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THE SET OF FIBER-BUNCHED COCYCLES WITH

NONVANISHING LYAPUNOV EXPONENTS OVER A

PARTIALLY HYPERBOLIC MAP IS OPEN

LUCAS BACKES, MAURICIO POLETTI, AND ADRIANA SÁNCHEZ

Abstract. We prove that the set of fiber-bunched SL(2,R)-valued Hölder
cocycles with nonvanishing Lyapunov exponents over a volume preserving,
accessible and center-bunched partially hyperbolic diffeomorphism is open.
Moreover, we present an example showing that this is no longer true if we do
not assume accessibility in the base dynamics.

1. Introduction

Given an invertible measure preserving transformation f : (M,µ) → (M,µ) of a
standard probability space and a measurable function A : M → GL(d,R) we define
the linear cocycle over f by the dynamically defined products

An(x) =







A(fn−1(x)) . . . A(f(x))A(x) if n > 0
Id if n = 0
(A−n(fn(x)))−1 = A(fn(x))−1 . . . A(f−1(x))−1 if n < 0.

(1)

The simplest examples of linear cocycles are given by derivative transformations of
smooth dynamical systems: the cocycle generated by A(x) = Df(x) over f is called
the derivative cocycle. Taking as an example the hyperbolic theory of Dynamical
Systems where one can understand certain dynamical properties of f by studying
the action of Df on the tangent space, one can hope that by studying properties of
linear cocycles one can also deduce some properties of f . Nevertheless, the notion
of linear cocycle is much more general and flexible, and arises naturally in many
other situations as in the spectral theory of Schrödinger operators, for instance.

In this short note we are interested in the asymptotic behavior of An(x). More
precisely, we are interested in understanding certain regularity properties of Lya-
punov exponents. These objects measure the asymptotic rates of contractions and
expansions along different directions and are one of the most fundamental notions
in dynamical systems.

It is well known that, in general, Lyapunov exponents can be very sensitive as
functions of the cocycle. For instance, Bochi [5, 6] proved that in the space of
SL(2,R)-valued continuous cocycles over an aperiodic map, if a cocycle is not hy-
perbolic, then it can be approximated by cocycles with zero Lyapunov exponents.
In particular, there are cocycles with positive Lyapunov exponents that are accu-
mulated by cocycles with zero Lyapunov exponents. Moreover, Bocker and Viana

Date: November 15, 2017.
2010 Mathematics Subject Classification. Primary: 37H15, 37A20; Secondary: 37D25.
Key words and phrases. Lyapunov exponents, Partially hyperbolic systems, Linear cocyles.

1

http://arxiv.org/abs/1706.08505v2


2 LUCAS BACKES, MAURICIO POLETTI, AND ADRIANA SÁNCHEZ

[7] constructed an example over a hyperbolic map showing that the same phenom-
enon can happen in the Hölder realm. Furthermore, when the base dynamic is far
from being hyperbolic, for example, when f is a rotation on the circle, Wang and
You [14], showed that having non-zero Lyapunov exponents is not an open property
even in the C∞ topology.

In order to construct their example, Bocker and Viana exploited the fact that
the cocycle is not fiber-bunched. In fact, it was shown by Backes, Butler and
Brown [3] that in the fiber-bunched setting over a hyperbolic map the Lyapunov
exponents vary continuously with respect to the cocycle and, in particular, cocycles
with positive Lyapunov exponents can not be approximate by cocycles with zero
Lyapunov exponents.

In the present work we are interested in understanding the case when the cocycle
still have some regularity properties, namely, it is fiber-bunched but the base dy-
namics exhibit some mixed behaviour of hyperbolicity and non-hyperbolicity, that
is, the map f is partially hyperbolic. In fact, we show that if f is chaotic enough
and A is fiber-bunched then the Bochi phenomenon can not occur. More precisely,
(see Section 2 for detailed definitions),

Theorem 1.1. If (f, µ) is a volume preserving partially hyperbolic accessible and
center-bunched diffeomorphism and A : M → SL(2,R) is a Hölder continuous
fiber-bunched map with nonvanishing Lyapunov exponents, then A can not be accu-
mulated by cocycles with zero Lyapunov exponents.

Moreover, we show that the accessibilty assumption in the previous result is
necessary. More precisely,

Theorem 1.2. There exists a volume preserving partially hyperbolic and center-
bunched diffeomorphism f and a Hölder continuous fiber-bunched map A with non-
zero Lyapunov exponents which is approximated by cocycles with zero Lyapunov
exponents.

2. Statements

Let f : M → M be a Cr, r ≥ 2, diffeomorphism defined on a compact manifold
M , µ an ergodic f -invariant Borel probability measure and let A : M → SL(2,R)
be an α-Hölder continuous map. This means that there exists a constant C > 0
such that

‖A(x) −A(y)‖ ≤ Cd(x, y)α

for all x, y ∈ M where ‖A‖ denotes the operator norm of a matrix A, that is,
‖A‖ = sup{‖Av‖/‖v‖; ‖v‖ 6= 0}. Let Hα(M) denote the space of all such α-
Hölder continuous maps. We endow this space with the α-Hölder topology which
is generated by the norm

‖A‖α = sup
x∈M

‖A(x)‖ + sup
x 6=y

‖A(x)−A(y)‖

d(x, y)α
.

2.1. Lyapunov exponents. It follows from the subadditive ergodic theorem of
Kingman [9] that there exists a full µ-measure set Rµ ⊂ M , whose points are
called µ-regular points, such that for every x ∈ Rµ the limits

λu(A, x) = lim
n→∞

1

n
log ‖An(x)‖ and λs(A, x) = lim

n→∞

1

n
log

∥

∥(An(x))−1
∥

∥

−1



THE SET OF COCYCLES WITH NONVANISHING LYAPUNOV EXPONENTS IS OPEN 3

exist. We call such limits Lyapunov exponents. Moreover, when λu(A, x) 6= λs(A, x)
it follows from a famous theorem of Oseledets [11] that there exists a decomposi-
tion R

2 = Eu,A
x ⊕ Es,A

x , called the Oseledets decomposition, into vector subspaces
depending measurably on x such that for every x ∈ Rµ,

A(x)E∗,A
x = E∗,A

f(x) and λ∗(A, x) = lim
n→±∞

1

n
log ‖An(x)v‖ (2)

for every non-zero v ∈ E∗,A
x and ∗ ∈ {u, s}. Furthermore, since the Lyapunov

exponents are f -invariant, ergodicity of µ implies that they are constant for every
x ∈ Rµ. In this case we write λu(A, x) = λu(A, µ) and λs(A, x) = λs(A, µ).

