2.1 Holomorphic functions in Banach spaces

A function $f \colon U \to N$ is said to be holomorphic at a point $a \in U$ if there exist $r>0$ and an n-linear symmetric continuous map $T_n \colon M \times \cdots \times M \to N$ , for $n \in \mathbb {N}$ ( $T_0$ is a constant), such that

$$ \begin{align*} f(x) = \sum_{n=0}^\infty T_n(x-a)^n, \end{align*} $$

for every $x \in B(a,r) \subset U$ , where $T_n y^n$ denotes $T_n (y,y,\ldots ,y)$ . If f is holomorphic at every point of U, then f is said to be holomorphic on U.

If M and N are Banach spaces over $\mathbb {R}$ and $f:U \rightarrow N$ satisfies the definition above at the point $a \in U$ , we will say that f is real analytic at a.

We introduce the following notation:

$$ \begin{align*}U(a,b) := \{ z \in \mathbb{C} \colon a+zb \in U \} .\end{align*} $$

Definition 2.1. A map $f \colon U \to N$ is said to be Gâteux holomorphic (or G-holomorphic) if, for every $a \in U$ and for every $b \in M$ , the map

$$ \begin{align*} z \mapsto f(a+zb) \end{align*} $$

is holomorphic on $U(a,b) \subset \mathbb {C}$ .

It is clear that every holomorphic map is also G-holomorphic. However, the converse in general is not true when M is infinite-dimensional. The following theorem provides a criterion for when the converse holds.

The variation of a complex measure $\mu $ is defined as

$$ \begin{align*} |\mu|(E) := \sup_{\pi} \sum_{A \in \pi} |\mu(A)| \end{align*} $$

where the supremum is taken over all partitions $\pi $ of a measurable set E into a countable number of disjoint measurable sets. An important property is that the variation of a complex measure is also a measure, whose values are defined on the set of non-negative extended real numbers.

Another characterization of the variation of a complex measure is the following:

$$ \begin{align*} |\mu|(E) = \sup \bigg \{ \bigg |\!\int_{E} f(g) \, d\mu (g) \bigg | : f \in L^{\infty} (\mu) \text{ and } \|f\|_{\infty} \le 1 \bigg\}. \end{align*} $$

Note that if $f \in L^1 (\mu )$ and E is a measurable set, then

$$ \begin{align*} \bigg | \!\int_E f(x) \, d\mu (x) \bigg | \le \int_E |f(x)| \, d|\mu| (x). \end{align*} $$

Let $\Sigma $ be a compact metric space. The total variation of a complex measure is defined as $\|\mu \| \colon = |\mu |(\Sigma )$ . If a measure satisfies $\|\mu \| < \infty $ , then we say that $\mu $ is finite or that it is of bounded variation. In this case, $|\mu |$ is a finite non-negative real-valued measure.

We will consider $\Sigma $ to be a compact (but possibly infinite) space of symbols. We denote by $\mathcal {M} (\Sigma )$ the set of complex-valued measures over $\Sigma $ with bounded variation. The set $\mathcal {M}(\Sigma )$ , endowed with the total variation norm, will play the role of the Banach space M.

In this paper, we consider slightly more general definitions of holomorphy and G-holomorphy, in which the domain could also be a translation of a Banach subspace. The following construction shows how one can transfer the holomorphic structure from Banach spaces to affine subspaces via translation.

Let $V \subset M$ be a closed subspace, $v_0 \in M$ , and consider a closed affine subspace $V_0 = V + v_0$ of M. Let $U_0 \subset V_0$ be an open set of $V_0$ . We consider a function $f_0: U_0 \rightarrow N$ to be holomorphic (G-holomorphic) at $x_0 \in U_0$ if there exists a function $f\colon U=U_0 - v_0 \rightarrow N$ which is holomorphic (G-holomorphic) at $x_0 - v_0$ , such that $f (x) = f_0 (x+v_0)$ for every $x\in U$ . Moreover, if f is holomorphic (G-holomorphic) at every point of its domain, then so is $f_0$ . We will call $V_0$ an affine Banach space.

It is then immediate that Theorem 2.1 also holds in this context.

2.2 Markov operators and convolution of measures

Next, we present the definition of the convolution of measures in a more general setting for measures on groups.

Definition 2.2. Let G be a topological group that acts on a Polish space S. Let $\mu $ be a measure in G and $\nu $ be a measure in S. Then we define the convolution of $\mu $ and $\nu $ as the measure $\mu *\nu $ on S such that

$$ \begin{align*} (\mu*\nu)(E) = \int_G \int_S \mathrm{1}_E(gx) \, d\nu(x)\, d\mu(g) \end{align*} $$

for every measurable set $E \subset S$ . By standard arguments of measure theory, in the same context,

$$ \begin{align*} \int_S f(x) \, d(\mu*\nu)(x) = \int_G \int_S f(gx) \, d\nu(x)\, d\mu(g) \end{align*} $$

for every measurable bounded function $f:S \rightarrow \mathbb {C}$ .

Given $k\geq 2$ and a measure $\mu \in \mathrm {Prob}(G)$ , we define by

$$ \begin{align*} \mu^{*k} := \mu*\cdots *\mu \quad (k~\text{times)} \end{align*} $$

the kth convolution of $\mu $ with itself.

Let us make three observations about the convolutions of measures. First, the convolution is associative, which makes $\mu ^{*k}$ well defined. Second, it is a known result that the expression for the iterates of the cocycle $A^n (\omega )$ has distribution $\mu ^{*n}$ . Lastly, it should be noted that $\mathrm {supp} \; \mu ^{*n} \not = \mathrm {supp} \; \mu ^{*m}$ for $n \not = m$ in general. Indeed, even if $\mu = \delta _A$ , for $A \in \mathrm {GL}_d (\mathbb {R})$ , we have that $\mu ^{*k} = \delta _{A^k}$ for every $k \in \mathbb {N}$ , which are generally different measures for different values of k. Instead, the convolution tends to ‘spread out’ the support of the measure, making it contain matrices with arbitrarily big and small norms in most cases. In fact, it is clear from the definition of Lyapunov exponents in terms of convolutions that if the support of $\mu ^{*n}$ is bounded from above (that is, there exists $K>0$ such that $\|A\| \le K$ for every $A \in \mathrm {supp} \; \mu ^{*n}$ , for every $n \in \mathbb {N}$ ), then $L_1 (\mu ) \le 0$ . However, even though the support of a measure is not preserved through convolution, another important property holds: if the support of $\mu $ is a compact set in $\mathrm {GL}_d (\mathbb {R})$ , then $\mathrm {supp} \; \mu ^{*n}$ is also a compact set in $\mathrm {GL}_d (\mathbb {R})$ for every $n \in \mathbb {N}$ (moreover, the upper and lower bounds of those sets are related). These observations will be used implicity throughout the text.

