straightforwardly following the strategy used in Subsection 6.2.

Choose $N_{0}\geqslant 1$ such that $2^{-N_{0}}<\varepsilon$ and let $\zeta>0$
be given by Lemma 10.4. Given an arbitrary
${\underline{\omega}}^{(0)}\in\Sigma_{\kappa}$ and $0<\varepsilon\ll\zeta$, we
claim that there exists a point ${\underline{\omega}}\in\Sigma_{\kappa}$ so
that
$d_{\Sigma_{\kappa}}({\underline{\omega}},{\underline{\omega}}^{(0)})<\varepsilon$
but the Cesàro averages
$\frac{1}{n}\sum_{j=0}^{n-1}\varphi_{A}(F_{A}^{j}({\underline{\omega}},v))$
diverge. Let $v_{1}\in\mathbf{P}E_{\underline{\kappa}}^{+}$ and
$v_{2}\in\mathbf{P}E_{\underline{\kappa}}^{-}$ and take a sequence of positive
integers $(n_{k})_{k\geqslant 1}$ satisfying (6.4). We first consider the pair

$\big{(}\,(\omega^{(0)}_{1},\omega^{(0)}_{2},\dots,\omega^{(0)}_{N_{0}})\;,\;A^{(N_{0})}({\underline{\omega}}^{(0)})\cdot
v\,\big{)}\in\\{1,2,\dots,\kappa\\}^{N_{0}}\times\mathbb{R}^{d}.$

As the projectivized maps are bi-Lipschitz, the strong projective
accessibility together with Lemma 6.1 implies that there exist a constant
$K(L^{-2n_{1}}\varepsilon)\geqslant 1$, a fibered transition time $0\leqslant
p_{1}\leqslant K(L^{-2n_{1}}\varepsilon)$ and a finite word
$(\omega^{(1)}_{1},\omega^{(1)}_{2},\dots,\omega^{(1)}_{p_{1}})\in\\{1,2,\dots,\kappa\\}^{p_{1}}$
so that
$v_{1}:=A^{(p_{1})}(\omega^{(1)}_{1},\omega^{(1)}_{2},\dots,\omega^{(1)}_{p_{1}})(A^{(N_{0})}({\underline{\omega}}^{(0)})\cdot
v)\in D_{A}$. By Lemma 10.4,

$\log\frac{\|A^{(n_{1})}(\underline{\kappa})\cdot
v_{1}\|}{\|v_{1}\|}\geqslant\theta^{-1}n_{1}.$

Now, using once more the strong accessibility property, there exists
$0\leqslant p_{2}\leqslant K(L^{-2n_{2}}\varepsilon)$ and a sequence
$(\omega^{(2)}_{1},\omega^{(2)}_{2},\dots,\omega^{(2)}_{p_{2}})\in\\{1,2,\dots,\kappa\\}^{p_{2}}$
so that

$v_{2}:=A^{(p_{2})}(\omega^{(2)}_{1},\omega^{(2)}_{2},\dots,\omega^{(2)}_{p_{2}})(v_{1})\in
C^{-}$

and $v_{2}\in
B(v_{-},L^{-2n_{2}}\varepsilon)\subset\mathbf{P}(\mathbb{R}^{d}\cap C^{-})$
(here and throughout, by some abuse of notation, we keep denoting by $v$ the
equivalence class of the vector $v\in\mathbb{R}^{d}$ in
$\mathbf{P}\mathbb{R}^{d}$ ). Since the maps $f_{j}$ are $L$-Lipschitz,
$v_{-}$ is a fixed point for $f_{\kappa}$, we conclude that
$f_{\kappa}^{j}(v_{2})\in B(v_{-},\zeta)$ for every $0\leqslant j\leqslant
n_{2}$. Equivalently, $A^{j}(\underline{\kappa})\cdot v_{2}\in C^{-}$ for
every $0\leqslant j\leqslant n_{2}$. Using Lemma 10.4 once more,

$\log\frac{\|A^{(n_{2})}(\underline{\kappa})\cdot
v_{2}\|}{\|v_{2}\|}\geqslant\theta n_{2}.$

Repeating this procedure inductively and uniform continuity of $\varphi_{A}$
we obtain that the sequence

