}^{k}{\nu_{i}}$

  * •

Each $\nu_{i}$ is a $\mu^{\tau}$-stationary ergodic probability measure which
is moreover non-degenerate in the projective subspace $V_{\nu_{i}}$ generated
by its support.

###### Proof.

Let $g_{1},\cdots,g_{k}\in H_{\mu}$ be such that
$H_{\mu}/H_{\mu}^{o}=\\{g_{i}H_{\mu}^{o}:i=1,\cdots,k\\}$. The group $H_{\mu}$
acts diagonally on $X:=H_{\mu}/H_{\mu}^{o}\times\operatorname{P}(V)$; so that
the $\mu$-random walk on $H_{\mu}$ induces a Markov chain on $X$. Namely the
$n$th step of this Markov chain starting at $(gH_{\mu}^{o},x)$ is
$(b_{n}\cdots b_{1}gH_{\mu}^{\circ},b_{n}\cdots b_{1}x)$. For every
$i\in\\{1,\cdots,k\\}$, denote
$Y_{i}:=\\{g_{i}H_{\mu}^{\circ}\\}\times\operatorname{P}(V)$. Since
$H_{\mu}^{o}$ is normal in $H_{\mu}$, $H_{\mu}^{o}$ stabilizes each $Y_{i}$.
Note that each $Y_{i}$ is a recurrent subset of $X$ thanks to the
irreducibility of the Markov chain on $H_{\mu}/H_{\mu}^{o}$. Moreover, for
every $i\in\\{1,\cdots,k\\}$, the Markov chain on $Y_{i}$ induced by the
Markov chain on $X$ projects on $\operatorname{P}(V)$ to the
$\mu^{\tau}$-Markov chain on $\operatorname{P}(V)$.

By Chacon–Ornstein theorem, for $\nu$-a.e. every $x\in\operatorname{P}(V)$,
the sequence of probability measures
$\frac{1}{n}\sum_{i=1}^{n}{\mu^{i}\ast\delta_{x}}$ converges weakly to $\nu$.
Let $x$ be such a $\nu$-generic point of $\operatorname{P}(V)$. By compactness
of $X$, the sequence of probability measures
$\frac{1}{n}\sum_{i=1}^{n}{\mu^{i}\ast\delta_{(H_{\mu}^{o},x)}}$ has a limit
point $\eta$ which is a stationary probability measure for the Markov chain on
$X$. Its projection on $H_{\mu}/H_{\mu}^{o}$ is $\mu$-stationary; it is then
the uniform probability measure on $H_{\mu}/H_{\mu}^{o}$. Disintegrating
$\eta$ over $H_{\mu}/H_{\mu}^{o}$, we get that
$\eta=\frac{1}{k}\sum_{i=1}^{k}{\delta_{g_{i}H_{\mu}^{o}}\otimes\nu_{i}}$
where each $\nu_{i}$ is a probability measure on $\operatorname{P}(V)$. In
particular, $\nu=\frac{1}{k}\sum_{i=1}^{k}{\nu_{i}}$ and the restriction of
$\nu$ to $Y_{i}$ is $\nu_{i}$. Now it follows from [4, Lemma 3.4] that
$\nu_{i}$ is stationary for the Markov chain restricted to $Y_{i}$, and that
it is ergodic because $\nu$ is ergodic for the Markov chain on $X$. We deduce
that each $\nu_{i}$ is $\mu^{\tau}$-stationary and ergodic. The fact that each
$\nu_{i}$ is non-degenerate in $\operatorname{P}(V_{\nu_{i}})$ is a general
fact that we recall in Lemma 2.3 below. ∎

######  Lemma 2.3.

Let $\mu$ be a probability measure on $\operatorname{GL}(V)$ such that
$\Gamma_{\mu}$ is Zariski-connected. Then each $\mu$-stationary ergodic
probability measure $\nu$ on $\operatorname{P}(V)$ is non-degenerate in
$\operatorname{P}(V_{\nu})$.

###### Proof.

By standard arguments due to Furstenberg (see for instance the proof of [8,
Proposition 2.3]), one can find a subspace $W$ of $V$ charged by $\nu$, of
minimal dimension for this property, such that the orbit $\Lambda$ of $W$ by
$\Gamma_{\mu}$ is finite. The set of elements $g\in\operatorname{GL}(V)$ that
stabilize $\Lambda$ being Zariski-closed, we deduce that $H_{\mu}$ stabilizes
also $\Lambda$. Since $H_{\mu}$ is Zariski-connected, it must stabilize each
element of $\Lambda$ (otherwise it would have a proper finite index algebraic
subgroup) so that $\Lambda=\\{W\\}$. By ergodicity of $\nu$, $V_{\nu}\subset
W$. We conclude by minimality of $W$. ∎

###### Proof of Theorem 1.1 (using Theorem 2.1).

• (Ergodicity) Suppose each ergodic component $\nu_{e}$ of $\nu$ has the
property that $V_{\nu_{e}}$ is completely reducible. Then, $V_{\nu}$ is
completely reducible. Indeed, by finite-dimensionality, $V_{\nu}$ is a sum of
finitely many $V_{\nu_{e}}$’s. But a sum of completely reducible spaces is
completely reducible.
• (Non-degenerate $\nu$ and Zariski-connectedness) Given an ergodic
$\mu$-stationary probability $\nu$, applying Lemma 2.2, we obtain finitely
many $\mu^{\tau}$-stationary and ergodic measures $\nu_{1},\ldots,\nu_{k}$,
each non-degenerate in projective subspace generated by its support and where
$k=[H_{\mu}:H_{\mu}^{o}]$ and $\nu=\frac{1}{k}\sum_{i=1}^{k}\nu_{i}$.
Therefore, $V_{\nu}=V_{\nu_{1}}+\ldots+V_{\nu_{k}}$. We now claim that

  1. (1)

$\overline{\Gamma}^{Z}_{\mu^{\tau}}=H_{\mu}^{o}$.

  2. (2)

$\mu^{\tau}$ has finite first moment.

  3. (3)

The measure $\mu^{\tau}$ has the property that every
$\Gamma_{\mu^{\tau}}$-invariant subspace has the top Lyapunov exponent
$\lambda_{1}(\mu^{\tau})$.

Once these claims are established, it follows from Theorem 2.1 that each
$V_{\nu_{i}}$ is $H_{\mu}^{o}$-completely reducible and hence $V_{\nu}$ is
$H_{\mu}^{o}$-completely reducible. Since we are working in characteristic
zero, the complete reducibility of $V_{\nu}$ as $H_{\mu}$-space is equivalent
to its complete reducibility as $H_{\mu}^{o}$-space (see e.g. [19, Lemma
3.1]).

It remains therefore to prove Claims (1), (2), and (3) above. For (1), it is
enough to show that $\Gamma_{\mu^{\tau}}=\Gamma_{\mu}\cap H_{\mu}^{o}$. The
inclusion $\subset$ is trivial. For the other inclusion, observe that
$\Gamma_{\mu}\cap H_{\mu}^{o}$ is open in $\Gamma_{\mu}$ because $H_{\mu}^{o}$
has finite index in $H_{\mu}$. Hence for every $g\in\Gamma_{\mu}\cap
H_{\mu}^{o}$ and every neighborhood $O$ of $g$, we have
$\beta(\limsup\\{b_{n}\cdots b_{1}\in O\\})>0$ and hence
$\beta(\limsup\\{b_{\tau(n)}\cdots b_{1}\in O\\})>0$. Hence
$g\in\Gamma_{\mu^{\tau}}$.
For (2), this is showed in [6, Corollary 5.6].
Finally, we show (3). By [12, 15], the assumption on $\mu$ is equivalent to
saying that for every non-zero vector $x$ of $V$ we have
$\lim_{n\to\infty}{\frac{1}{n}\log\|b_{n}\cdots b_{1}x\|}=\lambda_{1}(\mu)$.
By the law of large numbers, this implies that for every non-zero vector $x$
of $V$, almost surely $\lim_{n\to\infty}{\frac{1}{n}\log\|b_{\tau(n)}\cdots
b_{1}x}\|=\mathbb{E}(\tau)\lambda_{1}(\mu)$, where $\tau(n)$ is defined
inductively by $\tau(n):=\tau(n-1)+\tau\circ\theta^{\tau(n-1)}$ and
$\tau(0)=0$. Again, by [12, 15], this implies the desired property for
$\mu^{\tau}$. ∎

##  3\. Non-degenerate stationary probability measures on the projective space

The goal of this section is to deduce some consequences of the existence of a
non-degenerate $\mu$-stationary probability measure on the $\mu$-random walk
and on the structure of the semigroup $\Gamma_{\mu}$ generated by the support
of $\mu$. These are expressed in the following result.