2.2. Partial Hyperbolicity. A diffeomorphism f : M → M of a compact Cr

manifold M , r ≥ 1, is said to be partially hyperbolic if there exists a non-trivial
splitting of the tangent bundle

TM = Es ⊕ Ec ⊕ Eu

invariant under the derivative Df , a Riemannian metric ‖ · ‖ on M , and positive
continuous functions ν, ν̂, γ, γ̂ with ν, ν̂ < 1 and ν < γ < γ̂−1 < ν̂−1 such that,
for any unit vector v ∈ TxM ,

‖Df(x)v‖ < ν(x) if v ∈ Es(x),

γ(x) <‖Df(x)v‖ < γ̂(x)
−1

if v ∈ Ec(x),

ν̂(x)−1 <‖Df(x)v‖ if v ∈ Eu(x).

All three sub-bundles Es, Ec, Eu are assumed to have positive dimension. We say
that f is center-bunched if

ν < γγ̂ and ν̂ < γγ̂.

We need this hypothesis because we are going to use the results of [1]. From now
on, we take M to be endowed with the distance d : M ×M → R associated to such
a Riemannian structure.

Suppose that f : M → M is a partially hyperbolic diffeomorphism, then the
stable and unstable bundles Es and Eu are uniquely integrable and their integral
manifolds form two transverse continuous foliations Ws and Wu, whose leaves are
immersed sub-manifolds of the same class of differentiability as f . These foliations
are referred to as the strong-stable and strong-unstable foliations. They are invariant
under f , in the sense that

f(Ws(x)) = Ws(f(x)) and f(Wu(x)) = Wu(f(x)),

where Ws(x) and Wu(x) denote the leaves of Ws and Wu, respectively, passing
through any x ∈ M . We say that f is accessible if M and ∅ are the only su-
saturated sets. This means that, except of ∅, M is the only set that is a union of
entire strong-stable and strong-unstable leaves.

2.3. Fiber-bunched cocycles. Let f : M → M be a Cr partially hyperbolic map
on a compact manifold M and A : M → SL(2,R) be an α-Hölder continuous map.
We say that the cocycle generated by A over f is fiber-bunched if

‖A(x)‖‖A(x)−1‖ν(x)α < 1 and ‖A(x)‖‖A(x)−1‖ν̂(x)α < 1

for every x ∈ M . As a shorthand for this notion, since our base dynamics f is going
to be fixed, we simply say that A is fiber-bunched. Observe that this is an open
condition in Hα(M).



4 LUCAS BACKES, MAURICIO POLETTI, AND ADRIANA SÁNCHEZ

2.4. Main results. The main results of this note are the following. Recall that a
measure µ is in the Lebesgue class if it is generated by a volume form.

Theorem A. Let f : M → M be a Cr, r ≥ 2, partially hyperbolic, volume preserv-
ing, center-bunched and accessible diffeomorphism defined on a compact manifold
M and µ an ergodic f -invariant measure in the Lebesgue class. If A ∈ Hα(M) is
fiber-bunched and λu(A, µ) > λs(A, µ) then A can not be accumulated by cocycles
with zero Lyapunov exponents.

We observe that a similar result can be stated in terms of GL(2,R)-valued co-
cycles changing ‘cocycles with zero Lyapunov exponents’ by ‘cocycles with just one
Lyapunov exponent’. Indeed, by continuity of A and connectedness ofM (which fol-
lows from the accessibility), either det(A(x)) > 0 for every x ∈ M or det(A(x)) < 0
for every x ∈ M . Suppose we are in the first case (the other case can be easily
deduced from this one). Then, given A : M → GL(2,R) consider gA : M → R de-

fined by gA(x) = (detA(x))
1
2 and B : M → SL(2,R) such that A(x) = gA(x)B(x).

Therefore,

λu/s(A, µ) = λu/s(B, µ) +

∫

log(gA(x)) dµ(x),

and consequently,

λu(A, µ) = λs(A, µ) ⇐⇒ λu(B, µ) = 0 = λs(B, µ).

As already mentioned at the introduction, we also present an example show-
ing that the accessibilty assumption in the previous theorem is necessary. More
precisely,

Theorem B. There exists a volume preserving partially hyperbolic and center-
bunched diffeomorphism f and a Hölder continuous fiber-bunched map A with non-
zero Lyapunov exponents which is approximated by cocycles with zero Lyapunov
exponents.

In light of the previous results, we are lead to make the following conjecture which
is in the same spirit as the conjectures proposed by Viana [13] in the hyperbolic
setting.

Conjecture 2.1. Under the assumptions of Theorem A the Lyapunov exponents
of Hölder continuous SL(2,R)-valued cocycles vary continuously in the set of fiber-
bunched cocycles.

As a consequence of [10, Corollary 4] (see also [1]) it follows that the previous
conjecture is true in an open and dense subset of the fiber-bunched elements of
Hölder continuous SL(2,R)-valued cocycles giving more evidences of its veracity.

3. Preliminary results

In this section we recall some classical notions and present some useful results
that are going to be used in the proof of our main theorem. Let f : M → M ,
A ∈ Hα(M) and µ be as in Theorem A.



THE SET OF COCYCLES WITH NONVANISHING LYAPUNOV EXPONENTS IS OPEN 5

3.1. Accessibility and holonomies. Given x, y ∈ M , we write x ∼s y when-
ever y ∈ Ws(x). Observe that this is an equivalence relation and moreover, is
f -invariant. That is, if x ∼s y then f(x) ∼s f(y). Analogously, we write x ∼u z if
z ∈ Wu(x).

An su-path from x to y is a path connecting x and y which is a concatenation of
finitely many subpaths, each of which lies entirely in a single leaf of Ws or a single
leaf of Wu. Every sequence of points x = z0, z1, . . . , zn = y, such that zi ∼∗ zi+1

for ∗ = s or u, and i = 0, . . . , n−1 defines a unique su-path. An su-loop or a closed
su-path is an su-path beginning and ending at the same point. If γ1 is an su-path
given by z0, . . . , zn and γ2 is an su-path given by z′0, z

′
1, . . . , z

′
m, with z′0 = zn, we

define γ1 ∧ γ2 as the su-path given by z0, . . . , zn, z
′
1, . . . , z

′
m.