${\underline{\omega}}=([{\underline{\omega}}_{0}]_{[-N_{0},N_{0}]},\underbrace{\kappa,\kappa,\dots\kappa}_{n_{1}},\omega^{(1)}_{1}\omega^{(1)}_{2}\dots\omega^{(1)}_{{p_{1}}},\underbrace{\kappa,\kappa,\dots\kappa}_{n_{2}},\omega^{(2)}_{1}\omega^{(2)}_{2}\dots\omega^{(2)}_{{p_{2}}},\underbrace{\kappa,\kappa,\dots\kappa}_{n_{3}},\ldots)$

is an element in $\Sigma_{\kappa}^{+}$ according to which

$\liminf_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi_{A}(F_{A}^{j}({\underline{\omega}},v))<\limsup_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi_{A}(F_{A}^{j}({\underline{\omega}},v)).$

As $\varepsilon>0$ and ${\underline{\omega}}^{(0)}\in\Sigma_{\kappa}$ where
chosen arbitrary this proves that the set of points
${\underline{\omega}}\in\Sigma_{\kappa}$ for which $v$ has Lyapunov non-
typical behavior along the orbit of ${\underline{\omega}}$ is dense in
$\Sigma_{\kappa}$. As mentioned before, the modifications to obtain a Baire
residual subset of $\Sigma_{\kappa}$ along the strategy used in Subsection 6.2
are left as an easy but thorough exercise to the interested reader. The second
statement in the theorem is an immediate consequence of the first one. Indeed,
by separability, there exists a countable and dense subset
$\mathscr{D}\subset\mathbf{P}\mathbb{R}^{d}$. Then,
$\mathscr{R}=\bigcap_{v\in\mathscr{D}}\;\mathscr{R}_{v}$ is Baire generic in
$\Sigma_{\kappa}$ (it is a countable intersection of Baire generic subsets of
$\Sigma_{\kappa}$) which satisfies the requirements of the theorem.

######  Remark 10.5.

In the special context of $SL(2,\mathbb{R})$-cocycles satisfying the
assumptions of Theorem A, the top Lyapunov exponent associated to the maximal
entropy measure for the shift is strictly positive (cf. Theorem 4 (f) in [26])
and, consequently, $h_{*}(\varphi_{A})=\log\kappa$ (recall Corollary 10.3(i)).
While Theorem 1.1 in [74] guarantees that the set of points in
$\Sigma_{\kappa}$ for which the top Lyapunov exponent is not well defined have
full topological entropy, our arguments show that there exist a full
topological entropy subset of $\Sigma_{\kappa}$ whose Lyapunov exponents are
not well defined for a dense set of directions in $\mathbb{R}^{2}$.

###  10.3. A dichotomy leading to Lyapunov irregular behavior

Let us prove Theorem B. Assume that $A:\Sigma_{\kappa}\to SL(3,\mathbb{R})$ is
a locally constant hyperbolic cocycle and that its hyperbolic spliting
$\Sigma_{2}\times\mathbb{R}^{3}=E^{s}\oplus E^{u}$ satisfies $\dim E^{s}=2$.
By the $C^{0}$-robustness of uniform hyperbolicity, which can be defined in
terms of cone fields, there exists a $C^{0}$-open neighborhood
$\mathscr{U}\subset C_{\text{loc}}(\Sigma_{\kappa},SL(3,\mathbb{R}))$ formed
by uniformly hyperbolic cocycles. Moreover, in the simpler context of locally
constant cocycles, it is immediate to check that the proof of [6, Theorem 2.8]
guarantees there exists a $C^{0}$-open and dense subset
$\mathscr{V}\subset\mathscr{U}$ such that
$\mathscr{V}=\mathscr{V}_{1}\cup\mathscr{V}_{2}$, each $\mathscr{V}_{i}$ is a
$C^{0}$-open set and either: (i) a cocycle $B\in\mathscr{V}_{1}$ has one-
dimensional finest dominated splitting (ii) for each $B\in\mathscr{V}_{2}$
there exist two periodic points
${\underline{\omega}},\tilde{\underline{\omega}}\in\Sigma_{\kappa}$ of periods
$p,\tilde{p}\geqslant 1$, respectively, such that
$B^{p}(\tilde{\underline{\omega}})\mid_{E^{s}_{B,\tilde{\underline{\omega}}}}$
has two real eigenvalues and
$B^{p}({\underline{\omega}})\mid_{E^{s}_{B,{\underline{\omega}}}}$ has a
complex eigenvalue (notice that the $C^{1}$-topology in the case of
diffeomorphisms in [6] translates as $C^{0}$-topology for the linear
derivative cocycles $Df$). The hyperbolic subspace
${E^{s}_{B,{\underline{\omega}}}}$ varies continuously with the cocycle $B$
and the restriction of $B^{p}$ to that subspace is completely determined by
its complex eigenvalue (up to conjugacy). More precisely, for each
$B\in\mathscr{V}_{2}$ there exists $M_{B}\in GL(3,\mathbb{R})$ so that
$M_{B}\mid_{\mathbb{R}^{2}\times\\{0\\}}=E^{s}_{B,{\underline{\omega}}}$ and