######  Theorem 3.1.

Let $\mu$ be a probability measure on $\operatorname{GL}(V)$ with finite first
moment and $\nu$ a non-degenerate $\mu$-stationary probability measure on
$\operatorname{P}(V)$. Suppose that $H_{\mu}$ is Zariski-connected. Then every
irreducible $H_{\mu}$-subspace $W$ of $V$ such that
$\lambda_{1}(W)=\lambda_{1}(V)$ admits a $H_{\mu}$-invariant complement in
$V$.

Before proceeding with the proof, note that when $\nu$ is a non-degenerate
probability measure on $\operatorname{P}(V)$ and $\pi$ is a non-zero
endomorphism of $V$, since $\nu(\operatorname{P}(\ker\pi))=0$, the pushforward
$\pi\nu$ is a well-defined probability measure on $\operatorname{P}(V)$.
Moreover, by non-degeneracy, it is easy to see that

$\langle\operatorname{supp}(\pi\nu)\rangle=\operatorname{P}(\operatorname{Im}(\pi)),$
(3.1)

where we denoted by $\langle\operatorname{supp}(\pi\nu)\rangle$ the projective
subspace generated by the support of $\pi\nu$.

In the sequel, we will repeatedly use the following lemma. It follows at once
from dominated convergence theorem.

######  Lemma 3.2.

Let $V$ be a vector space, $\nu$ a non-degenerate probability measure on
$\operatorname{P}(V)$, and $h_{n}$ a sequence in $\operatorname{End}(V)$ that
converges to some non-zero $h\in\operatorname{End}(V)$. Then
$h_{n}\nu\underset{n\to+\infty}{\longrightarrow}h\nu$ weakly. ∎

###  3.1. Consequences on the random walk

The main outputs of this subsection are Lemma 3.3 and Lemma 3.5.

We start by recalling a fundamental result of Furstenberg [10] and
Guivarc’h–Raugi [13] (see also [8, II, Lemma 2.1]): for
$\beta\otimes(\sum_{i=0}^{\infty}{2^{-i-1}\mu^{i}})$-almost every $(b,g)\in
B\times\Gamma_{\mu}$,

$b_{1}\cdots b_{n}g\nu\overset{\textrm{weakly}}{\to}\nu_{b}$ (3.2)

where $\nu_{b}$ is a probability measure on $\operatorname{P}(V)$ such that
$\int{\nu_{b}d\beta(b)}=\nu$.

An immediate consequence is the following.

######  Lemma 3.3.

Let $\mu$ be a probability measure on $\operatorname{GL}(V)$, $W$ a
$\Gamma_{\mu}$-invariant subspace of $V$, and $\nu$ a non-degenerate
$\mu$-stationary probability measure on $\operatorname{P}(V)$. Then for
$\beta$-almost every $b\in B$, every limit point $\pi_{b}$ of
$\frac{b_{1}\cdots b_{n}}{\|b_{1}\cdots b_{n}\|}$ satisfies
$\operatorname{Im}(\pi_{b})\not\subset W$. Moreover, if $\mu$ has finite first
moment, then $\lambda_{1}(V/W)\geqslant\lambda_{1}(W)$ (or equivalently
$\lambda_{1}(V/W)=\lambda_{1}(V)$).

###### Proof.

Since $\nu=\int{\nu_{b}d\beta(b)}$ and $\nu([W])=0$ (as $\nu$ is non-
degenerate in $\operatorname{P}(V)$), there exists $B^{\prime}\subset B$ such
that $\beta(B^{\prime})=1$ and $\nu_{b}([W])=0$ for every $b\in B^{\prime}$.
By (3.2), restricting if necessary to a subset of $B^{\prime}$ of
$\beta$-measure $1$, we can assume $b_{1}\cdots b_{n}\nu\to\nu_{b}$ weakly for
every $b\in B^{\prime}$. Let $b\in B^{\prime}$ and $\pi_{b}$ a limit point of
$\frac{b_{1}\cdots b_{n}}{\|b_{1}\cdots b_{n}\|}$. Since $\nu$ is non-
degenerate in $\operatorname{P}(V)$ (and $\pi_{b}\neq 0$), it follows from
Lemma 3.2 that $b_{1}\cdots b_{n}\nu\to\pi_{b}\nu$. By (3.2), we deduce that
$\pi_{b}\nu=\nu_{b}$. Hence the support of $\nu_{b}$ is included in (is
actually equal to) $\operatorname{Im}(\pi_{b})$. But $\nu_{b}([W])=0$. Hence
$\operatorname{Im}(\pi_{b})\not\subset W$. This proves the first claim. The
claim about Lyapunov exponents follows from the fact that if
$\lambda_{1}(V/W)<\lambda_{1}(W)$ then all limits points of $\frac{b_{1}\cdots
b_{n}}{\|b_{1}\cdots b_{n}\|}$ have their image included in $W$ (a convenient
way to see this is by writing a matrix representation of elements of
$\Gamma_{\mu}$ in upper triangular form with the top-left block representing
the action on $W$). ∎

To state and prove the next result (Lemma 3.5), we introduce some further
definitions.

######  Definition 3.4.

Let $S\subset\operatorname{End}(V)$. We denote by $C(S)$ the closed subset
$\overline{\mathbb{R}_{>0}S}$ of $\operatorname{End}(V)$.

In other words, $C(S)$ is the set of all possible limits in
$\operatorname{End}(V)$ of $\epsilon_{n}g_{n}$ with $\epsilon_{n}>0$ and
$g_{n}\in S$. Observe also that if $\Gamma$ is a semigroup, $C(\Gamma)$ is a
(closed) semigroup.

######  Lemma 3.5.

Let $\mu$ be a probability measure on $\operatorname{GL}(V)$ and $\nu$ a non-
degenerate $\mu$-stationary probability on $\operatorname{P}(V)$. Then for
$\beta$-almost every $b=(b_{i})_{i\in\mathbb{N}}\in B$, denoting
$\Pi_{b}:=\\{\pi\in\operatorname{End}(V)\setminus\\{0\\}:\pi\nu=\nu_{b}\\}$,
the following hold:

  1. (1)

$\Pi_{b}=C(\Pi_{b})\setminus\\{0\\}$ and all the endomorphisms in $\Pi_{b}$
have the same image.

  2. (2)

For every $\gamma\in C(\Gamma)$ and every limit point $\pi_{b}$ of
$\frac{b_{1}\cdots b_{n}}{\|b_{1}\cdots b_{n}\|}$ such that $\pi_{b}\gamma\neq
0$, we have $\pi_{b}\gamma\in\Pi_{b}$.

  3. (3)

For every $\pi\in\Pi_{b}$ and $\gamma\in C(\Gamma)$ such that
$(\pi\gamma)^{2}\neq 0$, $\pi\gamma$ is diagonalizable over $\mathbb{C}$ with
all of its non-zero eigenvalues having the same modulus.

###### Proof of Lemma 3.5.

  1. (1)

The first claim follows from Lemma 3.2. The second one follows from (3.1): the
image of any element in $\Pi_{b}$ is equal to the subspace generated by the
support of $\nu_{b}$.