We say that an su-path γ defined by the sequence x = z0, z1, . . . , zn = y is
a (K,L)-path if n ≤ K and dW∗(zi+1, zi) ≤ L for every i = 1, . . . , n − 1 where
dW∗ is the distance induced by the Riemannian strucutre on the submanifold W∗

for ∗ = s, u. For simplicity we write x ∼∗
L y if dW∗(zi+1, zi) ≤ L for every i =

1, . . . , n − 1. Observe that, by the compactness of M and continuity of stable
manifolds of bounded size, the space of (K,L)-paths is compact. In particular,

Lemma 3.1. [15, Lemma 4.5] There exist constants K and L such that every pair
of points in M can be connected by an (K,L)-path.

For every pair of points x, y ∈ M so that x ∼s y, our fiber-bunched assumption
assures that the limit

Hs,A
xy = lim

n→+∞
An(y)−1 ◦An(x)

exists (see [1, Proposition 3.2]). Moreover, for every L > 0,

(x, y, A) → Hs,A
xy is continuous on Ws

L ×Hα(M)

where Ws
L = {(x, y) ∈ M ×M ;x ∼s

L y} (see [1, Remark 3.4]). In particular,

Remark 3.2. Given a sequence {Ak}k∈N converging to A in Hα(M), since Ws
L is

compact,

{Ws
L ∋ (x, y) → Hs,Ak

xy }k∈N

is equi-continuous for k sufficiently large.

The family of maps Hs,A
xy is called an stable holonomy for the cocycle (A, f). It

is easy to verify that (see [1, Proposition 3.2]) for x ∼s y and z ∼s y,

Hs,A
xx = Id and Hs,A

xy = Hs,A
zy ◦Hs,A

xz

and

Hs,A
fj(x)fj(y) = Aj(y)Hs,A

xy Aj(x)−1 ∀j ≥ 0.

Similarly, for x ∼u y we define the unstable holonomy Hu,A
xy as the stable

holonomies for (A−1, f−1). If γ is the su-path defined by the sequence z0, z1, . . . , zn
then we write HA

γ = H∗,A
zn−1zn ◦ . . . ◦H∗,A

z0z1 for ∗ ∈ {s, u}.

3.2. Disintegrations and su-invariance. We say that a measure m on M × P
1

projects on µ if π∗m = µ where π is the canonical projection π : M × P
1 →

M . Observe that any such measure admits a disintegration with respect to the
partition {{x}×P

1}x∈M and the measure µ, that is, there exists a family of measures
{mx}x∈M on {{x} × P

1}x∈M so that for every measurable B ⊂ M × P
1,



6 LUCAS BACKES, MAURICIO POLETTI, AND ADRIANA SÁNCHEZ

• x → mx(B) is measurable,
• mx({x} × P

1) = 1 and
• m(B) =

∫

M
mx(B ∩ ({x} × P

1))dµ(x).

Moreover, such disintegrantion is essentially unique [12]. Identifying each fiber
{x} × P

1 with P
1, we can think of x → mx as a map from M to the space of

probability measures on P
1 endowed with the weak∗ topology.

Let FA : M × P
1 → M × P

1 be the map given by

FA(x, v) = (f(x), [A(x)v])

and m be an FA-invariant measure projecting on µ. We say that m is s-invariant
if there exists a total measure set M s ⊂ M such that for every x, y ∈ M s satisfying
x ∼s y we have Hs,A

xy ∗
mx = my. Such measure m is also known as an s-state.

Analogously, we say that m is u-invariant (or an u-state) if the same is true replac-
ing stable by unstable in the previous definition. We say that m is su-invariant if
it is simultaneously s-invariant and u-invariant. The main property of su-ivariant
measures is the following

Proposition 3.3. [1, Theorem D] Any FA-invariant measure m projecting on µ
which is su-invariant admits a disintegration {mx}x∈M for which M s = Mu = M
and so that mx depends continuously on the base point x ∈ M in the weak∗ topology.

3.3. Trivial holonomies on su-loops. In this section we explain how in certain
specific situations we can perform a change of coordinates that makes the cocycle
(A, f) constant without changing its Lyapunov exponents.

Let us assume that HA
γ = id for every su-loop γ with at most 3K legs and each

of them with length at most L. Recall that we call such loops (3K,L)-loops. In
particular, HA

γ = id for every su-loop γ. Indeed, observe initially that if γ is a
(2K,L)-path from x to y then, by Lemma 3.1, there exists a (K,L)-path γ′ from x
to y so that HA

γ = HA
γ′ . In fact, if −γ′ denotes the path γ′ with opposite orientation

then γ ∧ (−γ′) is a (3K,L)-loop and

HA
γ ◦ (HA

γ′)−1 = HA
γ ◦HA

−γ′ = HA
γ∧(−γ′) = id .

Hence, HA
γ = HA

γ′ . Now, taking any su-loop γ with an arbitrary number of legs
whose lengths are at most L we can decompose it as γ = γ1 ∧ · · · ∧ γk, where every
γi is a (K,L)-path. In particular, γk−1 ∧ γk is a (2K,L)-path and by the previous
argumment we can replace it by a (K,L)-path γ′

k−1 with the same starting and

ending points and, so that HA
γk−1∧γk

= HA
γ′

k−1

. Thus, taking γ′ = γ1 ∧ · · · ∧ γk−2 ∧

γ′
k−1 we have that γ and γ′ have the same starting and ending points andHA

γ = HA
γ′ .

Repeating this procedure a finite number of times we get some (K,L)-loop γ′′ such
that HA

γ = HA
γ′′ = id. Finally, observing that any su-loop γ can be transformed

into an su-loop with legs of size at most L just by breaking one “large” leg into
several with smaller sizes we conclude that HA

γ = id for every su-loop proving our

claim. As a consequence we get that if γ is an su-path connecting x and y then HA
γ

does not depend on γ. In fact, if γ1 and γ2 are su-paths connecting x and y then
γ1 ∧ (−γ2) is an su-loop and thus HA

γ1
◦ (HA

γ2
)−1 = HA

γ1
◦HA

−γ2
= HA

γ1∧(−γ2)
= id

as claimed. Let us denote this common value simply by HA
xy. From the properties

of the holonomies and the fact that any two points x, y ∈ M can be connected by
a (K,L)-path it follows that

• HA
yzH

A
xy = HA

xz,



THE SET OF COCYCLES WITH NONVANISHING LYAPUNOV EXPONENTS IS OPEN 7

• A(y)HA
xy = HA

f(x)f(y)A(x),

• A → HA
xy is uniformly continuous for any pair of points x, y ∈ M and

• ‖HA
xy‖ ≤ N for some N > 0 and any x, y ∈ M .