$(M_{B}^{-1}BM_{B})\mid_{\mathbb{R}^{2}\times\\{0\\}}=\lambda_{B}\begin{pmatrix}\cos\theta_{B}&-\sin\theta_{B}\\\
\sin\theta_{B}&\cos\theta_{B}\end{pmatrix},$

where $|\lambda_{B}|<1$ and $\theta_{B}\neq 0$. It makes sense to consider
submersion

$\varrho:\mathscr{V}_{2}\to\mathbb{R}\quad\text{given by}\quad
B\mapsto\rho(B^{p}({\underline{\omega}})\mid_{E^{s}_{B,{\underline{\omega}}}}):=\theta_{B},$

where $\rho(B^{p}({\underline{\omega}})\mid_{E^{s}_{\underline{\omega}}})$
denotes the rotation number of the projectivization of
$B^{p}({\underline{\omega}})\mid_{E^{s}_{\underline{\omega}}}$ on
$\mathbf{P}{E^{s}_{\underline{\omega}}}\simeq\mathbf{S}^{1}$. In particular,
$\\{B\in\mathscr{V}_{2}\colon\rho(B^{p}({\underline{\omega}})\mid_{E^{s}_{\underline{\omega}}})\in\mathbb{Q}\\}$
is a meager subset of $\mathscr{V}_{2}$. As $B\in SL(3,\mathbb{R})$, the
eigenvalue $\lambda_{B}$ is uniquely determined by the real eigenvalue on the
unstable subbundle. Hence, up to conjugacy the matrix
$B\mid_{E^{s}_{B,{\underline{\omega}}}}$ is completely determined by
$\theta_{B}$ and, consequently,

$\\{B\in\mathscr{V}_{2}\colon\rho(B^{p}({\underline{\omega}})\mid_{E^{s}_{B,{\underline{\omega}}}})\in\mathbb{Q}\\}$

is a zero Haar measure subset of $\mathscr{V}_{2}$. In particular, there
exists a $C^{0}$-Baire residual and full Haar measure subset
$\mathscr{R}\subset\mathscr{U}$ so that for every $B\in\mathscr{R}$ either:

  1. (1)

there are two periodic points for $\sigma$ whose projective matrices
correspond to a irrational rotation and a Morse-Smale circle diffeomorphism,
or

  2. (2)

has a finest dominated splitting in three one-dimensional subbundles.

Up to a continuous change of coordinates we may assume that the stable bundle
$E^{s}_{B,{\underline{\omega}}}$ does not depend on $\omega$, meaning that it
coincides with a single eigenspace $E^{s}_{B}$. In the first case the cocycle
has strong projective accessibility and by the last statement of Theorem A,
making use of the previous identification of the subbundle $E^{s}$ with a
single subspace $\mathbb{E}_{B}^{s}$, there exists a dense subset
$\mathscr{D}\subset\mathbf{P}\mathbb{E}^{s}$ and a Baire residual subset
$\mathscr{R}\subset\Sigma_{\kappa}$ so that

$\liminf_{n\to\infty}\frac{1}{n}\log\|B^{n}({\underline{\omega}})v\|<\limsup_{n\to\infty}\frac{1}{n}\log\|B^{n}({\underline{\omega}})v\|$

holds for every ${\underline{\omega}}\in\mathscr{R}$ and every
$v\in\mathscr{D}$, as claimed. This completes the proof of the theorem.