  2. (2)

First we show that for $\beta$-almost every $b\in B$, the following holds for
every $g\in\Gamma_{\mu}$,

$\pi_{b}g\nu=\nu_{b}.$ (3.3)

Indeed, for $\beta$-almost every $b\in B$ and for all $g\in\Gamma_{\mu}$,
since $\nu$ is non-degenerate and $\pi_{b}g\neq 0$, it follows from Lemma 3.2
that

$b_{1}\cdots b_{n}g\nu\to\pi_{b}g\nu.$ (3.4)

Combining (3.2) and (3.4) we get that for
$\beta\otimes(\sum_{i=1}^{\infty}{2^{-i}\mu^{i}})$-almost every $(b,g)\in
B\times\Gamma_{\mu}$,

$\pi_{b}g\nu=\nu_{b}.$

By Fubini, there exists a $\beta$-full measure subset $B^{\prime}\subseteq B$
such that for every $b\in B^{\prime}$ the previous identity holds for
$(\sum_{i=1}^{\infty}{2^{-i}\mu^{i}})$-almost every $g\in\Gamma_{\mu}$. Since,
for each $b\in B^{\prime}$, the set of $g$ in $\Gamma_{\mu}$ such that the
previous identity holds is closed (see Lemma 3.2) and the support of
$(\sum_{i=1}^{\infty}{2^{-i}\mu^{i}})$ is $\Gamma_{\mu}$, the equality (3.3)
follows. Using this, we deduce from (1) that $\pi_{b}\gamma\in\Pi_{b}$
whenever $\gamma\in C(\Gamma)$ and $\pi_{b}\gamma\neq 0$.

  3. (3)

We restrict to a subset of $B$ of $\beta$-full measure where (2) holds. Let
now $\gamma\in C(\Gamma)$ such that $(\pi_{b}\gamma)^{2}\neq 0$. Since
$\gamma,\pi_{b}\in C(\Gamma)$ and $C(\Gamma)$ is a semigroup,
$\gamma\pi_{b}\gamma$ also belongs to $C(\Gamma)$ so that, by (2),
$(\pi_{b}\gamma)^{2}\in\Pi_{b}$. In particular by (1),

$\operatorname{Im}(\pi_{b}\gamma)=\operatorname{Im}((\pi_{b}\gamma)^{2}).$

This implies that $\pi_{b}\gamma$ is not nilpotent. Therefore, for every
$k\in\mathbb{N}$, $(\pi_{b}\gamma)^{k}\neq 0$ and since
$\gamma(\pi_{b}\gamma)^{k-1}\in C(\Gamma)$, we deduce from (2) that
$(\pi_{b}\gamma)^{k}\in\Pi_{b}$ for every $k\in\mathbb{N}$. Let
$(\pi_{b}\gamma)^{\infty}$ be any limit point of
$\frac{(\pi_{b}\gamma)^{k}}{\|(\pi_{b}\gamma)^{k}\|}$. We have
$(\pi_{b}\gamma)^{\infty}\in C(\Pi_{b})\setminus\\{0\\}$ so that by (1),
$(\pi_{b}\gamma)^{\infty}\in\Pi_{b}$, and

$\operatorname{Im}((\pi_{b}\gamma)^{\infty})=\operatorname{Im}(\pi_{b}\gamma).$
(3.5)

Writing $\pi_{b}\gamma$ in its Jordan canonical form, we deduce from (3.5)
that all the non-zero eigenvalues of $\pi_{b}\gamma$ have the same modulus and
that their respective geometric multiplicity coincide with their algebraic
one, ending the proof.

∎

###  3.2. Recurrence properties of random walks

In this section we record a recurrence property of random matrix products
which will be of crucial use for the construction of a projection map in the
next section.

######  Lemma 3.6.

(Recurrence of random walks on spaces with the same Lyapunov) Let $\mu$ be a
probability measure on a topological group $G$ and
$\rho:G\to\operatorname{GL}(V)$ and
$\rho^{\prime}:G\to\operatorname{GL}(V^{\prime})$ two strongly irreducible
representations of $G$ such that $\rho_{\ast}\mu$ and
$\rho^{\prime}_{\ast}\mu$ have finite first moment and same top Lyapunov
exponents. Then for $\beta$-almost every $b\in B$, there exists a subsequence
$(n_{k})_{k\in\mathbb{N}}$ such that

$\inf_{k\geqslant 0}\frac{\|\rho(b_{1}\cdots
b_{n_{k}})\|}{\|\rho^{\prime}(b_{1}\cdots b_{n_{k}})\|}>0.$ (3.6)

###### Proof.

For every $g\in G$ we denote by $\rho^{t}(g)\in\operatorname{End}(V^{*})$ the
transpose linear map on the dual space $V^{*}$ i.e.
$\rho^{t}(g)f(v)=f(\rho(g)v)$. Similarly one defines $\rho^{\prime
t}(g)\in\operatorname{End}(V^{\prime*})$. We equip $V^{*}$ and $V^{\prime*}$
with the dual norms so that $\|\rho(g)\|=\|\rho^{t}(g)\|$ and
$\|\rho^{\prime}(g)\|=\|\rho^{\prime t}(g)\|$ for every $g\in G$. Let
$H_{\mu}$ be the subgroup of
$\operatorname{GL}(V^{*})\times\operatorname{GL}(V^{\prime*})$ image of
$\zeta:g\mapsto(\rho^{t}(g),\rho^{\prime t}(g))$. The group $H_{\mu}$ acts on
$X:=\operatorname{P}(V^{*})\times\operatorname{P}(V^{\prime*})$. By
compactness of $X$, we can find a $\zeta_{\ast}\mu$-stationary ergodic
probability measure $\nu$ on $X$. Consider now the dynamical system
$(Y,\hat{T},\eta)$ where $Y:=B\times X$,
$\hat{T}(b,x):=(Tb,\zeta(b_{1})x),\eta=\beta\otimes\nu$ and $T:B\to B$ is the
shift map. The measure $\eta$ is $\hat{T}$-ergodic [6, Proposition 2.14]. For
every $g\in G$ and $x=([v],[v^{\prime}])\in X$ we let
$\sigma(g,([v],[v^{\prime}])):=\log\frac{\|\rho^{t}(g)v\|\|v^{\prime}\|}{\|v\|\|\rho^{\prime
t}(g)v^{\prime}\|}$ and for every $b=(b_{i})_{i\in\mathbb{N}}\in B$,
$f(b,x):=\sigma(b_{1},x)$. Denoting $q$ and $q^{\prime}$ the projections from
$X$ to $\operatorname{P}(V^{*})$ and $\operatorname{P}(V^{\prime*})$
respectively, we have

$\int_{X}{fd\eta}=\int_{G\times\operatorname{P}(V^{*})}{\log\frac{\|\rho^{t}(g)v\|}{\|v\|}d\mu(g)d(q_{\ast}\nu)([v])-\int_{G\times\operatorname{P}(V^{\prime*})}{\log\frac{\|\rho^{t}(g)v\|}{\|v\|}d\mu(g)d(q^{\prime}_{\ast}\nu)([v])}}.$

The projection maps $q:X\to\operatorname{P}(V^{*})$ and
$q^{\prime}:X\to\operatorname{P}(V^{\prime*})$ being $H_{\mu}$-equivariant,
the measures $q_{\ast}\eta$ and $q^{\prime}_{\ast}\eta$ are respectively
$\rho^{t}_{\ast}\mu$ and $\rho^{\prime t}_{\ast}\mu$-stationary probability
measures on $\operatorname{P}(V)$ and $\operatorname{P}(V^{*})$. Since
$H_{\mu}$ acts irreducibly on $V^{*}$ and $V^{\prime*}$ (because $\rho$ and
$\rho^{\prime}$ are irreducible representations), it follows from [12] that
all stationary probability measures in $\operatorname{P}(V^{*})$ (resp.
$\operatorname{P}(V^{\prime*})$) have the same cocycle average. Moreover, for
iid random matrix products in $\operatorname{GL}_{d}(\mathbb{C})$ the Lyapunov
exponents of a probability measure $\mu$ are the same as the Lyapunov
exponents of its pushforward by the transpose map (see [18] for a more general
statement). Therefore, since by assumption the Lyapunov exponents of $V$ and
$V^{\prime}$ are the same, it follows that
$\lambda_{1}(\rho^{t}_{\ast}\mu)=\lambda_{1}(\rho^{\prime t}_{\ast}\mu)$.
Hence

$\int{fd\eta}=0.$

By Atkinson’s result [2] we deduce that for $\eta$-almost every $(b,x)$, there
exists an increasing subsequence $(n_{k})_{k\in\mathbb{N}}$ such that for
every $k\in\mathbb{N}$

$\sum_{i=1}^{n_{k}}{f(\hat{T}^{i}(b,x))}\geqslant-1.$ (3.7)

By Fubini, we can then find $x=([v],[v^{\prime}])$ such that (3.7) holds for
$\beta$-almost every $b\in B$ (with $n_{k}$ depending on $b$). But $\sigma$ is
a cocycle for the right action (i.e.
$\sigma(gh,x)=\sigma(h,\zeta(g)x)+\sigma(g,x)$). Hence for every
$n\in\mathbb{N}$