Let $\mu \in \mathcal {M}(\Sigma )$ . Consider the operator $Q_\mu \colon L^\infty (\mathbb {P}(\mathbb {R}^d)) \to L^\infty (\mathbb {P}(\mathbb {R}^d))$ associated to $\mu $ as follows:

$$ \begin{align*} Q_\mu(\varphi)(\hat{v}) = \int_{\Sigma} \varphi(\hat{g}\hat{v}) \, d\mu(g), \end{align*} $$

where $\hat {g}\colon \mathbb {P}(\mathbb {R}^d) \to \mathbb {P}(\mathbb {R}^d)$ is the projective action of g defined by $\widehat {gv} =\hat {g}\hat {v}$ and $\mathbb {P}(\mathbb {R}^d)$ denotes the projective space of dimension d over $\mathbb {R}$ . When $\mu \in \mathrm {Prob}(\Sigma )$ , the operator $Q_\mu $ is called the Markov operator. Note that the Markov operator $Q_\mu $ is a positive, bounded, linear operator that preserves constant functions. A general reference for this and related concepts in this section is [Reference Duarte and Klein13].

We may also consider small perturbations of $\mu \in \mathrm {Prob}(\Sigma )$ given by complex measures $\nu \in \mathcal {M}(\Sigma )$ and their associated operators $Q_\nu $ . Although $Q_\mu $ is a Markov operator, $Q_{\nu }$ may not share the same properties. For instance, it does not fix constants when $\nu $ is not a probability measure. However, in this work we show that many results from the theory of Markov operators from [Reference Baraviera and Duarte3, Reference Duarte and Klein12] still hold for their complex analogues.

Moreover, we say that a measure $\eta \in \mathrm {Prob}(\mathbb {P}(\mathbb {R}^d))$ is stationary with respect to $\mu $ , or $\mu $ -stationary, if it satisfies

$$ \begin{align*} \eta(B) = \int \eta(\hat{g}^{-1}(B))\, d\mu(g) \end{align*} $$

for every measurable set $B \subset \mathbb {P}(\mathbb {R}^d)$ . It turns out that a measure $\eta $ is $\mu $ -stationary if, and only if, for every continuous function $\varphi \in C(\mathbb {P}(\mathbb {R}^d))$ ,

$$ \begin{align*} \int_{\mathbb{P}(\mathbb{R}^d)} Q_{\mu}(\varphi)\, d\eta = \int_{\mathbb{P}(\mathbb{R}^d)} \varphi\, d\eta. \end{align*} $$

Lemma 2.1. Let $\mu \in \mathcal {M} (\Sigma )$ be a complex measure with bounded variation. Then

$$ \begin{align*} Q_{\mu^{*n}} = Q^n_{\mu}. \end{align*} $$

Given $\hat {p}, \hat {q} \in \mathbb {P}(\mathbb {R}^d)$ , denote by $\delta \colon \mathbb {P}(\mathbb {R}^d) \times \mathbb {P}(\mathbb {R}^d)\to [0,\infty )$ the projective distance on $\mathbb {P}(\mathbb {R}^d)$ :

(2.1) $$ \begin{align} \delta(\hat{p}, \hat{q}) := \frac{\|p \wedge q\|}{\|p\| \|q\|}. \end{align} $$

Definition 2.3. Let $\mu \in \mathcal {M} (\Sigma )$ be a complex measure of bounded variation and $0< \alpha < 1$ . We define the average $\alpha $ -Hölder constant of the projective action $\hat {g_0} \colon \mathbb {P}(\mathbb {R}^d) \to \mathbb {P}(\mathbb {R}^d)$ by

$$ \begin{align*} k_\alpha(\mu) := \sup_{v_1 \neq v_2} \int_{\Sigma} \bigg(\frac{\delta( \hat{g}_0\hat{v}_1 , \hat{g}_0\hat{v}_2 )}{\delta(\hat{v}_1, \hat{v}_2)} \bigg)^\alpha \, d| \mu |(g_0). \end{align*} $$

Notice that, since $\mathrm {supp} \; \mu \subseteq \Sigma $ , one can remove the domain of integration $\Sigma $ , or choose any other compact set that contains the support of $\mu $ on the definition above, without affecting the value of $k_\alpha (\mu )$ . This fact is important since we wish to understand the behaviour of $k_\alpha (\mu ^{*n})$ as n increases and, as previously stated, $\mathrm {supp} \; \mu ^{*n} \not \subseteq \Sigma $ for large n.

Given $\mu , \nu \in \mathcal {M} (\Sigma )$ , we denote $|\nu | \le |\mu |$ if $| \nu | (E) \le | \mu |(E)$ for every measurable set E.

Proof. Let $\varphi \in L^{\infty } (\mu * \nu )$ and E be a measurable set. Then

$$ \begin{align*} \bigg |\!\int_{E} \varphi(x) \, d \mu*\nu (x) \bigg | &= \bigg | \!\int_{\Sigma} \int_{g^{-1}(E)} \varphi(gx) \, d \nu (x)\, d\mu (g) \bigg | \\ &\le \int_{\Sigma} \int_{g^{-1}(E)} |\varphi(gx)| \, d |\nu| (x) d|\mu| (g) \\ &= \int_{E} |\varphi(x)| \, d |\mu|*|\nu| (x). \end{align*} $$

By imposing the restriction $\|\varphi \|_{\infty } \le 1$ and taking the supremum on both sides, it follows that $|\mu * \nu |(E)\le |\mu |*|\nu | (E)$ . Since E is arbitrary, it follows that $|\mu *\nu | \le |\mu |*|\nu |$ . Moreover, applying the inequality above multiple times with $\nu = \mu $ concludes the result.

As a consequence of the previous lemma, if $\mu $ is a compactly supported measure of bounded variation, then $\mu ^{*n}$ is also a compactly supported measure of bounded variation, for every $n \in \mathbb {N}$ . In particular, the value $k_{\alpha } (\mu ^{*n})$ is well defined for every $n \in \mathbb {N}$ .

Lemma 2.3. The sequence $k_\alpha (\mu ^{*n})$ is sub-multiplicative:

$$ \begin{align*} k_\alpha(\mu^{*(m+n)}) \leq k_\alpha(\mu^{*m})\, k_\alpha(\mu^{*n}). \end{align*} $$