###  10.4. The absence of Lyapunov irregular behavior is rare

In the case of locally constant cocycles one can actually show that the
absence of a large set of points with Lyapunov irregular behavior leads to a
strong rigidity of the cocycle, namely it is cohomologous to a constant matrix
cocycle. Recall that two cocycles $A,B:\Sigma_{\kappa}\to\mathscr{G}$ over the
shift $\sigma$ are _cohomologous_ if there exists a change of variables
$C:\Sigma_{\kappa}\to\mathscr{G}$ so that

$A({\underline{\omega}})=C(\sigma({\underline{\omega}}))B({\underline{\omega}})C({\underline{\omega}})^{-1}\quad\text{for
every }{\underline{\omega}}\in\Sigma_{\kappa}.$

We are interested in describing geometric conditions under which the dichotomy
in the context of Theorem B is mutually exclusive. We observe that, in
general, periodic points having the same eigenvalues is a condition
substantially weaker than saying that the cocycle is cohomologous to a
constant cocycle (see e.g. [84]). In other words, the abelian and non-abelian
Livšic theorems differ substantially. In the three-dimensional context of
Theorem B, due to the diagonalization and absence of nilpotent terms this
turns out to be the case. More precisely:

######  Proposition 10.6.

Let $\mathscr{R}\subset\mathscr{U}$ be the Baire residual subset given by
Theorem B and let $A\in\mathscr{R}$ be such that it admits a finest dominated
splitting in three one-dimensional subbundles. The following are equivalent:

  1. (1)

the set of Lyapunov non-typical points is empty,

  2. (2)

all periodic points
${\underline{\omega}},\widetilde{{\underline{\omega}}}\in\Sigma_{\kappa}$ have
the same Lyapunov spectrum,

  3. (3)

every cocycle $A\in\mathscr{R}$ is cohomologous to a constant cocycle with
simple Lyapunov spectrum.

If any of the three properties above fails then the set of Lyapunov non-
typical points is a Baire generic subset of $\Sigma_{\kappa}$.

###### Proof.

Let $A\in\mathscr{R}$ be as above. We prove the implications
$(1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(1)$ separately.

$(1)\Rightarrow(2)$

As $A\in\mathscr{R}$ admits a finest dominated splitting in three one-
dimensional subbundles $T\mathbb{R}^{3}=E^{u}\oplus E^{s,1}\oplus E^{s,2}$,
each of the subbundles is invariant by the action of the cocycle and varies
Hölder continuously with the base point. This makes the cocycles
diagonalizable. Indeed, for each ${\underline{\omega}}\in\Sigma_{\kappa}$
there exists a base
$\mathcal{B}_{1}=(v_{1}({\underline{\omega}}),v_{2}({\underline{\omega}}),v_{3}({\underline{\omega}}))$
of $\mathbb{R}^{3}$, varying Hölder continuously with ${\underline{\omega}}$,
so that the change of basis matrix $C({\underline{\omega}})$ between the
canonical basis and $\mathcal{B}_{1}$ is such that the continuous cocycle
$\tilde{A}({\underline{\omega}}):=C(\sigma({\underline{\omega}}))^{-1}\,A({\underline{\omega}})\,C({\underline{\omega}})$
preserves the canonical axes, hence

$\tilde{A}({\underline{\omega}})=\begin{pmatrix}a_{1}({\underline{\omega}})&0&0\\\
0&a_{2}({\underline{\omega}})&0\\\
0&0&a_{3}({\underline{\omega}})\end{pmatrix},\qquad{\underline{\omega}}\in\Sigma_{\kappa}.$
(10.4)

Assume now that there exist periodic points
${\underline{\omega}},\widetilde{{\underline{\omega}}}\in\Sigma_{\kappa}$ with
different Lyapunov spectrum. As the shift $\sigma$ satisfies the specification
property the Birkhoff averages of some of the observables $\log a_{i}(\cdot)$
($i=1,2,3$) differ at those periodic points, then the set of irregular points
is actually Baire generic, hence non-empty (see e.g. [5]).

$(2)\Rightarrow(3)$

As the cocycle $A$ is diagonalizable, we will assume it is conjugate to
$\tilde{A}$ given by (10.4). In this context, the assumption is equivalent to
say that the Birkhoff averages of the continuous observable $\log
a_{i}(\cdot)$ is equal to $\lambda_{i}$ at all periodic points, for $i=1,2,3$.
Then, the classical Livšic theorem for Hölder continuous real valued cocycles
ensures that for each $i=1,2,3$ there exists a Hölder continuous function
$u_{i}:\Sigma_{\kappa}\to\mathbb{R}$ so that $\log
a_{i}=u_{i}-u_{i}\circ\sigma+\lambda_{i}$ or, equivalently,
$a_{i}=e^{-u_{i}\circ\sigma}e^{\lambda_{i}}e^{u_{i}}$. Altogether, taking