$\sum_{i=1}^{n}{f(\hat{T}^{i}(b,x))}=\log\frac{\|\rho^{t}(b_{1}\cdots
b_{n})v\|\|v^{\prime}\|}{\|\rho^{\prime t}(b_{1}\cdots
b_{n})v^{\prime}\|\|v\|}.$

We deduce that there exists $B_{1}\subset B$ such that $\beta(B_{1})=1$ and
for every $b\in B_{1}$, there exists a subsequence $(n_{k})_{k\in\mathbb{N}}$
such that

$\inf_{k\geqslant 0}\frac{\|\rho^{t}(b_{1}\cdots b_{n_{k}})v\|}{\|\rho^{\prime
t}(b_{1}\cdots b_{n_{k}})v^{\prime}\|}>0.$ (3.8)

On the other hand, the representations $\rho$ and $\rho^{\prime}$ are strongly
irreducible, so are their transpose maps. Hence by [9, III, Proposition 3.2
(c)], there exists $B_{2}\subset B$ such that $\beta(B_{2})=1$ and for every
$b\in B_{2}$,

$\inf_{n\in\mathbb{N}}\frac{\|\rho^{t}(b_{1}\cdots
b_{n})v\|}{\|\rho^{t}(b_{1}\cdots
b_{n})\|}>0\qquad\text{and}\qquad\inf_{n\in\mathbb{N}}\frac{\|\rho^{\prime
t}(b_{1}\cdots b_{n})v\|}{\|\rho^{\prime t}(b_{1}\cdots b_{n})\|}>0.$ (3.9)

Combining (3.8) and (3.9), we deduce that for every $b\in B_{1}\cap B_{2}$,
there exists a subsequence $(n_{k})_{k\in\mathbb{N}}$ such that (3.6) holds
with $\rho$ and $\rho^{\prime}$ replaced by $\rho^{t}$ and $\rho^{\prime t}$
respectively. Hence (3.6) holds too. ∎

###  3.3. Consequence on the semigroup generated by the support of $\mu$

The goal of this part is to prove a key technical ingredient (Proposition 3.7)
for the proof of Theorem 3.1. We introduce the following notation. To any pair
$(S,\pi)$ where $S$ is a subset of $\operatorname{End}(V)$ and $\pi$ an
endomorphism of $V$, we associate the subset

$\pi S\pi=\\{\pi\gamma\pi|\gamma\in S\\}$

of $\operatorname{End}(V)$ whose restriction to $\operatorname{Im}(\pi)$
yields a subset of $\operatorname{End}(\operatorname{Im}(\pi))$. We denote by
$\langle\pi S\pi\rangle$ the semigroup of $\operatorname{GL}(V)$ generated by
$\pi S\pi$.

The key technical result for the proof of Theorem 3.1 is the following.

######  Proposition 3.7.

Let $\mu$ be a probability measure on $\operatorname{GL}(V)$ with finite first
moment and $\nu$ a non-degenerate $\mu$-stationary probability on
$\operatorname{P}(V)$. Let $W_{0}=\\{0\\}\subset W_{1}\subset\cdots\subset
W_{r-1}\subset W_{r}=V$ be a Jordan–Hölder decomposition of $V$ as
$\Gamma_{\mu}$-module. For every $i\in\\{1,\cdots,r\\}$ such that
$\lambda_{1}(W_{i}/W_{i-1})=\lambda_{1}(\mu)$, there exists $\pi\in
C(\Gamma_{\mu})$ such that $\pi^{2}=\pi$, $\pi(W_{i})\not\subset W_{i-1}$,
$\operatorname{Im}(\pi)\not\subset W_{r-1}$ and there exists a non-empty
Zariski-open subset $O$ of $H_{\mu}$ such that $\langle\pi(\Gamma_{\mu}\cap
O)\pi\rangle_{|_{\operatorname{Im}(\pi)}}$ is a subsemigroup of
$\operatorname{GL}(\operatorname{Im}(\pi))$ whose projection to
$\operatorname{PGL}(\operatorname{Im}(\pi))$ has compact closure.

The endomorphism $\pi$ will be constructed using random walks.

###### Proof.

The semigroup $C(\Gamma_{\mu})>\Gamma_{\mu}$ stabilizes the Jordan–Hölder
decomposition of $V$. Hence it acts naturally on each quotient $W_{k}/W_{k-1}$
inducing a homomorphism
$\rho_{k}:C(\Gamma_{\mu})\to\operatorname{End}(W_{k}/W_{k-1})$. Let
$i\in\\{1,\cdots,r\\}$ such that $\lambda_{1}(W_{i}/W_{i-1})=\lambda_{1}(V)$.
By Lemma 3.3, we have
$\lambda_{1}(V/W_{r-1})=\lambda_{1}(V)=\lambda_{1}(W_{i}/W_{i-1})$. We can
then find a $\beta$-generic $b\in B$ satisfying simultaneously all the
conclusions of Lemmas 3.3, 3.5 and 3.6 applied to the representations of
$H_{\mu}$ on $W_{i}/W_{i-1}$ and $V/W_{r}$ (these representations are strongly
irreducible because they are irreducible and $H_{\mu}$ is Zariski-connected).

  1. (i)

By Lemma 3.6 there exists a subsequence $(n_{k})_{k\in\mathbb{N}}$ such that
$\inf_{k\in\mathbb{N}}{\frac{\|\rho_{i}(b_{1}\cdots
b_{n_{k}})\|}{\|\rho_{r}(b_{1}\cdots b_{n_{k}})\|}}>0$. Passing to a further
subsequence we can assume without loss of generality that $\frac{b_{1}\cdots
b_{n_{k}}}{\|b_{1}\cdots b_{n_{k}}\|}$ converges to an endomorphism $\pi_{b}$
of $V$ as $k\to\infty$. Clearly, $\pi_{b}\in C(\Gamma_{\mu})$ and by Lemma
3.3, $\rho_{r}(\pi_{b})\neq 0$. Hence
$\inf_{k\in\mathbb{N}}{\frac{\|\rho_{r}(b_{1}\cdots
b_{n_{k}})\|}{\|b_{1}\cdots b_{n_{k}}\|}}>0$ and consequently,
$\inf_{k\in\mathbb{N}}{\frac{\|\rho_{i}(b_{1}\cdots
b_{n_{k}})\|}{\|b_{1}\cdots b_{n_{k}}\|}}>0$. This implies that
$\rho_{i}(\pi_{b})\neq 0$.

  2. (ii)

For $k\in\\{i,r\\}$, it follows from (i) and the irreducibility of the action
of $\Gamma_{\mu}$ on $W_{k}/W_{k-1}$ that the set $S_{k}:=\\{g\in
H_{\mu}:g\operatorname{Im}(\rho_{k}(\pi_{b})))\subset\ker(\rho_{k}(\pi_{b}))\\}$
is a proper Zariski-closed subvariety of $H_{\mu}$. By Zariski-connectedness
of the group $H_{\mu}$ (or equivalently its topological irreducibility),
$S_{i}\cup S_{r}$ is a proper closed subvariety of $H_{\mu}$. Since
$\Gamma_{\mu}$ is Zariski-dense in $H_{\mu}$, we deduce the existence of an
element $g\in\Gamma_{\mu}$ satisfying $\rho_{k}((\pi_{b}g)^{2})\neq 0$ for
$k\in\\{i,r\\}$. In particular, we have $(\pi_{b}g)^{2}\neq 0$.

  3. (iii)

In view of (ii), we are in a position to apply Lemma 3.5 to $\pi_{b}g$, and
(using its notation) deduce that $\pi_{b}g\in\Pi_{b}$ and that $\pi_{b}g$ is a
non-zero endomorphism of $V$ diagonalizable over $\mathbb{C}$ with all of its
non-zero eigenvalues having the same modulus, say $r>0$. Since
$C(\Gamma_{\mu})$ is a semigroup, $\pi_{b}g\in C(\Gamma_{\mu})$. Hence,
$\pi_{b}g$ preserves each $W_{k}$ and the endomorphism that it induces on each
quotient $W_{k}/W_{k-1}$ has either the same property (all non-zero
eigenvalues have modulus equal to $r$) or is equal to zero. In particular, for
every $n\geqslant 0$, $(\pi_{b}g)^{n}\neq 0$ and by Lemma 3.5 (2),
$(\pi_{b}g)^{n}\in\Pi_{b}$.