Proof. By a straightforward computation,

$$ \begin{align*} \begin{aligned} k_\alpha&(\mu^{(m+n)}) = \sup_{v_1 \neq v_2} \int \bigg(\frac{\delta( \hat{g}\hat{v}_1 , \hat{g}\hat{v}_2 )}{\delta(\hat{v}_1, \hat{v}_2)} \bigg)^\alpha \, d| \mu^{*(m+n)} |(g) \\&\le \sup_{v_1 \neq v_2} \int \bigg(\frac{\delta( \hat{g}\hat{v}_1 , \hat{g}\hat{v}_2 )}{\delta(\hat{v}_1, \hat{v}_2)} \bigg)^\alpha \, d| \mu^{*n}|*|\mu^{*m} |(g) \\&= \sup_{v_1 \neq v_2} \iint \bigg(\frac{\delta( \hat{g_2} \hat{g_1}\hat{v}_1 , \hat{g_2} \hat{g_1}\hat{v}_2 )}{\delta(\hat{v}_1, \hat{v}_2)} \bigg)^\alpha \, d|\mu^{*m} |(g_1) d |\mu^{*n}| (g_2)\\&= \sup_{v_1 \neq v_2} \iint \bigg(\frac{\delta( \hat{g_2} \hat{g_1}\hat{v}_1 , \hat{g_2} \hat{g_1}\hat{v}_2 )}{\delta(\hat{g_1}\hat{v}_1, \hat{g_1}\hat{v}_2)} \cdot \frac{\delta( \hat{g_1}\hat{v}_1 , \hat{g_1}\hat{v}_2 )}{\delta(\hat{v}_1, \hat{v}_2)} \bigg)^\alpha d|\mu^{*m} |(g_1) d |\mu^{*n}| (g_2)\\&\le \sup_{v_1 \neq v_2} \int \bigg(\frac{\delta( \hat{g_1}\hat{v}_1 , \hat{g_1}\hat{v}_2 )}{\delta(\hat{v}_1, \hat{v}_2)} \bigg)^\alpha k_{\alpha} (\mu^{*n}) d |\mu^{*m}| (g_1) =k_{\alpha} (\mu^{*m}) k_{\alpha} (\mu^{*n}). \end{aligned}\\[-40pt] \end{align*} $$

Given $\varphi \in L^\infty (\mathbb {P}(\mathbb {R}^d))$ and $0 \leq \alpha \leq 1$ , we define

$$ \begin{align*} v_\alpha(\varphi) := \sup_{\hat{v}_1 \neq \hat{v}_2} \frac{| \varphi(\hat{v}_1) - \varphi(\hat{v}_2) |}{\delta( \hat{v}_1 , \hat{v}_2 )^\alpha}. \end{align*} $$

If $v_{\alpha }(\varphi ) < \infty $ then $\varphi $ is $\alpha $ -Hölder continuous. Let $C^\alpha (\mathbb {P}(\mathbb {R}^d))$ be the space of all Hölder continuous functions, which we endowed with its natural norm $\|.\|_\alpha = \|.\|_\infty +v_\alpha (.)$ .

Note that $v_0 (\varphi )= \sup _{\hat {v}_1 \neq \hat {v}_2} | \varphi (\hat {v}_1) - \varphi (\hat {v}_2) | \leq v_\alpha (\varphi )$ for every $\alpha>0$ .

Lemma 2.4. For every $n \geq 1$ , $\mu \in \mathcal {M} (\Sigma )$ and $\varphi \in C^\alpha (\mathbb {P}(\mathbb {R}^d))$ , the following inequality holds:

$$ \begin{align*} v_\alpha(Q_{\mu}^n(\varphi)) \leq k_\alpha(\mu^{*n})\, v_\alpha(\varphi). \end{align*} $$

Proof. Given any $\mu \in \mathcal{M}(\Sigma)$ , consider its associated operator $Q_\mu$ . Then, for every $n \in \mathbb{N}$ , $\hat{v}_1, \hat{v}_2 \in \mathbb{P}(\mathbb{R}^d)$ and $\varphi \in C^\alpha(\mathbb{P}(\mathbb{R}^d))$ , it holds that

$$ \begin{align*} \frac{| Q_{\mu}^n\varphi (\hat{v}_1) - Q_{\mu}^n\varphi (\hat{v}_2) |}{\delta(\hat{v}_1, \hat{v}_2)^\alpha} &= \frac{\big| \! \int \varphi(\hat{g}\hat{v}_1)\, d\mu^{*n}(g) - \int \varphi(\hat{g}\hat{v}_2)\, d\mu^{*n}(g) \big|}{\delta(\hat{v}_1, \hat{v}_2)^\alpha} \\ &= \bigg| \!\int \frac{ \varphi(\hat{g}\hat{v}_1) - \varphi(\hat{g}\hat{v}_2) }{\delta(\hat{v}_1, \hat{v}_2)^\alpha} \, d\mu^{*n}(g) \bigg| \\ & \leq \int \frac{ | \varphi(\hat{g}\hat{v}_1) - \varphi(\hat{g}\hat{v}_2)|}{\delta(\hat{v}_1, \hat{v}_2)^\alpha} \, d | \mu^{*n} | (g) \\ & \leq v_\alpha(\varphi) \int \frac{ \delta( \hat{g}\hat{v}_1 , \hat{g}\hat{v}_2 )^\alpha }{\delta(\hat{v}_1, \hat{v}_2)^\alpha} \, d | \mu^{*n} | (g). \end{align*} $$

We conclude the lemma by applying the supremum in $\hat {v}_1 \neq \hat {v}_2$ to both sides.

3.1 The convergence of the iterates of the Markov operator

A linear subspace $V \subset \mathbb {R}^d$ is called $\mu $ -invariant if $g V = V$ for $\mu $ -almost every. $g \in \mathrm { GL}_d(\mathbb {R})$ . One can restrict the cocycle to the subspace V and consider the induced cocycle, with top Lyapunov exponent $L_1(\mu |_V)$ . We call a probability measure $\mu $ quasi-irreducible if there is no proper $\mu $ -invariant linear subspace $V \subset \mathbb {R}^d$ such that $L_1(\mu |_V) < L_1(\mu )$ .

Note that the quasi-irreducibility of $\mu _0$ implies that Kifer’s non-random filtration [Reference Kifer21, Ch. 3, Theorem 1.2] is trivial, that is, for every $v \in \mathbb {R}^d \backslash \{ 0 \}$ and $\mu _0^{\mathbb {N}}$ -almost every $\{ g_n \}_n \in X$ ,

$$ \begin{align*} \lim_{n\to \infty} \frac{1}{n}\log \|g_{n-1} \cdots g_1g_0v\| = L_1(\mu_0). \end{align*} $$

Moreover, together with the hypothesis that $L_1(\mu _0)> L_2(\mu _0)$ , a consequence of the previous fact is that

(3.1) $$ \begin{align} \lim_{n\to \infty} \frac{1}{n} \int \log \|gv\| d\mu_0^{*n}(g) = L_1(\mu_0), \end{align} $$

with uniform convergence in $\hat {v} \in \mathbb {P}(\mathbb {R}^d)$ . For a proof of this consequence, see [Reference Durães17, Proposition 5.2.2].

Note that, given $g \in \mathrm {GL}(d)$ , $\|\wedge _2 g\| = s_1 (g) s_2 (g)$ , where $s_1(g)$ and $s_2(g)$ are the first and second singular values of a matrix.