Fix x ∈ M and, given y ∈ M , consider the following transformation

Â(y) = HA
f(y)xA(y)H

A
xy.

Then, Â2(y) = Â(f(y))Â(y) = HA
f2(y)xA(f(y))H

A
xf(y)H

A
f(y)xA(y)H

A
xy and conse-

quently Â2(y) = HA
f2(y)xA

2(y)HA
xy. More generally, Ân(y) = HA

fn(y)xA
n(y)HA

xy for

every n ∈ N and consequently (Â, f) and (A, f) have the same Lyapunov exponents.
Moreover, for any z, y ∈ M ,

Â(z)−1Â(y) =
(

HA
f(z)xA(z)H

A
xz

)−1

HA
f(y)xA(y)H

A
xy

=HA
zxA(z)

−1HA
xf(z)H

A
f(y)xA(y)H

A
xy

=HA
zxA(z)

−1HA
f(y)f(z)A(y)H

A
xy

=HA
zxA(z)

−1A(z)HA
yzHxy

=HA
zxH

A
yzHxy

=HA
zxH

A
xz

= id .

In particular, Â is constant and consequently its largest Lyapunov exponent is the
logarithm of the norm of the greatest eigenvalue of Â. Summarizing, if HA

γ = id for
every (3K,L)-loop γ then we can perform a change of coordinates that makes the
cocycle (A, f) constant without changing its Lyapunov exponents. This is going to
be used in Section 4.3.

3.4. SL(2,R) matrices and invariant measures on P
1. The following result

plays an important part in our proof below.

Proposition 3.4. For each n ∈ N, let Ln be a SL(2,R) matrix so that Ln
n→+∞
−−−−−→

id and let ηn be an Ln-invariant measure on P
1 so that ηn

n→+∞
−−−−−→ 1

2 (δp + δq) for

some p, q ∈ P
1 with p 6= q. Then for every n sufficiently large either Ln is hyperbolic

or Ln = id.

Proof. The proof is by contradiction. We start observing that as Ln converges to the
identity all the matrices have positive trace for n sufficiently large. Consequently,
if Ln is not the identity we have three posibilities: if the trace tr(Ln) > 2 then
the matrix Ln is hyperbolic, if tr(Ln) < 2 then the matrix Ln is elliptic and is

conjugated to a rotation of angle θn = arccos( tr(Ln)
2 ) and if tr(Ln) = 2 then the

matrix Ln is parabolic and is non diagonalizable with both eigenvalues equal to 1.
Suppose initially that all the matrices Ln have tr(Ln) < 2. In particular, for each

n ∈ N there exists Pn ∈ SL(2,R) so that Ln = P−1
n RθnPn where Rθn stands for the

rotation of angle θn. Moreover, since tr(Ln)
n→+∞
−−−−−→ 2, we get that θn

n→+∞
−−−−−→ 0.

Now, for each n ∈ N let us consider νn = Pn∗ηn which is an Rθn-invariant mea-

sure. We start observing that there exists a subsequence {nj}j so that νnj

j→+∞
−−−−→

Leb where Leb stands for the Lebesgue measure on P
1. Indeed, if θn is an irrational

number then we know that the only Rθn-invariant measure is Leb. In particular,



8 LUCAS BACKES, MAURICIO POLETTI, AND ADRIANA SÁNCHEZ

νn = Leb. Thus, if there are infinitely many values of n for which θn is an irrational
number we are done.

Suppose then that θn is a rational number for every n ∈ N. In particular,

Rθn is periodic and denoting by qn its period, since θn
n→+∞
−−−−−→ 0, we have that

qn
n→+∞
−−−−−→ +∞.
In what follows we make an abuse of notation thinking of P1 as [0, 1] identifying

the extremes of the interval.
Let ϕ : P1 → R be a continuous map and ε > 0. Since P

1 is compact, there
exists δ > 0 so that | ϕ(x) − ϕ(y) |< ε whenever d(x, y) < δ. Thus, taking n ≫ 0

so that qn > 1
δ we get that | ϕ(x) − ϕ( j

qn
) |< ε for every x ∈ [ j

qn
, j+1

qn
) and

j = 0, 1, . . . , qn − 1. In particular,

∣

∣

∣

∣

∣

1

νn([
j
qn
, j+1

qn
))

∫
j+1

qn

j
qn

ϕdνn − ϕ(
j

qn
)

∣

∣

∣

∣

∣

< ε.

Now, observing that νn([
j
qn
, j+1

qn
)) = 1

qn
for every j = 0, 1, . . . , qn−1 once νn is Rθn -

invariant, summing the previous expression for j from 0 up to qn − 1 and dividing
both sides by qn we get that

∣

∣

∣

∣

∣

∣

∫ 1

0

ϕdνn −
1

qn

qn−1
∑

j=0

ϕ(
j

qn
)

∣

∣

∣

∣

∣

∣

< ε.

On the other hand, since ϕ is Riemann integrable,

lim
n→∞

1

qn

qn−1
∑

j=0

ϕ(
j

qn
) =

∫

ϕdLeb

which implies that νn
n→+∞
−−−−−→ Leb as claimed. So, restricting to a subsequence, if

necessary, we may assume that νn
n→+∞
−−−−−→ Leb.

We now analyse the accumulation points of ηn = P−1
n ∗νn. If {P−1

n }n stay in a
compact set of SL(2,R) then, taking a subsequence if necessary, we may assume
that there exists P ∈ SL(2,R) so that P−1

n → P . In particular, limn→∞ ηn = P∗Leb
which contradicts our assumption since P∗Leb is non-atomic. If

∥

∥P−1
n

∥

∥ → ∞ then
we can work on the compactification of quasi-projective transformations (see [13]
or [8, Section 6.1]). In particular, restricting to a subsequence, if necessary, we
have that P−1

n → Q, where Q is defined outside some kernel (a one dimensional
subspace) and the image Im(Q) ⊂ P

1 of Q is a one dimensional subspace. Thus,
as the kernel has zero Lebesgue measure we can apply [2, Lemma 2.4] to conclude
that

lim
n→∞

P−1
n ∗νn = Q∗Leb = δIm(Q)

which is a contradiction. Consequently, Ln may be elliptic only for finitely many
values of n.