$U({\underline{\omega}})=\begin{pmatrix}e^{u_{1}({\underline{\omega}})}&0&0\\\
0&e^{u_{2}({\underline{\omega}})}&0\\\
0&0&e^{u_{3}({\underline{\omega}})}\end{pmatrix}$

we conclude that

$[C(\sigma({\underline{\omega}}))U(\sigma({\underline{\omega}}))]^{-1}\,A({\underline{\omega}})\,[C({\underline{\omega}})U({\underline{\omega}})]=\begin{pmatrix}\lambda_{1}&0&0\\\
0&\lambda_{2}&0\\\ 0&0&\lambda_{3}\end{pmatrix}\qquad$

is a constant matrix for every ${\underline{\omega}}\in\Sigma_{\kappa}$, which
proves that $A$ is cohomologous to a constant cocycle.

$(3)\Rightarrow(1)$

Assume there exists a matrix $B_{0}\in\mathscr{G}$ and a measurable
$C:\Sigma_{\kappa}\to\mathscr{G}$ so that
$A({\underline{\omega}})=C(\sigma({\underline{\omega}}))\,B_{0}\,C({\underline{\omega}})^{-1}$
for every ${\underline{\omega}}\in\Sigma_{\kappa}$. In this case
$A^{(n)}({\underline{\omega}})=C(\sigma^{n}({\underline{\omega}}))B_{0}^{n}C({\underline{\omega}})^{-1}$
for every $n\geqslant 1$, and the Lyapunov exponents of $A$ are well defined
at all points and coincide with the logarithm of the modulus of the
eigenvalues of the matrix $B_{0}$.

Finally, the last statement in the corollary is a consequence of [5]. ∎

###  10.5. Projective accessibility and strong irreducibility

In this short subsection we provide simple examples to relate the assumptions
of Furstenberg theorem (Theorem 2.1) and Theorem A. The first example shows
that the strong projective accessibility property on the cocycle is a
sufficient but not necessary condition for the irregular behavior to be
prevalent in the projective space.

######  Example 10.7.

Consider the following $SL(3,\mathbb{R})$ matrices

$A_{1}=\begin{pmatrix}-3&0&0\\\ 0&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\\
0&\frac{1}{\sqrt{3}}&0\end{pmatrix},\quad A_{2}=\begin{pmatrix}3&0&0\\\
0&\frac{1}{3}\cos\theta&-\frac{1}{3}\sin\theta\\\ \vskip 3.0pt plus 1.0pt
minus
1.0pt0&\frac{1}{3}\sin\theta&\;\frac{1}{3}\cos\theta\end{pmatrix}\quad\text{and}\quad
A_{3}=\begin{pmatrix}-2&0&0\\\ 0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\\
0&\frac{1}{\sqrt{2}}&0\end{pmatrix},$

where $\theta\notin\mathbb{Q}$. Let $A:\Sigma_{2}\to SL(3,\mathbb{R})$ be the
locally constant linear cocycle determined by $A\mid_{[i]}=A_{i}$, $i=1,2$. It
is clear that for every Bernoulli probability measure $\nu$ on $\Sigma_{2}$
one has that $\lambda_{+}(A,\nu)=\log 3>0$ and,

$\lim_{n\to\infty}\frac{1}{n}\log\|A^{n}({\underline{\omega}})v\|=\log
3\,\quad\text{for
\emph{every}}\;{\underline{\omega}}\in\Sigma_{2}\;\text{and}\;v\in\mathbb{R}^{3}\setminus<e_{1}>,$

where $<e_{1}>$ stands for the one dimensional subspace generated by
$e_{1}=(1,0,0)$. In particular, there is no Lyapunov irregular behavior
concerning the top Lyapunov exponent and all vectors there exists an open and
dense subset of $\mathbf{P}\mathbb{R}^{3}$ formed by vectors whose Lyapunov
exponent is well defined. Still, Theorem A guarantees that there exists a
dense subset $\mathscr{D}\subset\mathbf{P}(\\{0\\}\times\mathbb{R}^{2})$ so
that for for each $v\in\mathscr{D}$ there exists a Baire generic subset of
points ${\underline{\omega}}\in\Sigma_{2}$ so that
$\lim_{n\to\infty}\frac{1}{n}\log\|A^{n}({\underline{\omega}})v\|$ does not
exist.