Lemma 3.1. For every $\mu \in \mathcal {M} (\Sigma )$ and every $\alpha>0$ ,

$$ \begin{align*} k_\alpha(\mu) \leq \sup_{ {\hat{v} \in \mathbb{P}(\mathbb{R}^d)}} \int_{\Sigma} \bigg({\frac{s_1( {g}) s_2( {g})}{\|{g} {v}\|^2}}\bigg)^\alpha \;d |\mu| (g). \end{align*} $$

Proof. Recall that

$$ \begin{align*} \| gp \wedge gq \| \le s_1(g)s_2(g)\|p \wedge q\|. \end{align*} $$

Hence, by (2.1), given $\alpha> 0$ , two points $\hat {p}, \hat {q} \in \mathbb {P}(\mathbb {R}^d)$ and any $g \in \Sigma $ , we have that

$$ \begin{align*} \bigg[ \frac{\delta(\hat{g} \hat{p}, \hat{g} \hat{q})}{\delta(\hat{p}, \hat{q})} \bigg] ^\alpha &\le \bigg[ (s_1(g)s_2(g))\frac{\|p\| \|q\|}{\|gp\| \|gq\|} \bigg]^\alpha \\ &\leq \frac{[ s_1(g)s_2(g)]^\alpha}{2} \bigg[ \frac{1}{\|gp\|^{2\alpha}} +\frac{1}{\|gq\|^{2\alpha}} \bigg] \end{align*} $$

where in the last inequality we use that the geometric mean is less or equal than the arithmetic mean.

Note that if we integrate with respect to the measure $|\mu |$ and take the supremum in $\hat {p} \neq \hat {q}$ on both sides of this inequality, we conclude the lemma.

Proposition 3.2. Assume that $\mu _0 \in \mathrm {Prob}(\Sigma )$ is quasi-irreducible and $L_1(\mu _0)>L_2(\mu _0)$ . Then there exist $0< \alpha \leq 1$ , $\theta>1$ , $C>0$ and a neighbourhood $V \subset \mathcal {M} (\Sigma )$ of $\mu _0$ with respect to the total variation distance such that, for every $n \in \mathbb {N}$ and for every $\mu \in V$ ,

(3.2) $$ \begin{align} k_\alpha(\mu^{*n}) \leq C \theta^{-n}. \end{align} $$

Proof. We follow the argument in [Reference Duarte and Klein12]. By equation (3.1),

$$ \begin{align*} \lim_{n\to \infty} \frac{1}{n}\int \log \|g \, v\|^{-2} d\mu_0^{*n}(g) = -2 L_1(\mu_0), \end{align*} $$

with uniform convergence in $v\in \mathbb {S}^{d-1}$ .

Hence, choosing $\epsilon>0$ and choosing n sufficiently large, we have that

$$ \begin{align*} \int \log \Vert g \, v \Vert ^{-2} d\mu_0^{*n}(g) \leq n (-2L_1(\mu_0) +\epsilon). \end{align*} $$

Given a matrix $g\in \mathrm {GL}_d(\mathbb {R})$ , let $\wedge _2 g \in \mathrm {GL}_{d \choose 2} (\mathbb {R})$ denote the second exterior power of g. Note that $\|\wedge _2 g\| = s_1 (g) \, s_2 (g)$ .

Then, for n large enough and for all $\omega \in \Sigma $ , we have

$$ \begin{align*} \frac{1}{n}\int \log (s_1(g) \, s_2(g))\, d\mu_0^{*n}(g) &= \frac{1}{n} \, \int \log \|(\wedge_2 \, g)\| d\mu_0^{*n}(g) \\ & \le L_1 (\wedge_2 \, g, \mu_0) + \epsilon = L_1(\mu_0) + L_2(\mu_0) + \epsilon . \end{align*} $$

Choosing $\epsilon = \tfrac 14(L_1(\mu _0) - L_2(\mu _0))>0$ and combining the previous estimates, for all $v\in \mathbb {S}^{d-1}$ , we get

$$ \begin{align*} \int \log \bigg( \frac{s_1( g) s_2( g)}{ \Vert gv \Vert ^{2}} \bigg)\, d\mu_0^{*n}(g) &\le n (L_1(\mu_0) + L_2(\mu_0) + \epsilon) + n \, (-2 L_1 (\mu_0)+\epsilon) \\ & = - n \, (L_1(\mu_0) - L_2(\mu_0)-2\epsilon) = \\ &= - \frac{n}{2} \, (L_1(\mu_0) - L_2(\mu_0)) < -1 , \end{align*} $$

since $L_1(\mu _0)>L_2(\mu _0)$ and provided that n is large enough.

Using the inequality $ \exp x \leq 1+x+({x^2}/{2}) \exp | x |, $ we conclude that for every $v \in \mathbb {S}^{d-1}$ and n large enough,

$$ \begin{align*} & \int\! \bigg({\frac{s_1( g) s_2( g)}{\|{g} {v}\|^2}}\bigg)^\alpha d\mu_0^{*n}(g) = \int \exp \bigg( \alpha \log \frac{ s_1( g) s_2( g)}{ \Vert gv \Vert ^{2}} \bigg)\, d\mu_0^{*n}(g) \\&\quad \le 1+ \int \bigg( \alpha \log \frac{s_1( g) s_2( g)}{ \Vert gv \Vert ^{2}} \bigg)\, d\mu_0^{*n}(g) \\&\qquad + \int \bigg( \frac{\alpha^2}{2}\log^2 \frac{s_1( g) s_2(g)}{ \Vert gv \Vert ^{2}} \exp \bigg| \frac{ \alpha \log s_1( g) s_2( g)}{ \Vert gv \Vert ^{2}} \bigg| \bigg)\, d\mu_0^{*n}(g) \\&\quad \le 1- \alpha +C \frac{\alpha^2}{2} \end{align*} $$

for some finite constant C that depends only on $\mu _0$ and n.

Thus, fixing $n_0$ sufficiently large and considering $\alpha $ small enough, we conclude by Lemma 3.1 that

$$ \begin{align*} k_{\alpha}(\mu_0^{*n_0}) \leq 1 - \alpha +C \frac{\alpha^2}{2} < 1. \end{align*} $$

Note that, for a fixed n, the quantity $\int ({({s_1( g) s_2( g)})/{\|g {v}\|^2}})^\alpha \, d|\mu _0|^{*n}(g)$ , which bounds $k_\alpha (\mu _0^{*n})$ from above, depends continuously on the measure. Then the previous inequality extends to a neighbourhood of $\mu _0$ . There exist $\kappa <1$ and a neighbourhood $V \subset \mathcal {M}(\Sigma )$ of $\mu _0$ with respect to the total variation distance, such that $k_\alpha (\mu ^{*n_0}) \leq \kappa <1$ for every $\mu \in V$ .

Because of the sub-multiplicative property of $k_\alpha $ , we conclude that there exist $C>0$ and $\theta>1$ such that inequality (3.2) holds for every $n \in \mathbb {N}$ .