To conclude the proof it remains to rule out the cases when tr(Ln) = 2 and the
matrix are non diagonalizable for infinitely many values of n. So, suppose Ln is non
diagonalizable and both of its eigenvalues are 1 for every n. Then by the Jordan’s
normal decomposition we have

Ln = P−1
n

(

1 1
0 1

)

Pn



THE SET OF COCYCLES WITH NONVANISHING LYAPUNOV EXPONENTS IS OPEN 9

for some Pn ∈ GL(2,R). Consequently, the only invariant measure for Ln is atomic

and have only one atom contradicting the fact that ηn
n→+∞
−−−−−→ 1

2 (δp+δq). Thus, Ln

can be parabolic and different from id only for finitely many values of n concluding
the proof of the proposition. �

3.5. PSL(2,R) cocycles. Let us consider the projective special linear group given
by PSL(2,R) = SL(2,R)/{±Id}. That is, given A,B ∈ SL(2,R) let ∼ be the
equivalence relation given by A ∼ B if and only if A = B or A = −B. Given
A ∈ SL(2,R), let [[A]] = {B ∈ SL(2,R);B ∼ A} be the equivalence class of A with
respect to ∼. Then, PSL(2,R) = {[[A]];A ∈ SL(2,R)}. Observe that the norm ‖·‖
on SL(2,R) naturally induces a norm, which we are going to denote by the same
symbol, on PSL(2,R): given A ∈ SL(2,R), ‖[[A]]‖ := ‖A‖ = ‖−A‖.

Given A : M → SL(2,R) let us consider à : M → PSL(2,R) given by Ã(x) =
[[A(x)]]. By Kingman’s subadditive ergodic theorem [9] and the ergodicity of µ it
follows that the limit

L(Ã, µ) = lim
n→+∞

1

n
log ‖Ãn(x)‖

exists and is constant for µ-almost every x ∈ M . In particular, since ‖An(x)‖ =

‖Ãn(x)‖ for every x ∈ M and n ∈ N, we get that λu(A, µ) = L(Ã, µ). Another

simple observation is that for every v ∈ P
1, [A(x)v] = [Ã(x)v] and, consequently,

the action induced by A on P
1 coincide with the action of à on P

1. Moreover,

HÃ
γ = [[HA

γ ]] ∈ PSL(2,R) is well defined and have similar properties with respect

to à as those of HA
γ with respect to A described in Section 3.3. In particular, a

similar conclusion to that of Section 3.3 holds for à whenever HÃ
γ = [[id]] for every

(3K,L)-loop γ: we can perform a change of coordinates that makes the cocycle

(Ã, f) constant without changing L(Ã, µ). Consequently, denoting this new cocycle

by ˆ̃A, it follows that L(Ã, µ) is equal to logarithm of the norm of the greatest

eigenvalue of any representative of ˆ̃A.
Furthermore, the results of Section 3.4 also have a counterpart for PSL(2,R)

cocycles. In order to state it, recall that a sequence {L̃n}n in PSL(2,R) is said to

converge to L̃ ∈ PSL(2,R) if there are representatives L and Ln in SL(2,R) of L̃

and L̃n, respectively, so that the sequence {Ln}n converges to L in SL(2,R).

Proposition 3.5. For each n ∈ N, let L̃n ∈ PSL(2,R) be so that L̃n
n→+∞
−−−−−→ [[id]]

and let ηn be an L̃n-invariant measure on P
1 so that ηn

n→+∞
−−−−−→ 1

2 (δp+δq) for some

p, q ∈ P
1 with p 6= q. Then for every n sufficiently large either L̃n is hyperbolic or

L̃n = [[id]].

This result follows easily from Proposition 3.4: for every L̃n ∈ PSL(2,R) we can

take a representative of L̃n in SL(2,R) with positive trace and apply the aforemen-
tioned result to these representatives.

4. Proof of the main result

Let f : M → M , A : M → SL(2,R) and µ be given as in Theorem A and
suppose there exists a sequence {Ak}k∈N in Hα(M) with λu(Ak, µ) = λs(Ak, µ) = 0

for every k ∈ N and such that Ak
k→+∞
−−−−−→ A.



10 LUCAS BACKES, MAURICIO POLETTI, AND ADRIANA SÁNCHEZ

For each k ∈ N, letmk be an ergodic FAk
-invariant probability measure onM×P

1

projecting on µ where FAk
is defined similarly to FA. Passing to a subsequence if

necessary, we may assume that the sequence {mk}k converges in the weak∗ topology
to some measure m which is, as one can easily check, FA-invariant and projects on
µ. In order to prove Theorem A we are going to analyse these families of measures
and its respective disintegrations.

4.1. Continuity and convergence of conditional measures. It follows from
Remark 3.2 and [1, Theorem C] and its proof that

Corollary 4.1. For every k sufficiently large there exists an su-invariant disinte-
gration {mk

x : x ∈ M} of mk with respect to the partition {{x} × P
1 : x ∈ M} and

µ such that

{M ∋ x → mk
x}k≫0 is equi-continuous.

As an application of this corollary we get that

Proposition 4.2. The measure m is su-invariant and admits a continuous dis-
integration {mx}x∈M with respect to {{x} × P

1}x∈M and µ so that mk
x converges

uniformly on M to mx.

In order to prove the previous proposition we need the following auxiliary result.

Lemma 4.3. Let X and Y be compact metric spaces, µ a Borel probability measure
on X and {νk}k∈N be a sequence of probability measures on X × Y projecting on µ
and converging in the weak∗ topology to some measure ν. Then for every measurable
function ρ : X → R and every continuous function ϕ : Y → R,

lim
k→∞

∫

ρ× ϕdνk =

∫

ρ× ϕdν.

Proof. Given ε > 0 let ρ̂ : X → R be a continuous function so that
∫

X |ρ̂− ρ|dµ <
ε

2 supϕ . Take k0 ∈ N such that for every k > k0,

∣

∣

∣

∣

∫

ρ̂× ϕdνk −

∫

ρ̂× ϕdν

∣

∣

∣

∣

<
ε

2
.