  4. (iv)

Let $re^{i\theta_{1}},\cdots re^{i\theta_{s}}$ be the non-zero eigenvalues of
$\pi_{b}g$. We can find a sequence $(m_{k})$ such that
$\theta_{j}^{m_{k}}\underset{k\to+\infty}{\longrightarrow}0\;(\text{mod}\;2\pi)$
for every $j\in\\{1,\cdots,s\\}$. Hence, we can find a limit point
$\pi_{b}^{\infty}$ of $r^{-n}(\pi_{b}g)^{n}$ which is a projection
endomorphism verifying for every $k\in\\{1,\cdots,r\\}$,
$\rho_{k}(\pi_{b}^{\infty})\neq 0$ if and only if $\rho_{k}(\pi_{b}g)\neq 0$.
Since, by (ii), the latter condition is verified for $k\in\\{i,r\\}$, this
gives $\pi_{b}^{\infty}(W_{k})\not\subset W_{k-1}$.

  5. (v)

From now on, we set $\pi:=\pi_{b}^{\infty}$. Observe that, since by Lemma 3.5
(1) $\Pi_{b}$ is closed in $\operatorname{End}(V)\setminus\\{0\\}$ and $\pi$
is a non-zero limit point of $r^{-n}(\pi_{b}g)^{n}$’s which are all in
$\Pi_{b}$ by (iii), it follows that $\pi\in\Pi_{b}\setminus\\{0\\}$. Moreover,
$\pi\in C(\Gamma_{\mu})$.

  6. (vi)

Let $O$ be the set of elements $h\in H_{\mu}$ such that $\pi h\pi\neq 0$. It
is a Zariski-open subset of $H_{\mu}$ and it is non-empty as it contains the
identity element (as $\pi$ is a non-zero projection by (iv)).

  7. (vii)

We now claim that $\pi$ plays the same role as $\pi_{b}$ in Lemma 3.5 (2), in
other words, for any $\gamma\in C(\Gamma_{\mu})$ such that $\pi\gamma\neq 0$,
we have $\pi\gamma\in\Pi_{b}$. Indeed, let $\gamma\in C(\Gamma_{\mu})$ be any
endomorphism such that $\pi\gamma\neq 0$. We have
$\pi=\lim_{k\to\infty}{r^{-m_{k}}(\pi_{b}g)^{m_{k}}}$ for some sequence
$m_{k}$ and hence for sufficiently large $k$, $(\pi_{b}g)^{m_{k}}\gamma\neq 0$
so that by Lemma 3.5 (2)
$r^{-m_{k}}(\pi_{b}g)^{m_{k}}\gamma=r^{-m_{k}}\pi_{b}(g\pi_{b})^{m_{k}-1}g\gamma\in\Pi_{b}$.
Hence $\pi\gamma\in C(\Pi_{b})\setminus\\{0\\}$. By (1) of the same lemma,
$\pi\gamma\in\Pi_{b}$ as claimed.

  8. (viii)

Let $\gamma\in O\cap\Gamma_{\mu}$. Since $C(\Gamma_{\mu})$ is a semigroup, and
by (v) $\pi\in C(\Gamma_{\mu})$, we have $\gamma\pi\in C(\Gamma_{\mu})$. It
follows from (vii) and the definition of $O$ that
$\pi\gamma\pi\in\Pi_{b}\setminus\\{0\\}$. By Lemma 3.5 (1)
$\operatorname{Im}(\pi\gamma\pi)=\operatorname{Im}(\pi)$. But since $\pi$ is a
projection,
$\operatorname{Im}((\pi\gamma\pi)_{|{\operatorname{Im}(\pi)}})=\operatorname{Im}(\pi\gamma\pi)$.
Hence the restriction of $\pi\gamma\pi$ to $\operatorname{Im}(\pi)$ is an
invertible endomorphism of $\operatorname{Im}(\pi)$. In other words
$\langle\pi(\Gamma_{\mu}\cap
O)\pi\rangle_{|_{\operatorname{Im}(\pi)}}\subset\operatorname{GL}(\operatorname{Im}(\pi))$.

  9. (ix)

We claim that $\nu_{b}$ is invariant under the action of
$\langle\pi(\Gamma_{\mu}\cap O)\pi\rangle$. Indeed, it follows from (viii)
that $0\not\in\langle\pi(\Gamma_{\mu}\cap O)\pi\rangle$ and that
$\langle\pi(\Gamma_{\mu}\cap O)\pi\rangle\subset C(\Gamma_{\mu})$. Since each
element of $\langle\pi(\Gamma_{\mu}\cap O)\pi\rangle$ is invariant by left
multiplication by $\pi$, (vii) yields that

$\langle\pi(\Gamma_{\mu}\cap O)\pi\rangle\subset\Pi_{b}\setminus\\{0\\}.$

Let $\gamma\in\langle\pi(\Gamma_{\mu}\cap O)\pi\rangle$. We have
$\gamma\nu=\nu_{b}$. But by (v) $\pi\nu=\nu_{b}$. Since $\pi$ is a projection
and $\gamma$ is also invariant by right multiplication by $\pi$, we deduce
that $\gamma\nu_{b}=\nu_{b}$.

  10. (x)

Now we conclude the proof. It only remains to show that the closure of
$\langle\pi\Gamma_{\mu}\pi\rangle_{|_{\operatorname{Im}(\pi)}}$ in
$\textrm{PGL}(\operatorname{Im}(\pi))$ is a compact group (indeed the
projection map $\pi$ was constructed in (iv) and it satisfies the claims made
about its image in the statement, and the Zariski open subset $O$ of $H_{\mu}$
was constructed in (vi)). To see the compactness, first observe that $\nu_{b}$
is a non-degenerate probability measure on
$\operatorname{P}(\operatorname{Im}(\pi))$, as it follows from the fact that
$\pi\in\Pi_{b}\setminus\\{0\\}$ and the non-degeneracy of $\nu$ (see (3.1)).
By a classical result of Furstenberg [10] (see also [21, Corollary 3.2.2]),
the stabilizer in $\operatorname{PGL}(E)$ (for any vector space $E$) of any
non-degenerate probability measure on $\operatorname{P}(E)$ is a compact
group. Hence $\textrm{Stab}(\nu_{b})$ is a compact group. It follows from
(viii) and (ix), that $\langle\pi(\Gamma_{\mu}\cap
O)\pi\rangle_{|_{\operatorname{Im}(\pi)}}$ is a semigroup included in a
compact group. Its closure is then a compact group and the proof is complete.

∎

###  3.4. Proof of Theorem 3.1

The proof is in two steps.

(i) We first show the following particular case: every proper irreducible
subspace admits a non-zero invariant subspace that is in direct sum. To prove
this, let $\nu$ be a non-degenerate $\mu$-stationary probability measure on
$\operatorname{P}(V)$ and $W$ a proper $\Gamma_{\mu}$-irreducible invariant
subspace of $V$ with top Lyapunov exponent, as in the statement. We can find a
Jordan–Hölder decomposition $\\{0\\}=W_{0}\subset W_{1}\subset\cdots\subset
W_{r}=V$ of $V$ such that $W_{1}=W$ (necessarily $r\neq 1$ as $W$ is proper).
Proposition 3.7 applies with $i=1$, and it yields a projection $\pi\in
C(\Gamma_{\mu})$ such that $\pi(W)\neq\\{0\\}$ and
$\operatorname{Im}(\pi)\not\subset W$, and a Zariski-open subset $O$ of
$H_{\mu}$ such that $\overline{\pi(O\cap\Gamma_{\mu})\pi}$ is a compact
subgroup $H$ of $\operatorname{GL}(\operatorname{Im}(\pi))$. In particular,
$\operatorname{Im}(\pi)$ is $H$-completely reducible. But
$\pi(W)=W\cap\operatorname{Im}(\pi)$ is $H$-invariant (because, being in
$C(\Gamma_{\mu})$, $\pi$ stabilises $W$). Hence there exists a direct
$H$-invariant complement $W^{\prime}$ of $\pi(W)$ in $\operatorname{Im}(\pi)$.
The $H$-invariance of $W^{\prime}$ is equivalent to saying that

$\forall\gamma\in O\cap\Gamma_{\mu},\gamma
W^{\prime}\subset\pi^{-1}(W^{\prime}).$ (3.10)

Since $H_{\mu}$ is Zariski-connected, $\Gamma_{\mu}\cap O$ is Zariski-dense in
$H_{\mu}$ and since the set of elements $\gamma\in H_{\mu}$ verifying $\gamma
W^{\prime}\subset\pi^{-1}(W^{\prime})$ is Zariski-closed, it follows from
(3.10) that

$\forall g\in H_{\mu},\quad gW^{\prime}\subset\pi^{-1}(W^{\prime}).$ (3.11)

Consider now the $H_{\mu}$-invariant subspace $E:=\textrm{Span}(\\{gv:g\in
H_{\mu},v\in W^{\prime}\\})$. By (3.11), we have $W\not\subset E$. Since $W$
is irreducible, we deduce that $W\cap E=\\{0\\}$. Finally $E\neq 0$ as
otherwise $W^{\prime}=\\{0\\}$, contradicting $\operatorname{Im}(\pi)\neq W$.
This finishes the claim at the beginning of the proof.