Corollary 3.3. Assume that $\mu _0 \in \mathrm {Prob}(\Sigma )$ is quasi-irreducible and $L_1(\mu _0)>L_2(\mu _0)$ . Then there exist $0< \alpha \leq 1$ , $\theta>1$ , $C>0$ and a neighbourhood $V \subset \mathcal {M} (\Sigma )$ of $\mu _0$ with respect to the total variation distance, such that, for every $n \in \mathbb {N}$ , for every $\mu \in V$ and every $\varphi \in C^\alpha (\mathbb {P}(\mathbb {R}^d))$ ,

$$ \begin{align*} v_\alpha(Q_{\mu}^n \varphi) \leq C v_\alpha(\varphi) \theta^{-n}. \end{align*} $$

Proof. By Proposition 3.2, there exist $0< \alpha \leq 1$ , $\theta>1$ , $C>0$ and a neighbourhood $V \subset \mathcal {M}(\Sigma )$ of $\mu _0$ with respect to the total variation distance, such that, for every $n \in \mathbb {N}$ and for every $\mu \in V$ , we have that $k_\alpha (\mu ^{*n}) \leq C \theta ^{-n}$ . Together with Lemma 2.4, we conclude that

$$ \begin{align*} v_\alpha(Q_{\mu}^n(\varphi)) \leq k_\alpha(\mu^{*n})\, v_\alpha(\varphi) \leq C v_\alpha(\varphi) \theta^{-n}.\\[-38pt] \end{align*} $$

When $\mu $ is a probability measure on $\Sigma $ , a consequence of Corollary 3.3 is that there exist $\alpha \in (0,1]$ , $\theta>1$ and $C<\infty $ such that, for every $n \in \mathbb {N}$ and every $\varphi \in C^\alpha (\mathbb {P}(\mathbb {R}^d))$ ,

(3.3) $$ \begin{align} \bigg\|Q_{\mu}^n \varphi - \int \varphi \, d\eta_{\mu}\bigg\|_\alpha \leq C\theta^{-n} \|\varphi\|_\alpha, \end{align} $$

where $\eta _\mu $ is the unique $\mu $ -stationary measure on $\mathbb {P}(\mathbb {R}^d)$ . That is because

$$ \begin{align*} \bigg\|\varphi - \int \varphi \, d\eta\bigg\|_{\infty} &= \bigg| \varphi(\hat{v}) - \int \varphi(\hat{p}) \, d\eta(\hat{p}) \bigg| \\ &\leq \int| \varphi(\hat{v}) - \varphi(\hat{p})| \, d\eta(\hat{p}) \leq v_0(\varphi) \leq v_\alpha(\varphi). \end{align*} $$

Therefore, since $\eta _\mu $ is $Q_\mu $ stationary,

$$ \begin{align*} \bigg\|Q^n_\mu \varphi - \int \varphi \;d\eta_\mu\bigg\|_\infty &= \bigg\|Q^n_\mu \varphi - \int Q_\mu^n \varphi \;d\eta_\mu\bigg\|_\infty \\ &\leq v_\alpha(Q^n_\mu \varphi) \leq C v_\alpha (\varphi) \theta^{-n}. \end{align*} $$

Moreover,

$$ \begin{align*} v_\alpha \bigg (Q^n_\mu \varphi - \int Q_\mu^n \varphi \;d\eta_\mu \bigg ) = v_\alpha(Q^n_\mu \varphi) \leq C\theta^{-n} \|\varphi\|_\alpha, \end{align*} $$

which concludes the argument.

It follows from inequality (3.3) that $\eta _\mu $ is the unique $\mu $ -stationary measure of this cocycle.

Consider the observable $\varphi \colon \mathbb {P}(\mathbb {R}^d) \to \mathbb {R}$ given by

(3.4) $$ \begin{align} \varphi(\hat{v}) = \int_{\Sigma} \log \frac{\|gv\|}{\|v\|} \, d\mu(g). \end{align} $$

If $\mu $ is a probability measure, then, by Furstenberg’s formula,

(3.5) $$ \begin{align} \int_{\mathbb{P}(\mathbb{R}^d)} \varphi(\hat{v}) \, d\eta_{\mu} = L_1(\mu). \end{align} $$

3.2 The domain of holomorphy

We now establish a holomorphic extension of the Lyapunov exponent $L_1$ . We approach this by showing that there exists an holomorphic function which depends on the measure $\mu $ and coincides with the Lyapunov exponent when $\mu $ is a probability. In particular, this fact guarantees that, under the hypothesis of the Theorem 1.1, the usual Lyapunov exponent is real analytic, which is a property that does not depend on the extension chosen.

We start by defining the domain where $L_1$ will be shown to be analytic.

Let $\mathcal {M}_0(\Sigma )$ be the set of finite complex measures that give measure $0$ to $\Sigma $ . Therefore, every $\mu \in \mathcal {M}_0(\Sigma )$ satisfies $\mu (\Sigma )=0$ .

Let $\mathcal {M}_1(\Sigma )$ denote the set of finite complex measures that give measure $1$ to $\Sigma $ . Note that $\mathcal {M}_1(\Sigma )$ is an affine Banach space of $\mathcal {M}(\Sigma )$ , namely $\mathcal {M}_1(\Sigma )= \mathcal {M}_0(\Sigma ) + \mu _0$ , for some $\mu _0 \in \mathrm {Prob}(\Sigma )$ . Therefore, $\mathcal {M}_1(\Sigma )$ can be endowed with an analytic structure as seen in §2.1.

We are going to prove that $L_1$ admits a holomorphic extension to $V \cap \mathcal {M}_1(\Sigma )$ , where V is the neighbourhood of $\mu _0$ from Proposition 3.2. In fact, all the proofs from the previous sections were done in $\mathcal {M}(\Sigma )$ , but could have been done directly in $\mathcal {M}_1(\Sigma )$ . Thus, from now on, we are going to consider the neighbourhood V to be in $\mathcal {M}_1(\Sigma )$ and we will write just V instead of $V \cap \mathcal {M}_1(\Sigma )$ .

Lemma 3.5. For every $\mu \in V \subset \mathcal {M}_1(\Sigma )$ , $\hat {v_1} \neq \hat {v_2} \in \mathbb {P}(\mathbb {R}^d)$ and $\varphi \in C^\alpha (\mathbb {P}(\mathbb {R}^d))$ , we have that

(3.6) $$ \begin{align} | Q^n_{\mu}\varphi(\hat{v}_1) - Q^n_{\mu}\varphi(\hat{v}_2) | \leq C v_\alpha (\varphi) \theta^{-n}. \end{align} $$

Proof. By Lemma 2.4, for every $\mu \in \mathcal {M}(\Sigma )$ and every $\varphi \in C^\alpha (\mathbb {P}(\mathbb {R}^d))$ , we have $v_\alpha (Q^n_{\mu }\varphi ) \leq v_\alpha (\varphi )k_\alpha (\mu ^{*n})$ . Therefore, we also have that, for every $\hat {v}_1 \neq \hat {v}_2 \in \mathbb {P}(\mathbb {R}^d)$ ,

$$ \begin{align*} | Q^n_{\mu}\varphi(\hat{v}_1) - Q^n_{\mu}\varphi(\hat{v}_2) | \leq v_\alpha(\varphi)k_\alpha(\mu^{*n}). \end{align*} $$

Then, by Proposition 3.2, we conclude the proof.