Then, for k > k0,
∣

∣

∣

∣

∫

ρ× ϕdνk −

∫

ρ× ϕdν

∣

∣

∣

∣

< supϕ

∫

X

|ρ̂− ρ|dµ+

∣

∣

∣

∣

∫

ρ̂× ϕdνk −

∫

ρ̂× ϕdν

∣

∣

∣

∣

< ε.

�

Proof of Proposition 4.2. For each k ∈ N, let {mk
x}x∈M be the disintegration of

mk given by Corollary 4.1. We start observing that for every continuous function
ϕ : P

1 → R, by Arezelà-Aslcoi’s theorem (recall Corollary 4.1), there exists a

subsequence of {
∫

P1 ϕdm
k
x}k such that

∫

P1 ϕdm
kj
x → Ix(ϕ) uniformly on M . Taking

a dense subset {ϕj}j∈N of the space C0(P1) of continuous functions ϕ : P1 → R and
using a diagonal argument, passing to a subsequence if necessary, we can suppose
that

∫

P1 ϕdm
k
x → Ix(ϕ) for every ϕ ∈ C0(P1). It is easy to see that Ix defines a

positive linear functional on C0(P1). Consequently, by Riesz-Markov’s theorem, for
every x ∈ M there exists a measure m̂x on P

1 such that Ix(ϕ) =
∫

ϕdm̂x.



THE SET OF COCYCLES WITH NONVANISHING LYAPUNOV EXPONENTS IS OPEN 11

On the other hand, letting {mx}x∈M be a disintegration of m with respect to
{{x}×P

1}x∈M and µ and invoking Lemma 4.3 it follows that for every continuous
function ϕ : P1 → R and any µ-positive measure subset D ⊂ M ,

∫

D

∫

P1

ϕdmk
xdµ =

∫

D×P1

ϕdmk →

∫

D×P1

ϕdm =

∫

D

∫

P1

ϕdmxdµ.

Consequently, mx = m̂x for µ almost every x ∈ M . Thus, extending mx = m̂x for
every x ∈ M we get a continuous disintegration of m such that mk

x → mx uniformly
on x ∈ M . In particular, by Remark 3.2 and the su-invariance of mk for every k it
follows that m is also su-invariant as claimed. �

From now on we work exclusively with the disintegrations {mk
x}x∈M and {mx}x∈M

of mk and m, respectively, given by Corollary 4.1 and the previous proposition.
Recall we are assuming λu(A, µ) > 0 > λs(A, µ). Thus, letting R

2 = Eu,A
x ⊕Es,A

x

be the Oseledets decomposition associated to A at the point x ∈ M , it follows from
Proposition 3.1 of [4] that for any FA-invariant measure m, its conditional measures
are of the form mx = aδEu,A

x
+ bδEs,A

x
for some a, b ∈ [0, 1] such that a + b = 1

where here and in what follows we abuse notation and identify a 1-dimensional
linear space E with its class [E] in P

1.

Lemma 4.4. There exist continuous and su-invariant functions which coincide
with x → Es,A

x , Eu,A
x for µ-almost every point. By su-invariance we mean that for

every (admissible) choice of x, y, z ∈ M , Hs,A
xy E∗

x = E∗
y and Hu,A

xz E∗
x = E∗

z for
∗ ∈ {s, u}.

From now on we think of Es,A
x and Eu,A

x as continuous functions defined for
every x ∈ M .

Proof. Recall mk is a FAk
-invariant measure such that mk → m. Since λu(Ak, µ) =

0 for every k ∈ N we get that
∫

ΦAk
dmk = 0 where ΦAk

: M × P
1 → R is given by

ΦAk
(x, v) = log ‖Ak(x)v‖

‖v‖ . On the other hand,
∫

ΦAk
dmk →

∫

ΦAdm.

Thus,
∫

ΦAdm = 0 which implies that the numbers a and b given above are strictly
larger than zero. Now, by Proposition 4.2 we know that {mx}x is su-invariant.
Consequently, since Eu,A

x is u-invariant and Es,A
x is s-invariant, it follows δEu,A

x
=

1
a (mx − bδEs,A

x
) is also s-invariant. Analogously, Es,A

x is u-invariant. In particular,

Eu,A
x and Es,A

x are su-invariant. Continuity follows easily (see [1, Theorem D]). �

4.2. Excluding the atomic case with a bounded number of atoms. In this
subsection we prove that mk

xk
can not have a bounded number of atoms (with

bound independent of k) for infinitely many values of k ∈ N and any xk ∈ M . In
order to do so, we need the following lemma.

Lemma 4.5. If mk
y has an atom for some y ∈ M , then there exists j = j(k) ∈ N

such that for every x ∈ M , there exist v1x, . . . v
j
x ∈ P

1 so that mk
x = 1

j

∑j
i=1 δvi

x
.

Proof. Let vy ∈ P
1 be such that mk

y(vy) = β > 0 and for every x ∈ M , let γx be

an su-path joining y and x. By the su-invariance of the disintegration {mk
x}k it

follows that mk
x(H

Ak
γx

vy) = β for every x ∈ M . Thus, considering L = {(x, vx) ∈



12 LUCAS BACKES, MAURICIO POLETTI, AND ADRIANA SÁNCHEZ

M × P
1; mk

x(vx) = β} we get that mk(L) =
∫

mk
x(L ∩ {x} × P

1)dµ ≥ β > 0.
Consequently, since L is FAk

-invariant and mk is ergodic it follows that mk(L) = 1.
In particular, mk

x(L ∩ {x} × P
1) = 1 for µ-almost every x ∈ M which implies that

mk
x = 1

j

∑j
i=1 δvi

x
, where 1

j = β (in particular, j does not depend on x). Finally, to

prove that this claim holds true for every x ∈ M , we just take some su-path from
a point in the total measure set and x and use the su-invariance. �

The proof is going to be by contradiction. So, passing to a subsequence and using
the previous lemma suppose mk

x has j(k) atoms and that the sequence {j(k)}k is
bounded. Restricting again to a subsequence, if necessary, we may assume that
j(k) is constant equal to some j ∈ N. In particular, since mx = 1