Consider, alternatively, the locally constant cocycle $B:\Sigma_{2}\to
SL(3,\mathbb{R})$ given by $B\mid_{[1]}=A_{1}$ and $B\mid_{[2]}=A_{3}$. In
this case, using that $\mathbb{R}^{3}=<e_{1}>\oplus<e_{1}>^{\perp}$ is a
dominated splitting for the cocycle, it is simple to verify that for each
$v\in\mathbf{P}\mathbb{R}^{3}\setminus<(1,0,0)>$

$\limsup_{n\to\infty}\frac{1}{n}\log\|A^{n}({\underline{\omega}})v\|=\log
3\quad\text{if and only
if}\quad\limsup_{n\to\infty}\frac{1}{n}\\#\\{0\leqslant j\leqslant
n-1\colon\sigma^{j}({\underline{\omega}})\in[1]\\}=1$

and

$\liminf_{n\to\infty}\frac{1}{n}\log\|A^{n}({\underline{\omega}})v\|=\log
2\quad\text{if and only
if}\quad\limsup_{n\to\infty}\frac{1}{n}\\#\\{0\leqslant j\leqslant
n-1\colon\sigma^{j}({\underline{\omega}})\in[1]\\}=0.$

Both properties on the right hand-side above hold for a Baire generic subset
of points in $\Sigma_{2}$ as a consequence of [5, Theorem 3.1].

######  Example 10.8.

Consider the $SL(2,\mathbb{R})$ matrices

$A_{1}=\begin{pmatrix}2&1\\\ 1&1\end{pmatrix}\quad\text{and}\quad
A_{2}=\begin{pmatrix}\cos\theta&-\sin\theta\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\sin\theta&\;\cos\theta\end{pmatrix},$

and take the locally constant cocycles $A,B:\Sigma_{2}\to SL(2,\mathbb{R})$
given by and $A\mid_{[1]}=B\mid_{[1]}=A_{1}$, $A\mid_{[2]}=A_{2}$ and
$B\mid_{[2]}=A_{2}\cdot A_{1}$, for some small $\theta\notin\mathbb{Q}_{+}$.
The cocycle $A$ is projectively accessible as $A_{2}$ induces due a minimal
rotation on $\mathbf{P}\mathbb{R}^{2}$, hence it is strongly irreducible. In
turn, $B\mid_{[1]}$ and $B\mid_{[2]}$ do not preserve any finite union of one-
dimensional subspaces, hence the cocycle $B$ is strongly irreducible.
Moreover, it is clear that $B$ is a hyperbolic cocycle and, consequently, the
projectivizations of these matrices preserve admit a proper attracting set in
$\mathbf{P}\mathbb{R}^{2}$. Thus $B$ is not strongly projectively accessible.

In the next example we derive an application to the context of symplectic
matrices.

######  Example 10.9.

Given a fixed $\kappa\geqslant 1$ consider the closed subset

$\displaystyle\mathscr{U}$
$\displaystyle=\\{(A_{1},\dots,A_{\kappa})\in\text{Sp}(4,\mathbb{R})^{\kappa}\colon\exists
1\leqslant i\leqslant\kappa\;\text{so that}\;\|A_{i}\|=1\\}$
$\displaystyle=\bigcup_{i=i}^{\kappa}\,\\{(A_{1},\dots,A_{\kappa})\in\text{Sp}(4,\mathbb{R})^{\kappa}\colon\|A_{i}\|=1\\},$
(10.5)

which is a finite union of closed smooth submanifolds of the Lie group
$\text{Sp}(4,\mathbb{R})^{\kappa}$, endowed with the Haar measure. We claim
that there exists a Baire generic and full Haar measure subset of
$\kappa$-uples $(A_{1},\dots,A_{\kappa})\in\mathscr{U}$ such that either:

  1. (1)

the closure of the subgroup of $\text{Sp}(4,\mathbb{R})^{\kappa}$ generated by
$\\{A_{1},\dots,A_{\kappa}\\}$ is compact; or

  2. (2)

there exists a Baire residual subset $\mathscr{R}\subset\Sigma_{\kappa}$ and a
dense subset $\mathscr{D}\subset\mathbf{P}\mathbb{R}^{d}$ so that the Lyapunov
exponents are not well defined along the direction determined by
$v\in\mathscr{D}$, for every ${\underline{\omega}}\in\mathscr{R}$.