Proposition 3.6. For every $\mu \in V \subset \mathcal {M}_1(\Sigma ),\hat {v_1} \neq \hat {v_2} \in \mathbb {P}(\mathbb {R}^d)$ and $\varphi \in C^{\alpha }(\mathbb {P}(\mathbb {R}^d))$ ,

$$ \begin{align*} | Q^{n+1}_{\mu}\varphi(\hat{v}_1) - Q^n_{\mu}\varphi(\hat{v}_2) | \leq C v_\alpha (\varphi) \theta^{-n} \|\mu\|. \end{align*} $$

Proof. Note that, for every $\hat {v}_1 \neq \hat {v}_2 \in \mathbb {P}(\mathbb {R}^d)$ ,

$$ \begin{align*} | Q^{n+1}_{\mu} \varphi (\hat{v}_1) - Q^{n}_{\mu} \varphi (\hat{v}_2) | = \bigg |\! \int_\Sigma Q^n_{\mu}\varphi (\hat{g} \hat{v}_1) \, d\mu - Q^{n}_{\mu} \varphi (\hat{v}_2) \bigg|. \end{align*} $$

Moreover, for every $\mu \in V \subset \mathcal {M}_1(\Sigma )$ and $\hat {v_1} \neq \hat {v_2} \in \mathbb {P}(\mathbb {R}^d)$ ,

$$ \begin{align*} \begin{aligned} &\bigg |\! \int_\Sigma Q^n_{\mu}\varphi(\hat{g}\hat{v}_1) \, d\mu - Q^{n}_{\mu} \varphi (\hat{v}_2) \bigg | = \bigg |\! \int_\Sigma Q^n_{\mu}\varphi(\hat{g} \hat{v}_1) - Q^{n}_{\mu} \varphi (\hat{v}_2) \, d\mu \bigg | \\ &\quad\leq \int_\Sigma | Q^n_{\mu}\varphi(\hat{g} \hat{v}_1) - Q^{n}_{\mu} \varphi (\hat{v}_2) | \, d |\mu | \leq Cv_\alpha (\varphi)\theta^{-n} \|\mu\|. \end{aligned}\\[-38pt] \end{align*} $$

Note that, for a fixed $\hat {v}$ and $\varphi $ given in equation (3.4), the sequence $\{ Q_\mu ^n \varphi (\hat {v}) \}_n$ is Cauchy. Therefore, its limit, denoted by $Q_\mu ^\infty \varphi (\hat {v})$ , exists. Also, by Lemma 3.5, this value does not depend on $\hat {v}$ . As a consequence, after fixing $\varphi $ , the limit $Q_{\mu }^{\infty } \varphi (\hat {v})$ can be seen as a complex-valued function depending only on $\mu $ . Moreover, note that $\mu \mapsto Q_\mu ^n \varphi (\hat {v}) $ is continuous. Since $\mu \mapsto Q_\mu ^\infty \varphi (\hat {v})$ is a uniform limit of continuous functions, it is also continuous. Furthermore, when $\mu $ is a probability measure, $Q_\mu ^\infty \varphi (\hat {v}) = L_1(\mu )$ , the top Lyapunov exponent (as shown in Remark 3.1).

It is worth observing that, if we fix the measure $\mu $ and a direction $\hat {v}$ , the expression $T_\mu \colon C^{\alpha } (\mathbb {P}(\mathbb {R}^d)) \rightarrow \mathbb {C}$ given by $T_\mu (\varphi ) = Q_\mu ^{\infty } \varphi (\hat {v})$ defines a continuous linear functional (which does not depend on $\hat {v}$ ). Therefore, by Riesz’s theorem, there exists a unique complex measure $\eta _\mu $ in $C^{\alpha } (\mathbb {P}(\mathbb {R}^d))$ such that

$$ \begin{align*} Q_\mu^{\infty} \varphi = \int \varphi \, d\eta_{\mu} \end{align*} $$

for every $\varphi \in C^{\alpha } (\mathbb {P}(\mathbb {R}^d))$ . Moreover, if $\mu $ is a probability, then $\eta _{\mu }$ is a stationary measure for $\mu $ .

We want to prove that $\mu \mapsto Q_\mu ^\infty \varphi (\hat {v})$ is holomorphic. For this, we are going to use Theorem 2.1. Since we already know that the limit is continuous, it suffices to prove that it is also G-holomorphic.

As stated, $\mathcal {M}_1(\Sigma )$ is not a proper Banach space, therefore we need to transfer the holomorphic structure from $\mathcal {M}_0(\Sigma )$ to it. Intuitively, G-holomorphy means being holomorphic along complex lines, hence to say that the map $\mu \mapsto Q_\mu ^\infty \varphi (\hat {v})$ from $ V \subset \mathcal {M}_1(\Sigma )$ to $\mathbb {C}$ is Gâteaux holomorphic means that, for all $\mu \in V$ , for all $\nu \in \mathcal {M}_0(\Sigma )$ , the map $z \mapsto Q_{\mu + z \nu }^\infty \varphi (\hat {v})$ is holomorphic on $V(\mu , \nu )=\{ z \in \mathbb {C} \colon \mu + z\nu \in V \}$ .

Consider measures $\mu _z$ of the form $\mu _z= \mu + z\nu $ , where $\mu \in V$ and $\nu $ is any finite complex measure with $\nu (\Sigma )=0$ . Note that, since $\mu \in \mathcal {M}_1(\Sigma )$ , we have that $\mu _z \in \mathcal {M}_1(\Sigma )$ for every $z \in \mathbb {C}$ . Consider small perturbations of the Markov operator in the following sense: for each $z \in \mathbb {C}$ , let the operator $Q_{\mu + z\nu } \colon L^\infty (\mathbb {P}(\mathbb {R}^d), \mathbb {C}) \to L^\infty (\mathbb {P}(\mathbb {R}^d), \mathbb {C})$ be defined by

$$ \begin{align*} Q_{\mu + z\nu}(\varphi)(\hat{v}) &= \int_\Sigma \varphi(\hat{g}\hat{v})\, d(\mu + z \nu)(g) \\ &= \int_\Sigma \varphi(\hat{g}\hat{v})\, d(\mu) + z\int_\Sigma \varphi(\hat{g}\hat{v})\, d(\nu) \\ &=Q_{\mu}(\varphi)(\hat{v})+zQ_{\nu}(\varphi)(\hat{v}). \end{align*} $$

Note that, for a fixed vector $\hat {v}$ , each $Q_{\mu _z}^n(\varphi )(\hat {v})$ is a polynomial of degree smaller than or equal to n; in particular, the map $z \mapsto Q_{\mu _z}^n(\varphi )(\hat {v})$ is holomorphic for $\mu _z \in V$ . Therefore, for every $z \in V(\mu , \nu )$ , the limit function is a uniform limit of holomorphic functions, hence $z \mapsto Q_{\mu _z}^\infty \varphi (\hat {v})$ is holomorphic. In other words, the Lyapunov exponent is G-holomorphic in the neighbourhood $V \subset \mathcal {M}_1(\Sigma )$ of $\mu _0$ . Together with the continuity, we conclude that it is indeed holomorphic. This concludes the proof of item $(1)$ of Theorem 1.1.