2δEs,A
x

+ 1
2δEu,A

x
,

for k sufficiently large mk
x has an even number of atoms. Thus, writing mk

x =
1
j

∑j
i=1 δvi

k
(x) and reordering if necessary we may suppose that vik(x) → Eu,A

x for i ≤
j
2 and vℓk(x) → Es,A

x for ℓ > j
2 . Moreover, such convergence is uniform. Observe now

that for each k ∈ N there exists some xk ∈ M such that Ak(xk)v
ik
k (xk) = vjkk (f(xk))

for some ik ≤ j
2 and jk > j

2 , otherwise the set L = ∪x∈M{x} × {v1k(x), . . . v
j
2

k (x)}
would be FAk

-invariant with measure

mk(L) =

∫

mk
x({v

1
k(x), . . . v

j
2

k (x)})dµ =
1

2
,

contradicting the ergodicity. Thus, restricting to a subsequence, if necessary, we
may assume without loss of generality that vikk (xk) = v1k(xk) and vjkk (xk) = vjk(xk)
for every k ∈ N and that xk → x. In particular,

A(x)Eu,A
x = lim

k→∞
Ak(xk)v

1
k(xk) = lim

k→∞
vjk(f(xk)) = Es,A

f(x),

a contradiction. Summarizing, we can not have a subsequence {ki}i so that the
sequence {j(ki)}i is bounded where j(k) stands for the number of atoms of mk

x

(which is independent of x ∈ M).

4.3. Conclusion of the proof. Given x ∈ M let γ be a non-trivial su-loop at
x. In particular, from Lemma 4.4 it follows that HA

γ E∗,A
x = E∗,A

x for ∗ ∈ {s, u}.

Consequently, either HA
γ is hyperbolic or HA

γ = ± id. If HA
γ is hyperbolic then,

since HAk
γ

k→+∞
−−−−−→ HA

γ , it follows that HAk
γ is also hyperbolic for every k ≫ 0.

Thus, since HAk
γ ∗

mk
x = mk

x, it follows that mk
x is atomic and has at most two

atoms for every k ≫ 0 but from Section 4.2 we know this is not possible. So, we get

that HA
γ = ± id for every su-loop at x and every x ∈ M and therefore HÃ

γ = [[id]]
for every su-loop at x and every x ∈ M . Consequently, from Proposition 3.5 we get
that either there exists a non-trivial su-loop γ at some point x ∈ M and a sequence

{kj}j going to infinite as j → +∞ so that H
Ãkj
γ is hyperbolic for every j and thus

H
Akj
γ is also hyperbolic for every j or HÃk

γ = [[id]] for every su-loop γ and every
k > kγ for some kγ ∈ N. Arguing as we did above we conclude that the first case

can not happen. So, all we have to analyse is the case when HÃk
γ = [[id]] for every

su-loop γ and every k > kγ for some kγ ∈ N.
If there exists k0 ∈ N so that kγ ≤ k0 for every su-loop γ then making the change

of coordinates given in Section 3.3 for every k > k0 (recall Section 3.5) we get the

that L(Ãk, µ) is equal to the logarithm of the norm of the greatest eigenvalue of



THE SET OF COCYCLES WITH NONVANISHING LYAPUNOV EXPONENTS IS OPEN 13

any representative of ˆ̃Ak(x), where
ˆ̃Ak(x) is a constant element of PSL(2,R), and

ˆ̃Ak(x) →
ˆ̃A(x). In particular,

λu(Ak, µ) = L(Ãk, µ)
k→+∞
−−−−−→ L(Ã, µ) = λu(A, µ)

which is a contradiction. Now, recalling that in order to perform the change of

coordinates in Section 3.3 it is enough to assume thatHÃk
γ = [[id]] for every (K ′, L′)-

loop γ for some K ′, L′ > 0, to conclude the proof of Theorem A, in view of the
previous argumment, we only have to show that we can not have kγ arbitrarly large
for (K ′, L′)-loops.

Let kγ be minimum for its defining property, that is, HÃk
γ = [[id]] for every

k > kγ and H
Ãkγ
γ 6= [[id]] and suppose that for each j ∈ N there exist xj ∈ M and

a (K ′, L′)-loop γj at xj so that kγj

j→+∞
−−−−→ +∞. Passing to a subsequence we may

assume xj
j→+∞
−−−−→ x and γj

j→+∞
−−−−→ γ where γ is an su-loop at x. This can be

done because each γj has at most K ′ legs and each of them with length at most

L′. In particular, if γj is defined by the sequence xj = zj0, z
j
1, . . . , z

j
nj

= xj then

nj ≤ K ′ for every j. Thus, passing to a subsequence we may assume nj = n ≤ K ′

for every j ∈ N and zji
j→+∞
−−−−→ xi for every i = 1, . . . , n and consequently γ is

the su-loop defined by the sequence x = x0, x1, . . . , xn = x. Now, since HÃ
γ =

[[id]], H
Ãkγj
γj

j→+∞
−−−−→ HÃ

γ and H
Ãkγj
γj 6= [[id]] it follows from Proposition 3.5 (recall

Proposition 4.2) that H
Ãkγj
γj is hyperbolic for every j ≫ 0 and thus H

Akγj
γj is also

hyperbolic for every j ≫ 0. Consequently, m
kγj
x is atomic and has at most two

atoms for every x ∈ M and every j ∈ N which again from Section 4.2 we know is
not possible concluding the proof of Theorem A.

Remark 4.6. We observe that Theorem A can also be proved using the technics of
couplings and energy developed in [3]. Maybe those ideas can be useful in proving
Conjecture 2.1. We chose to present the previous proof because it is shorter and
also different. It is also worth noticing that a similar result was obtained by Liang,
Marin and Yang [10, Theorem 6.1] for the derivative cocycle under the additional
assumption that f has a pinching hyperbolic periodic point. In our context, such a
hypothesis would immediately imply that all the conditional measures mk

x are atomic
with at most two atoms for every k ≫ 0. In particular, Theorem A would follow
from the results of Section 4.2.

5. Examples

At this section we present two examples of fiber-bunched cocycles with nonvan-
ishing Lyapunov exponents over a partially hyperbolic map which are accumulated
by cocycles with zero Lyapunov exponents.