(ii) Now we prove the theorem in full generality. We proceed by induction on
$\textrm{dim}(V)$. The result trivially holds when $\textrm{dim}(V)=1$. Let
$V,\mu,\nu$, and $W$ be as in Theorem 3.1. By step (i), there exists a non-
zero $H_{\mu}$-invariant subspace $E$ of $V$ which is in direct sum with $W$.
Let $V^{\prime}:=V/E$. Since $\nu([E])=0$ (as $\nu$ is non-degenerate and $E$
is proper), $\nu$ descends to a $\mu$-stationary probability measure
$\overline{\nu}$ on $\operatorname{P}(V/E)$ which is non-degenerate. Let
$q:V\to V/E$ denote the projection map. The subspace $q(W)$ of $V/E$ is a
$H_{\mu}$-invariant subspace of $V/E$ which is $H_{\mu}$-equivariantly
isomorphic to $W$ (as $W\cap E=\\{0\\}$). Thus $q(W)$ is irreducible and
$\lambda_{1}(q(W))=\lambda_{1}(\mu)=\lambda_{1}(V/E)$. The induction
hypothesis applied with the ambient vector space $V/E$ (satisfying
$\dim(V/E)<\dim(V)$), invariant subspace $q(W)$, non-degenerate stationary
measure $\overline{\nu}$ and random walk measure $\mu$ yields a
$G_{\mu}$-invariant complement $\tilde{F}$ of $q(W)$ in $V/E$. Let
$F:=q^{-1}(\tilde{F})$. This is a $G_{\mu}$-invariant subspace of $V$. Clearly
$V=W+F$ and, since $W\cap E=\\{0\\}$, we have $W\cap F=\\{0\\}$, concluding
the proof. ∎

##  4\. Proof of the main statements

###  4.1. Proof of Theorem 1.1

By §2 it is enough to prove Theorem 2.1. Let $\mu$ be a probability measure on
$\operatorname{GL}(V)$ such that $F_{2}(V)=\\{0\\}$ and assume that
$\Gamma_{\mu}$ is Zariski-connected. Let $\nu$ be a non-degenerate
$\mu$-stationary ergodic probability measure on $\operatorname{P}(V)$.
Consider an irreducible $\Gamma_{\mu}$-invariant subspace $W$ of $V$. By the
assumption $F_{2}(V)=\\{0\\}$, we have $\lambda_{1}(W)=\lambda_{1}(V)$ and
hence Theorem 3.1 provides an invariant complement $W^{\prime}$ of $W$ in $V$.
We have therefore shown that any irreducible subspace has an invariant
complement and this implies complete reducibility. ∎

######  Remark 4.1 ($V_{\nu}$ is a sum of same highest weight
representations).

Since by Corollary 1.2 the support of $\nu$ lives in a compact
$H_{\mu}$-orbit, every irreducible $H_{\mu}$-subrepresentation of $V_{\nu}$
has the same highest weight.

###  4.2. Proof of Theorem 1.5

As mentioned in the introduction, Theorem 1.5 is a slightly stronger version
of Theorem 1.1 (which additionally combines results from our previous work
[1]).

To prove it, we will begin by treating the particular case that does not
involve noise from several Furstenberg–Kifer–Hennion exponents (FKH exponents
for short) of $W$. Let us make these terms precise. For a probability measure
$\mu$ on $\operatorname{GL}(V)$ with finite first moment and a probability
measure $\nu$ on a $\operatorname{P}(V)$ the cocycle average of $\nu$ is the
scalar $\alpha(\nu):=\iint{\log\frac{\|gv\|}{\|v\|}d\mu(g)d\nu(\mathbb{R}v)}$.
A result of Furstenberg–Kifer [12] says that, when $\nu$ runs over all
$\mu$-stationary ergodic probability measures on $\operatorname{P}(V)$,
$\alpha(\nu)$ can take finitely many values
$\lambda_{1}(V)=\beta_{1}>\beta_{2}>\cdots>\beta_{k}=\beta_{\min}(V)$ and
these all are top Lyapunov exponents of the FKH spaces $F_{i}<V$ appearing in
Theorem 1.3 (and top Lyapunov exponents of the space $V_{\nu}$ generated by
the support of $\nu$ which is then included in the corresponding $F_{i}$).
These exponents $\beta_{i}$’s are called FKH exponents222Sometimes also called
deterministic exponents.. Saying that $F_{2}(V)=\\{0\\}$ (i.e. $V$ is critical
or equivalently $\beta_{\min}(V)=\lambda_{1}(V)$) is equivalent to saying that
there is only one cocycle average.

The particular case of Theorem 1.5 that we will start with (Theorem 4.2
below), corresponds to $\lambda_{1}(V_{\overline{\nu}})=\beta_{\min}(W)$. This
is the case for instance when $\Gamma_{\mu}$ acts irreducibly on both of $W$
and $V/W$ in Theorem 1.5.

######  Theorem 4.2 (Purely critical case).

Let $\mu$ be a probability measure on $\operatorname{GL}(V)$ with finite first
moment and $W$ a $\Gamma_{\mu}$-invariant subspace. Let $\overline{\nu}$ be a
$\mu$-stationary and ergodic probability measure on $\operatorname{P}(V/W)$
such that

$\alpha(\overline{\nu})\leqslant\beta_{\min}(W).$ (4.1)

Then, the following are equivalent:

  * (i)

There exists a $\mu$-stationary lift $\nu$ of $\overline{\nu}$ on
$\operatorname{P}(V)\setminus\operatorname{P}(W)$.

  * (ii)

There exists a $\Gamma_{\mu}$-invariant subspace $W^{\prime}$ of $V$ in direct
sum with $W$ such that $\operatorname{P}(W^{\prime}/W)$ is the projective
subspace generated by $\overline{\nu}$.

First, we begin by recording the following consequence of [1, Theorem 1.5].

######  Lemma 4.3 (Lift invariance of cocycle-average).

Let $\mu$ be a probability measure with finite first moment on
$\operatorname{GL}(V)$, $W$ a $\Gamma_{\mu}$-invariant subspace of $V$ and
$\overline{\nu}$ a $\mu$-stationary ergodic probability measure on
$\operatorname{P}(V/W)$. If $\nu$ is a $\mu$-stationary lift of
$\overline{\nu}$ then $\alpha(\overline{\nu})=\alpha(\nu)$.