We now prove item $(2)$ of Theorem 1.1. We drop the assumption of irreducibility of $\mu _0$ and instead we assume that $\mathrm {supp} \, \mu _0 = \Sigma $ . Let W be a non-trivial vector subspace of $\mathbb {R}^d$ that is invariant for $\mu _0$ -almost every matrix g. The measure $\mu _0$ defines the measures $ \mu _{0,W}$ and $\mu _{0,\mathbb {R}^d / W}$ in $\mathrm {GL}_d(W)$ and $\mathrm {GL}_d(\mathbb {R}^d / W)$ . Moreover, they induce the linear cocycles restricted to W and to $\mathbb {R}^d /W$ , with Lyapunov exponents $L_1(\mu _{0,W})$ and $L_1(\mu _{0,\mathbb {R}^d /W})$ .

By Lemma $3.6$ of [Reference Furstenberg and Kifer18], $L_1(\mu _0) = \max \{ L_1(\mu _{0,W}), L_1(\mu _{0, \mathbb {R}^d / W}) \}$ . Without loss of generality we may suppose that $L_1(\mu _0) = L_1(\mu _{0,W})$ . The other case is similar. We claim that, in fact, $L_1(\mu ) = L_1(\mu _W)$ for every $\mu $ in a neighbourhood of $\mu _0$ .

First note that $\mathrm {supp} \, \mu _0 = \Sigma $ implies that $gW =W \; \text { for all } g \in \Sigma $ . Therefore, every $\mu \in \mathrm {Prob}(\Sigma )$ also satisfies $gW=W$ for $\mu $ -almost every g. Hence, we have that $L_1(\mu ) = \max \{ L_1(\mu _W), L_1(\mu _{\mathbb {R}^d / W}) \}$ . Moreover, by Corollary B of [Reference Peres25], if $\mu _n \to \mu _0$ in the weak-star topology and $\mathrm {supp} \, \mu _n \subset \mathrm {supp} \, \mu _0$ for every n, then $L_1(\mu _n) \to L_1(\mu _0)$ .

Since $\mathrm {supp} \, \mu _W \subseteq \mathrm {supp} \, \mu _{0,W}$ and $\mathrm {supp} \, \mu _{\mathbb {R}^d / W} \subseteq \mathrm {supp} \, \mu _{0,\mathbb {R}^d /W}$ , we may use the same corollary to each induced cocycle to conclude that both $\mu \mapsto L_1(\mu _{W})$ and $\mu \mapsto L_1(\mu _{\mathbb {R}^d / W})$ are continuous maps. By assumption, $L_1(\mu _0) = L_1(\mu _{0,W})> L_1(\mu _{0, \mathbb {R}^d /W})$ . Thus, $L_1(\mu ) = L_1(\mu _{W})> L_1(\mu _{\mathbb {R}^d /W})$ for every $\mu $ sufficiently close to $\mu _0.$ This concludes the proof of the claim.

Note that the Lyapunov spectra of both $\mu _{0,W}$ and $\mu _{0,\mathbb {R}^d / W}$ are contained in the Lyapunov spectrum of $\mu _0$ . Since $L_1(\mu _0) $ is simple, the same holds for $L_1(\mu _{0,W})$ . Hence, if $\mu _{0,W}$ is irreducible, then the map $\mu \mapsto L_1(\mu _W)$ is holomorphic. Since $L_1(\mu ) = L_1(\mu _W)$ in a neighbourhood of $\mu _0$ , we conclude that $\mu \mapsto L_1(\mu )$ is also holomorphic in a neighbourhood of $\mu _0$ .

If $ \mu _{0,W}$ is not irreducible, there exists another non-trivial invariant subspace $W' \subset W$ . The measure $\mu _{0,V}$ defines measures $\mu _{0,W'}$ and $\mu _{0,W / W'}$ . Then we carry out the same procedure. Since the invariant subspaces are of decreasing dimension, this process must stop after a finite number of steps. Therefore, we conclude the proof of Theorem 1.1.

3.3 Corollaries and remarks

Let $\Sigma $ be an abstract compact space, $X = \Sigma ^{\mathbb {N}}$ , and $\sigma \colon X \to X$ be the forward shift on X. We fix a measurable and bounded function $A\colon \Sigma \to \mathrm {GL}_d(\mathbb {R})$ and denote also by A the locally constant (fiber) map $A \colon X \to \mathrm {GL}_d(\mathbb {R})$ given by $A((x_n)_{n \in \mathbb {N}})=A(x_0)$ .

Given $\mu \in \mathrm {Prob} (\Sigma )$ , let $\mu ^{\mathbb {N}}$ be the product (Bernoulli) measure on X. A random (Bernoulli) locally constant linear cocycle $F_A \colon X \times \mathbb {R}^d \to X \times \mathbb {R}^d$ relative to the product measure $\mu ^{\mathbb {N}}$ is a skew product transformation such that

$$ \begin{align*} F_A(\omega,v) = (\sigma(\omega),A(x)v). \end{align*} $$

Its iterates are given by

$$ \begin{align*} F_A^n(\omega,v)=(\sigma^n(\omega),A^n(\omega)v), \end{align*} $$

where $A^n(\omega ):=A(\omega _{n-1}) \cdots A(\omega _1)A(\omega _0)$ .

A seminal result from Furstenberg and Kesten states that under the integrability condition $\log ^+\|A^{\pm }\| \in L^1(\mu )$ , the limit

$$ \begin{align*} L_1(A, \mu) = \lim_n \frac{1}{n}\log \|A^n(\omega)\| \end{align*} $$

exists $\mu $ -almost everywhere and it is called the top Lyapunov exponent of this cocycle. Consider the pushforward measure on $\mathrm {Prob}(\mathrm {GL}_d(\mathbb {R}))$ given by $A_*\mu $ . By the boundedness of A, the support of $A_*\mu $ remains compact, and therefore its Lyapunov exponent is well defined. A straightforward computation shows that the Lyapunov exponent $L_1 (A_*\mu )$ associated to the measure $A_*\mu $ is equal to $L_1 (A, \mu )$ . Moreover, the application $A_* : \mathcal {M} (\Sigma ) \rightarrow \mathcal {M} (\mathrm {GL}_d (\mathbb {R}))$ is a linear continuous (and, therefore, analytic) mapping that preserves probabilities. Since the composition on analytic maps is analytic, it follows by Theorem 1.1 that the map $L_1 (A, \mu )= L_1 (A_* \mu )$ is analytic with respect to $\mu $ , which guarantees that the result holds for arbitrary locally constant linear cocycles.

A second corollary is an analogue of the finite case, in which the support of the measure is a fixed compact set on $\mathrm {GL}_d (\mathbb {R})$ and we look at the dependence on the probability weights. Let $\mu _0 \in \mathrm {Prob}(\Sigma )$ be a reference measure of full support. We restrict to the measures in $\Sigma $ which are absolutely continuous with respect to $\mu _0$ .