5.1. Proof of Theorem B. Let ω be an irrational number of bounded type and
f0 : S1 → S1 be given by f0(t) = t+2πω where S1 is the unit circle. Recently, Wang
and You [14, Theorem 1] constructed examples of cocycles A ∈ Cr(S1, SL(2,R))
over f0, for any r = 0, 1, . . . ,∞ fixed, with arbitrarily large Lyapunov exponents
which are approximated in the Cr-topology by cocycles with zero Lyapunov ex-
ponents. Let A0 : S1 → SL(2,R) be such a cocycle and {Ak}k be a sequence in



14 LUCAS BACKES, MAURICIO POLETTI, AND ADRIANA SÁNCHEZ

Cr(S1, SL(2,R)) converging to A so that λu(Ak, ν) = 0 for every k ∈ N where
ν denotes the Lebesgue measure on S1. Now, given f1 : N → N , a volume-
preserving Anosov diffeomorphism of a compact manifoldN , let us consider the map
f : M := S1 ×N → M given by f(t, x) = (f0(t), f1(x)) and let  : M → SL(2,R)

be given by Â(t, x) = A0(t). Thus, defining Âk(t, x) = Ak(t) and denoting by µ the

Lebesgue measure onM we have that limk→+∞ Âk = Â, λu(Âk, µ) = λu(Ak, ν) = 0

for every k ∈ N and λu(Â, µ) = λu(A0, ν) > 0. Consequently, since f is a volume-
preserving partially hyperbolic and center-bunched diffeomorphism and f1 may be
chosen so that (Â, f) is fiber-bunched, we complete the proof of Theorem B.

5.2. Random product cocycles. We now present another construction showing
that given any real number λ > 0, we have a fiber-bunched cocycle A over a
partially hyperbolic and center-bunched map f so that λu(A, µ) = λ which can be
approximated by cocycles with zero Lyapunov exponents. We start with a general
construction.

Let Σ = {1, . . . , k}Z be the space of bilateral sequences with k symbols and
σ : Σ → Σ be the left shift map. Given maps fj : K → K and Aj : K → SL(2,R)
for j = 1, . . . , k whereK is a compact manifold, let us consider f : Σ×K → SL(2,R)
and A : Σ×K → SL(2,R) given, respectively, by

f(x, t) = (σ(x), fx0
(t))

and

A(x, t) = Ax0
(t).

The random product of the cocycles {(Aj , fj)}
k
j=1 is then defined as the cocycle

over f which is generated by A. Observe that this definition generalizes the notion
of random products of matrices explaining our terminology. Indeed, taking K as
being a single point we recover the aforementioned notion.

Differently from the case of random products of matrices where one have conti-
nuity of Lyapunov exponents (see [3],[7], [13]), in the setting of random products
of cocycles Lyapunov exponents can be very ‘wild’. This is what we exploit to
construct our next example.

Let f0 : S1 → S1 and ν be as in the previous example and let A0 ∈ Cr(S1, SL(2,R))
be given by [14, Theorem 1] so that λu(A0, ν) > λ. Taking f1 : S1 → S1 to be
f1(t) = t and A1 : S1 → SL(2,R) given by A1(t) = id, let (A, f) be the random
product of the cocycles (A0, f0) and (A1, f1) as defined above. Thus, letting η be
the Bernoulli measure on Σ defined by the probability vector (p0, p1) where p0 is so
that p0λ

u(A0, ν) = λ and considering µ = η × ν, the cocycle generated by A over
f has positive Lyapunov exponents and is accumulated by cocycles with zero Lya-
punov exponents. Indeed, let {A0,k}k be a sequence in Cr(S1, SL(2,R)) converging
to A0 for which the cocycle (A0,k, f0) satisfies λ

u(A0,k, ν) = 0 for every k ∈ N whose
existence is guaranteed by our choice of A0 and [14, Theorem 1], {A1,k}k be the
sequence such that A1,k = id for every k ∈ N and (Ak, f) be the random product

of (A0,k, f0) and (A1,k, f1). It is easily to see that Ak
k→∞
−−−−→ A. Now, for µ-almost

every (x, t) ∈ Σ× S1,

λu(Ak, µ, x, t) = lim
n→∞

1

n
log ‖An

k (x, t)‖.



THE SET OF COCYCLES WITH NONVANISHING LYAPUNOV EXPONENTS IS OPEN 15

Thus, observing that An
k (x, t) = A

τn(x)
0,k (t) where

τn(x) = #
{

1 ≤ j ≤ n; σj(x)0 = 0
}

,

it follows that

λu(Ak, µ, x, t) = lim
n→∞

τn(x)

n

1

τn(x)
log

∥

∥

∥
A

τn(x)
0,k (t)

∥

∥

∥
= p0λ

u(A0,k, ν).

In partitular, λu(Ak, µ, x, t) is constant equal to λ
u(Ak, µ) for µ-almost every (x, t) ∈

Σ× S1. Analogously, λu(A, µ) = p0λ
u(A0, ν). Consequently,

λu(Ak, µ) = 0 for every k ∈ N and λu(A, µ) = λ > 0

as claimed. Observe that despite the fact of not being smooth, the map f is
partially hyperbolic in the sense of the expansion and contraction properties when
Σ is endowed with the usual metric. Moreover, it is center-bunched and the cocycle
A is fiber-bunched.

Acknowledgements

We thank to Karina Marin for many helpful comments and suggestions on this
work and also for pointing out a gap in a previous version of the concluding argu-
ment. The first author was partially supported by a CAPES-Brazil postdoctoral
fellowship under Grant No. 88881.120218/2016-01 at the University of Chicago.
The second author was partially supported by Université Paris 13. The third au-
thor was partially supported by Universidad de Costa Rica and CNPq-Brazil.

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16 LUCAS BACKES, MAURICIO POLETTI, AND ADRIANA SÁNCHEZ

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Departamento de Matemática, Universidade Federal do Rio Grande do Sul, Av.

Bento Gonçalves 9500, CEP 91509-900, Porto Alegre, RS, Brazil.

E-mail address: lhbackes@impa.br

LAGA – Université Paris 13, 99 Av. Jean-Baptiste Clément, 93430 Villetaneus,

France.

E-mail address: mpoletti@impa.br

IMPA – Estrada D. Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, Brazil.

E-mail address: asanchez@impa.br


	1. Introduction
	2. Statements
	2.1. Lyapunov exponents
	2.2. Partial Hyperbolicity
	2.3. Fiber-bunched cocycles
	2.4. Main results

	3. Preliminary results
	3.1. Accessibility and holonomies
	3.2. Disintegrations and su-invariance
	3.3. Trivial holonomies on su-loops
	3.4. SL(2,R) matrices and invariant measures on P1
	3.5. PSL(2,R) cocycles

	4. Proof of the main result
	4.1. Continuity and convergence of conditional measures
	4.2. Excluding the atomic case with a bounded number of atoms
	4.3. Conclusion of the proof

	5. Examples
	5.1. Proof of Theorem ??
	5.2. Random product cocycles

	Acknowledgements
	References