###### Proof.

It is clear that $\alpha(\nu)\geqslant\alpha(\overline{\nu})$. Without loss of
generality, we can suppose $\nu$ to be ergodic. The probability measure $\nu$
lives in the projective space of $\pi^{-1}(F_{\overline{\nu}})\supset W$ where
$\pi:V\to V/W$ is the canonical projection. Hence without loss of generality
we can assume that $V=\pi^{-1}(F_{\overline{\nu}})$ so that
$\lambda_{1}(V/W)=\alpha(\overline{\nu})$. Arguing by contradiction, assume
that $\alpha(\nu)>\lambda_{1}(V/W)$. By ergodicity of $\nu$, it follows from
[12, Theorem 3.9] that $\alpha(\nu)$ is some Lyapunov exponent of $V$. But by
[12, Lemma 3.6] $\lambda_{1}(V)=\max\\{\lambda_{1}(V/W),\lambda_{1}(W)\\}$,
necessarily $\alpha(\nu)$ is some top Lyapunov exponent of $W$ so that
$\lambda_{1}(W)>\alpha(\overline{\nu})$. It is enough to apply now Theorem [1,
Theorem 1.5]. ∎

###### Proof of Theorem 4.2.

By [1, Theorem 1.5] it suffices to treat the case
$\alpha(\overline{\nu})=\beta_{\min}(W)$. The implication
$(i)\Longrightarrow(ii)$ is trivial. To prove the other implication, suppose
that $\nu$ is some $\mu$-stationary lift of $\overline{\nu}$ on
$\operatorname{P}(V)\setminus\operatorname{P}(W)$. Denote for simplicity
$V^{\prime}:=V_{\nu}$ and $W^{\prime}:=V_{\nu}\cap W$. By Lemma 4.3,
$\alpha(\nu)=\beta_{\min}(W)$ so that $W^{\prime}\subset F_{k}(W)$ where
$F_{k}(W)$ is the smallest FKH space of $W$. By definition of the smallest FKH
exponent $\beta_{\min}(W)$, it follows that $F_{2}(W^{\prime})=\\{0\\}$. Since
$\beta_{\min}(W)=\lambda_{1}(V^{\prime})$, we deduce that
$F_{2}(V^{\prime})\cap W^{\prime}=\\{0\\}$. Since $\nu(F_{2}(V^{\prime}))=0$,
$V^{\prime}/F_{2}(V^{\prime})$ is the projective space generated by
$\tilde{\nu}$, the projection of $\nu$ on $V^{\prime}/F_{2}(V^{\prime})$.
Applying now Theorem 1.1 to $V^{\prime}/F_{2}(V^{\prime})$, we deduce that the
subspace $(W^{\prime}+F_{2}(V^{\prime}))/F_{2}(V^{\prime})$ admits a
$\Gamma_{\mu}$-invariant complement in $V^{\prime}/F_{2}(V^{\prime})$. Since
$F_{2}(V^{\prime})\cap W^{\prime}=\\{0\\}$, lifting this invariant complement
to $V^{\prime}$ gives rise to an invariant complement $U$ of $W^{\prime}$
inside $V^{\prime}$. Clearly $U\cap W=\\{0\\}$. Moreover $U\simeq
V^{\prime}/W^{\prime}$ is isomorphic as $\Gamma_{\mu}$-space to
$V_{\overline{\nu}}$. ∎

###### Proof of Theorem 1.5.

The deduction of the general case (Theorem 1.5) from the purely critical case
(Theorem 4.2) is done in the same way as in the proof [1, Theorem 1.5] using
[1, Theorem 5.1]. We indicate the arguments. The implication $(ii)\implies(i)$
follows immediately from [1, Theorem 1.1] (contracting case) applied to
$W^{\prime}$ as ambient space and $W^{\prime}\cap W$ as invariant subspace.
Now we prove $(i)\implies(ii)$. Let $\overline{\nu}$ be a $\mu$-stationary
ergodic probability measure on $\operatorname{P}(V/W)$ such that
$\alpha(\overline{\nu})\leqslant\lambda_{1}(W)$. Denote by $j$ the smallest
index such that $\beta_{j+1}(W)<\alpha(\overline{\nu})\leqslant\beta_{j}(W)$.
Without loss of generality we can assume $V/W=V_{\overline{\nu}}$. Projecting
$\nu$ to $\operatorname{P}(V/F_{j+1}(W))$, we get a $\mu$-stationary measure
$\tilde{\nu}$ on
$\operatorname{P}(V/F_{j+1}(W))\setminus\operatorname{P}(W/F_{j+1}(W))$ that
lifts $\overline{\nu}$. Since $\beta_{\min}(W/F_{j+1}(W))=\beta_{j}(W)$ (see
for instance [1, Corollary 3.4]), our assumption on $\overline{\nu}$ yields
$\beta_{\min}(W/F_{j+1}(W))\geqslant\alpha(\overline{\nu})$. Theorem 4.2
applied to $V/F_{j+1}(W)$ as ambient space and $W/F_{j+1}(W)$ as subspace
yields a $\Gamma_{\mu}$-invariant complement $\tilde{W}$ of $W/F_{j+1}(W)$ in
$V/F_{j+1}(W)$. Its preimage $W^{\prime}<V$ by the projection map $V\to
V/F_{j+1}(W)$ is a $\Gamma_{\mu}$-invariant subspace whose intersection with
$W$ is equal to $F_{j+1}(W)$ (hence of Lyapunov exponent
$\beta_{j}(W)<\alpha(\overline{\nu}))$ and such that
$V_{\overline{\nu}}=W^{\prime}/W^{\prime}\cap W$. ∎

######  Remark 4.4 (On uniqueness of lifts).

Keep the assumptions of Theorem 1.5. Suppose that there exists a
$H_{\mu}$-invariant subspace $W^{\prime}$ that intersects $W$ only in an
invariant subspace of slower expansion and such that $\overline{\nu}$ is
supported in $\operatorname{P}(W^{\prime}/W^{\prime}\cap W)$.

  1. (1)

Suppose $\lambda_{1}(W)>\alpha(\overline{\nu})$. Then $\overline{\nu}$ admits
a unique $\mu$-stationary lift $\nu$ on
$\operatorname{P}(V)\setminus\operatorname{P}(W)$ and the subspace generated
by $\nu$ is $\operatorname{P}(W^{\prime})$ [1, Theorem 1.5].

  2. (2)

Suppose that $\lambda_{1}(W)=\lambda(V/W)$. Although
$\operatorname{P}(W^{\prime})$ admits a unique $\mu$-stationary lift of
$\overline{\nu}$, $\overline{\nu}$ may admit many lifts on
$\operatorname{P}(V)\setminus\operatorname{P}(W)$ and these may even be non-
degenerate. An example is a Zariski-dense probability measure $\mu$ on
$H=\operatorname{SO}_{2}(\mathbb{R})\times\operatorname{SO}_{2}(\mathbb{R})$
acting on $V=\mathbb{R}^{2}\oplus\mathbb{R}^{2}$. For any
$x\in\operatorname{P}(V)$ that does not belong to one of the two invariant
$2$-planes, the $H$-orbit of $x$ is a proper algebraic variety in
$\operatorname{P}(V)$ which is not included in any proper projective subspace.
Being compact, each such orbit supports a (unique) stationary probability
measure.

######  Remark 4.5 (Affine case).

We illustrate Theorem 1.5 by showing how it recovers Bougerol–Picard’s result
[9] in the invertible case: if $\mu$ is a probability measure on the affine
group $\textrm{Aff}(\mathbb{R}^{d})$ such that $\Gamma_{\mu}$ does not
preserve an affine subspace of $\mathbb{R}^{d}$, then there exists a
$\mu$-stationary probability measure on $\mathbb{R}^{d}$ if and only if the
top Lyapunov exponent of the linear part is negative. The backward implication
(contracting case) being standard, we comment on the forward implication. It
is enough to embed $\textrm{Aff}(\mathbb{R}^{d})$ in
$\operatorname{GL}_{d+1}(\mathbb{R})$ in the usual way and to take $W$ as the
hyperplane in $V:=\mathbb{R}^{d+1}$ spanned by the first $d$ vectors of the
canonical basis. In this case the action on the quotient $V/W$ is trivial so
$\alpha(\overline{\nu})\leqslant\lambda_{1}(W)\Longleftrightarrow
0\leqslant\lambda_{1}(W)$. Moreover stationary measures on $\mathbb{R}^{d}$
(for the affine action) correspond to stationary measures on
$\operatorname{P}(V)\setminus\operatorname{P}(W)$ (for the linear action).
Affine irreducibility is equivalent to saying that any
$\Gamma_{\mu}$-invariant subspace of $V$ must be included in $W$. In
particular, a subspace $W^{\prime}$ fulfilling condition (ii) of Theorem 1.5
cannot exist, showing indeed the absence of stationary probability measures on
$\mathbb{R}^{d}$ in the critical/expanding situation.

Finally, one also recovers [7, Theorem 5.1] in a similar way (realizing the
linear action as a projective action). The extra conclusion of compactness in
[7, Theorem 5.1] follows from Corollary 1.2 (or more simply transience of
random walks on non-compact semisimple Lie groups, due to Furstenberg [10]).
We omit the details.