By the Radon–Nikodym theorem, this space is identified with the space $L^1 (\mu _0)$ of integrable complex functions with respect to $\mu _0$ through the map $I: L^1 (\mu _0) \rightarrow \mathcal {M} (\Sigma )$ given by

$$ \begin{align*} I(f)(E)= \int_E f(x) \,d \mu_0 (x) \end{align*} $$

for every measurable set E. The map I is an isomorphism between $L^1 (\mu _0)$ and the measures on $\mathcal {M} (\Sigma )$ which are absolutely continuous with respect to $\mu _0$ . Observe that, given $f,g \in L^1 (\mu _0)$ , it follows that

$$ \begin{align*} \|I(f)-I(g)\|_{\mathrm{TV}} \le \|f-g\|_{1} \le \|f-g\|_p, \end{align*} $$

where $\|.\|_{\mathrm {TV}}$ denotes the total variation norm, $\|.\|_1$ denotes the $L^1$ norm and $\|.\|_p$ denotes the $L^p$ norm, with $p \in [1,+\infty ]$ . This fact guarantees that, given $r>0$ , it follows that $B_p(f, r) \subset B_1 (f,r) \subset B_{\mathrm {TV}}(I(f), r)$ , where each of the previous sets denotes an open ball on its respective norm.

This observation, aligned with the Theorem 1.1, proves the following corollary.

Corollary 3.7. Let $\mu _0 \in \mathrm {Prob} (\Sigma )$ have full support and assume that $L_1(\mu _0)> L_2(\mu _0)$ . For $p \in [1,+\infty ]$ , define

$$ \begin{align*} L^p_1 (\mu_0):= \bigg \{ f:\Omega \rightarrow \mathbb{R} : f \in L^p (\mu_0) \text{ and } \int f (x)\, d\mu_0 (x)=1 \bigg \}. \end{align*} $$

Then the Lyapunov exponent $L_1: L^p_1 (\mu _0) \rightarrow \mathbb {R}$ is a real analytic function of f with respect to the $L^p$ norm in a neighbourhood of the constant function $1$ .

We now look at the set in which $L_1$ is analytical. We say that a measure $\mu \in \mathcal {M} (\mathrm {GL}_d (\mathbb {R}))$ is irreducible if there is no proper subspace $V \subset \mathrm {GL}_d (\mathbb {R})$ such that $gV = V$ for $\mu $ -almost every g. Notice that every irreducible measure is quasi-irreducible.

Observe that irreducibility is a dense property with respect to the total variation norm. Indeed, let $\mu _0$ be a probability in $\mathrm {GL}_d(\mathbb {R})$ , and let $\nu $ be another probability in $\mathrm {GL}_d(\mathbb {R})$ , with compact support. If $\nu $ is irreducible and given $\varepsilon> 0$ , then $\mu _{\varepsilon } = (1-\varepsilon )\mu + \varepsilon \nu $ is an irreducible probability in $\mathrm {GL}_d(\mathbb {R})$ . To see this, let V be a proper subspace of $\mathrm {GL}_d(\mathbb {R})$ . Since $\nu $ is irreducible, there exists a Borel set $B \subset \mathrm { GL}_d(\mathbb {R})$ such that $\nu (B)>0$ and $gV \not = V$ for every $g \in B$ . Notice that $\mu _{\varepsilon } (B) = (1-\varepsilon ) \mu (B) + \varepsilon \nu (B) \ge \varepsilon \nu (B)> 0$ , so it follows that $\mu _{\varepsilon }$ is irreducible.

Notice also that $\|\mu _{\varepsilon } - \mu \| = \|\varepsilon \nu - \varepsilon \mu \| \le 2 \varepsilon $ , so we can choose $\varepsilon $ sufficiently small such that $\mu _{\varepsilon }$ is arbitrarely close to $\mu $ . Moreover, $\mathrm {supp} \; \mu _{\epsilon } = \mathrm {supp} \; \mu \cup \mathrm {supp} \; \nu $ , so if $\mu $ has compact support, then $\mu _{\epsilon }$ also has compact support, and if $\mathrm {supp} \; \mu , \mathrm {supp} \; \nu \subset \Sigma $ , then $\mathrm {supp} \; \mu _{\epsilon } \subset \Sigma $ .

We also observe that, in [Reference Kifer20], Kifer proved that being irreducible was an open property on $\mathrm {Prob}(\mathrm {GL}_d(\mathbb {R}))$ with respect to the weak-star topology. Since the total variation norm generates a finer topology than the weak-star topology, it follows that being irreducible is also an open property with respect to the total variation norm. Therefore, by Theorem 1.1, we can conclude that $L_1$ is analytic on the set of compactly supported irreducible measures on $\mathrm {GL}_d (\mathbb {R})$ , which is a dense open set on the space $\mathrm {Prob} (\mathrm {GL}_d(\mathbb {R}))$ with respect to the total variation norm.

To conclude this section, we make the observation that the restriction of the probabilities to a compact set $\Sigma $ in Theorem 1.1 cannot be removed. Indeed, let $\mu \in \mathrm {Prob}(\mathrm {GL}(d))$ be a compactly supported measure with $L_1 (\mu )> L_2 (\mu )$ . Let $a, \varepsilon>0$ and consider the measure $\mu _{a,\varepsilon }:=(1-\varepsilon ) \mu + \varepsilon \delta _{aI}$ , where I is the identity matrix. By a previous comment, the measure $\mu _{a, \varepsilon }$ is a compactly supported measure such that $\|\mu _{a,\varepsilon } - \mu \| < 2\varepsilon $ . The identity

$$ \begin{align*} L_1 (\nu) + \cdots + L_d (\nu) = \int \log |\!\det g| \, d \nu (g), \end{align*} $$

which is true for any compactly supported measure $\nu $ on $\mathrm {GL}(d)$ , gives us

$$ \begin{align*} L_1 (\mu_{a, \varepsilon}) \ge \dfrac{L_1 (\mu_{a, \varepsilon}) + \cdots + L_d (\mu_{a, \varepsilon})}{d} = \dfrac{1}{d}\int \log |\!\det g| \, d \mu_{a, \varepsilon} (g). \end{align*} $$

Now, given $\delta> 0$ , let $0<\varepsilon < \delta /2$ , so that $\mu _{a, \varepsilon } \in B(\mu , \delta )$ for every $a>0$ . Observe that

$$ \begin{align*} \dfrac{1}{d}\int \log |\!\det g| \, d \mu_{a, \varepsilon} (g) = \varepsilon \log a+\dfrac{(1-\varepsilon)\int \log |\!\det g| \, d \mu (g)}{d}. \end{align*} $$

Then choose a sufficiently big such that

$$ \begin{align*} \log a> \dfrac{L_1 (\mu) +1 + (({\varepsilon-1})/{d})\int \log |\!\det g| \, d \mu (g)}{\varepsilon}. \end{align*} $$

The previous inequality then guarantees that $L_1 (\mu _{a, \varepsilon })> L_1 (\mu ) + 1$ . In particular, $L_1$ cannot be continuous in $\mu $ , much less analytic. The problem relies on the fact that, as we shrink the neighbourhood of $\mu $ , the value of $\varepsilon $ decreases, which causes the choice of a above to become increasingly large. Restricting ourselves to a compact set limits the size of a, which make this construction fail for small enough $\delta $ .