Using the decomposition (10.5), it is enough to prove that the previous
dichotomy holds for a Baire generic and full Haar measure subset of
$\\{(A_{1},\dots,A_{\kappa})\in\text{Sp}(4,\mathbb{R})^{\kappa}\colon\|A_{i}\|=1\\},$
for every $1\leqslant i\leqslant\kappa$. Every $4\times 4$ symplectic matrix
of norm one is parameterized in a one to one way (up to conjugacy) by their
eigenvalues $e^{\pm\theta_{1}{\bf i}}=a\pm{\bf i}\,b$ and
$e^{\pm\theta_{2}{\bf i}}=c\pm{\bf i}\,d$ (recall Example 5.8). Hence, as

$\\{(\theta_{1},\theta_{2})\in\mathbb{R}^{2}\colon\exists
m,n\in\mathbb{Z}\;\text{so that}\;(a\pm{\bf i}\,b)^{n}(c\pm{\bf
i}\,d)^{m}=1\\}$

is a meager and zero Lebesgue measure subset of $\mathbb{R}^{2}$ then the set
$\mathcal{S}_{i}$ of $\kappa$-uples
$(A_{1},\dots,A_{\kappa})\in\text{Sp}(4,\mathbb{R})^{\kappa}$ so that the
projectivization of the map $A_{i}$ is a minimal isometry is Baire generic and
has full Lebesgue measure in
$\\{(A_{1},\dots,A_{\kappa})\in\text{Sp}(4,\mathbb{R})^{\kappa}\colon\|A_{i}\|=1\\}.$
For such $\kappa$-uples the projective cocycle is strongly projectively
accessible. In particular either the closure of the subgroup of
$\text{Sp}(4,\mathbb{R})^{\kappa}$ generated by $\\{A_{1},\dots,A_{\kappa}\\}$
is compact or the assumptions of Theorem A are met, proving the dichotomy.

######  Example 10.10.

Let $d\geqslant 2$ and $A:\Sigma_{\kappa}\to\mathrm{SL}(d,\mathbb{R})$ be a
Hölder continuous hyperbolic cocycle admitting a hyperbolic spliting
$\mathbb{R}^{d}=E_{\underline{\omega}}^{s}\oplus E_{\underline{\omega}}^{u}$
for every ${\underline{\omega}}\in\Sigma_{\kappa}$, with $\dim
E_{\underline{\omega}}^{u}=1$ and such that the positive Lyapunov exponent of
all periodic points is a constant $\lambda>1$. The hyperbolic subbundles are
well known to be Hölder continuous [43]. Thus, Livšic’s theorem ensures that
the Hölder continuous observable
$\Sigma_{\kappa}\ni{\underline{\omega}}\mapsto\log\|A\mid_{E^{u}_{\underline{\omega}}}({\underline{\omega}})\|$
is cohomologous to the constant observable $\log\lambda$ by a Hölder
continuous function. In consequence, there exists an open and dense subset of
directions in the projectivized space for which the top Lyapunov exponent is
well defined at all points: for each ${\underline{\omega}}\in\Sigma_{\kappa}$
the following limit exists

$\lim_{n\to+\infty}\frac{1}{n}\log\|A^{(n)}(\underline{\omega})v\|=\log\lambda\quad\text{for
every}\;v\in(\\{{\underline{\omega}}\\}\times\mathbb{R}^{d})\setminus
E^{s}_{{\underline{\omega}}}.$

In consequence, the set of Lyapunov irregular points for the top Lyapunov
exponent is meager. We note that the limit may not exist if one considers
alternatively negative iteration, in which case the iteration all vectors that
do not lie in the one-dimensional sub-bundle $E^{s}_{{\underline{\omega}}}$
approximate the codimension one stable subbundle.

### Acknowledgments

This work was initiated during a postdoctoral position of GF at the Federal
University of Bahia, supported by INCTMat-Brazil. PV was partially supported
by the project ‘New trends in Lyapunov exponents’ (PTDC/MAT-PUR/29126/2017),
by CMUP (UID/MAT/00144/2013), and by Fundação para a Ciência e Tecnologia
(FCT) - Portugal, through the grant CEECIND/03721/2017 of the Stimulus of
Scientific Employment, Individual Support 2017 Call. The authors are grateful
to Z. Buczolich, M. Carvalho, D. Ma, P. Sun and X. Tian for useful comments
and references.

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