###  4.3. Proof of Corollary 1.6

First we show $(2.1)\Longrightarrow(2.2)$ together with the uniqueness
statement (1). Denote $F_{\mu}:=F_{2}(V_{\mathcal{O}})$. Suppose that there
exists a $\mu$-stationary probability measure $\nu$ on $\mathcal{O}$. We can
suppose that $\nu$ is ergodic, and since
$\mathcal{O}\cap\operatorname{P}(F_{\mu})=\emptyset$,
$\nu(\operatorname{P}(F_{\mu}))=0$. Let $\psi$ be the natural
$H_{\mu}$-equivariant projection
$\operatorname{P}(V_{\mathcal{O}})\setminus\operatorname{P}(F_{\mu})\to\operatorname{P}(V_{\mathcal{O}}/F_{\mu})$
and denote $\overline{\nu}:=\psi_{\ast}\nu$. This is a $\mu$-stationary
probability measure on the $H_{\mu}$-orbit $\psi(\mathcal{O})$ in
$\operatorname{P}(V_{\mathcal{O}}/F_{\mu})$. Observe that by [12],
$F_{2}(V_{\mathcal{O}}/F_{\mu})=\\{0\\}$ (in other words
$V_{\mathcal{O}}/F_{\mu}$ is critical). Therefore Corollary 1.2 implies that
$\psi(O)$ is closed. Now we show that
$\overline{\mathcal{O}}\setminus\mathcal{O}\subset\operatorname{P}(F_{\mu})$.
We argue by contradiction. Suppose this is not the case. Since
$\psi(\mathcal{O})$ is closed, this implies that
$\psi(\mathcal{\overline{O}}\setminus(O\cup\operatorname{P}(F_{\mu}))$ is a
non-empty $H_{\mu}$-invariant subset of the orbit $\psi(O)$. Hence
$\psi(\mathcal{\overline{O}}\setminus(O\cup\operatorname{P}(F_{\mu}))=\psi(\mathcal{O})$.
Consider now a $\overline{\nu}$-generic point
$\overline{x}\in\psi(\mathcal{O})$, i.e.
$\frac{1}{n}\sum_{i=1}^{n}{\mu^{i}\ast\delta_{\overline{x}}}$ converges weakly
to $\overline{\nu}$. We can find
$x\in\overline{\mathcal{O}}\setminus(\mathcal{O}\cup\operatorname{P}(F_{\mu}))$
such that $\psi(x)=\overline{x}$. By [1, Proposition 1.2] it follows that
$\frac{1}{n}\sum_{i=1}^{n}{\mu^{i}\ast\delta_{x}}\to\nu$. Since the orbit
$\mathcal{O}$ is locally closed (see for instance [16, 8.3]) and
$x\in\overline{\mathcal{O}}\setminus\mathcal{O}$, it follows from the
equidistribution above that
$\nu(\overline{\mathcal{O}}\setminus\mathcal{O})=1$, which contradicts
$\nu(\mathcal{O})=1$. This shows that $(2.1)\Longrightarrow(2.2)$. Now we show
the uniqueness of $\nu$. Clearly
$V_{\psi(\mathcal{O})}=V_{\mathcal{O}}/F_{\mu}$. Hence by Theorem 1.1,
$V_{\mathcal{O}}/F_{\mu}$ is completely reducible. Hence the uniqueness of
$\nu$ follows directly from the uniqueness of a $\mu$-stationary probability
measure on $\psi(O)$, by [5, Theorems 1.5 & 1.7], and then by the uniqueness
of lift statement in [1, Theorem 1.1].

Conversely, assume that $\psi(\mathcal{O})$ is compact and that
$\overline{\mathcal{O}}\setminus\mathcal{O}\subset F_{\mu}$. By compactness of
$\psi(\mathcal{O})$ there exists a $\mu$-stationary probability measure
$\overline{\nu}$ on $\psi(\mathcal{O})$. Let $x\in\mathcal{O}$ such that
$\psi(x)$ is a $\overline{\nu}$-generic point (i.e.
$\frac{1}{n}\sum_{k=1}^{n}\mu^{k}\ast\delta_{\psi(x)}\to\overline{\nu}$ as
$n\to\infty$). By [1, Theorem 1.1] there exists a unique $\mu$-stationary
probability measure $\nu$ on
$\operatorname{P}(V_{\mathcal{O}})\setminus\operatorname{P}(F_{\mu})$ lifting
$\nu$ and, by [1, Proposition 1.2], $x$ is generic for $\nu$. It follows that
$\nu$ is supported on $\overline{\mathcal{O}}$. Since
$\nu(\operatorname{P}(F_{\mu}))=0$ and
$\overline{\mathcal{O}}\setminus\mathcal{O}\subset F_{\mu}$, we get that
$\nu(\mathcal{O})=1$. ∎

## References

  * [1] R. Aoun and C. Sert.  Stationary probability measures on projective spaces 1: block-lyapunov dominated systems.  To appear in Math. Ann., 2023.
  * [2] G. Atkinson.  Recurrence of co-cycles and random walks.  J. London Math. Soc. (2), 13(3):486–488, 1976.
  * [3] Y. Benoist and C. Bruère.  Recurrence on affine Grassmannians.  Ergodic Theory Dynam. Systems, 39(12):3207–3223, 2019.
  * [4] Y. Benoist and J.-F. Quint.  Stationary measures and invariant subsets of homogeneous spaces (III).  Ann. of Math. (2), 178(3):1017–1059, 2013.
  * [5] Y. Benoist and J.-F. Quint.  Random walks on projective spaces.  Compos. Math., 150(9):1579–1606, 2014.
  * [6] Y. Benoist and J.-F. Quint.  Random walks on reductive groups.  Springer, Cham, 2016.
  * [7] P. Bougerol.  Tightness of products of random matrices and stability of linear stochastic systems.  Ann. Probab., 15(1):40–74, 1987.
  * [8] P. Bougerol and J. Lacroix.  Products of random matrices with applications to Schrödinger operators, volume 8 of Progress in Probability and Statistics.  Birkhäuser Boston, Inc., Boston, MA, 1985.
  * [9] P. Bougerol and N. Picard.  Strict stationarity of generalized autoregressive processes.  Ann. Probab., 20(4):1714–1730, 1992.
  * [10] H. Furstenberg.  Noncommuting random products.  Trans. Amer. Math. Soc., 108:377–428, 1963.
  * [11] H. Furstenberg and H. Kesten.  Products of random matrices.  Ann. Math. Statist., 31:457–469, 1960.
  * [12] H. Furstenberg and Y. Kifer.  Random matrix products and measures on projective spaces.  Israel J. Math., 46(1-2):12–32, 1983.
  * [13] Y. Guivarc’h and A. Raugi.  Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence.  Z. Wahrsch. Verw. Gebiete, 69(2):187–242, 1985.
  * [14] Y. Guivarc’h and A. Raugi.  Actions of large semigroups and random walks on isometric extensions of boundaries.  Ann. Sci. École Norm. Sup. (4), 40(2):209–249, 2007.
  * [15] H. Hennion.  Loi des grands nombres et perturbations pour des produits réductibles de matrices aléatoires indépendantes.  Z. Wahrsch. Verw. Gebiete, 67(3):265–278, 1984.
  * [16] J. E. Humphreys.  Linear algebraic groups.  Graduate Texts in Mathematics, No. 21. Springer-Verlag, New York-Heidelberg, 1975.
  * [17] H. Kesten and F. Spitzer.  Convergence in distribution of products of random matrices.  Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 67:363–386, 1984.
  * [18] E. S. Key.  Lyapunov exponents for matrices with invariant subspaces.  The Annals of Probability, pages 1721–1728, 1988.
  * [19] G. D. Mostow.  Fully reducible subgroups of algebraic groups.  Amer. J. Math., 78:200–221, 1956.
  * [20] Y. Peres.  Analytic dependence of Lyapunov exponents on transition probabilities.  In Lyapunov exponents (Oberwolfach, 1990), volume 1486 of Lecture Notes in Math., pages 64–80. Springer, Berlin, 1991.
  * [21] R. J. Zimmer.  Ergodic theory and semisimple groups, volume 81 of Monographs in Mathematics.  Birkhäuser Verlag, Basel, 1984.