Using Lemma 5.4 and ((S)) we see that $g_{N,n}^{(1)},g_{n}^{(1)}\in L^{1}(m)$
for each $N,n$. Moreover using these same facts

$\displaystyle\|g_{N,n}^{(1)}-g_{n}^{(1)}\|_{1}$ $\displaystyle\leq$
$\displaystyle\int\left(\left|\frac{\theta_{\omega,\varepsilon_{N},n}(t_{\omega}+\xi_{\omega,N})}{\nu_{\sigma^{-n}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-n}\omega,0})}-\theta_{\omega,0,n}t_{\omega}\right|\right)\
dm.$

The integrand goes to zero $\omega$ almost surely and it is bounded by
$C_{\varepsilon_{N}}(|t|_{\infty}+W)+|t|_{\infty}.$ We could therefor apply
dominated convergence to prove that the $L^{1}(m)$ difference goes to zero.

Referring to the first term in the $Y$-sum in (5.28), we may now apply Lemma
5.6 to conclude that

$\sum_{i=0}^{N-1}\frac{\theta_{\sigma^{i}\omega,\varepsilon_{N},n}\Delta_{\sigma^{i}\omega,\varepsilon_{N}}}{\lambda_{\sigma^{i}\omega,0}\nu_{\sigma^{-n+i}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-n+i}\omega,0})}=\frac{1}{N}\sum_{i=0}^{N-1}g_{N,n}^{(1)}(\sigma^{i}\omega)\to\int_{\Omega}g_{n}^{(1)}(\omega)\
dm(\omega)=\int_{\Omega}\theta_{\omega,0,n}t_{\omega}\ dm(\omega),$

as $N\to\infty$ for each $n$ and $m$-a.e. $\omega$.

Step 4: Estimating $g^{(2)}$. We now perform a similar analysis to the
previous step to control the terms in the sum (5.28) corresponding to
$\theta^{\prime}$. Again in preparation for applying Lemma 5.6, recall that

$g_{N,n}^{(2)}(\omega)=\frac{N\theta^{\prime}_{\omega,\varepsilon_{N},n}}{\lambda_{\omega,0}\nu_{\sigma^{-n}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-n}\omega,0})}$

and set $g_{n}^{(2)}(\omega)\equiv 0$ for each $n$. Using (5.24), (5.6),
(5.9), and ((C7’)), for sufficiently large $N$ we have

$\displaystyle\theta_{\omega,\varepsilon_{N},n}^{\prime}$ $\displaystyle:=$
$\displaystyle\sum_{k=1}^{n}\lambda^{-1}_{\sigma^{-k}\omega,0}(\lambda_{\sigma^{-k}\omega,0}-\lambda_{\sigma^{-k}\omega,\varepsilon})\nu_{\sigma\omega,0}((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0}))$
$\displaystyle\leq$
$\displaystyle\frac{C_{2}C_{3}\lambda_{\omega,0}}{N}\sum_{k=1}^{n}\lambda_{\sigma^{-k}\omega,0}^{-1}\left(\lambda_{\sigma^{-k}\omega,0}(t_{\sigma^{-k}\omega}+\xi_{\sigma^{-k}\omega,N})\right)\nu_{\omega,0}\left(\mathbbold{1}_{H_{\omega,\varepsilon_{N}}}\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon_{N}}^{k}\varphi_{\sigma^{-k}\omega,0}\right)$
$\displaystyle\leq$
$\displaystyle\frac{C_{2}C_{3}\lambda_{\omega,0}}{N}\sum_{k=1}^{n}(t_{\sigma^{-k}\omega}+\xi_{\sigma^{-k}\omega,N})\nu_{\omega,0}(H_{\omega,\varepsilon_{N}})\|\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon_{N}}^{k}\varphi_{\sigma^{-k}\omega,0}\|_{\infty,\omega}$
$\displaystyle\leq$
$\displaystyle\frac{C_{2}C_{3}\lambda_{\omega,0}}{N}\cdot\frac{C_{3}(t_{\omega}+\xi_{\omega,N})}{N}\sum_{k=1}^{n}(t_{\sigma^{-k}\omega}+\xi_{\sigma^{-k}\omega,N})\|\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon_{N}}^{k}\varphi_{\sigma^{-k}\omega,0}\|_{\infty,\omega}.$

Using Lemma 5.4, ((C5’)), and ((C4’)) we note that

(5.34)
$\displaystyle\|\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon_{N}}^{k}\varphi_{\sigma^{-k}\omega,0}\|_{\infty,\omega}$
$\displaystyle=$
$\displaystyle\|\nu_{\sigma^{-k}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-k}\omega,0})\varphi_{\omega,\varepsilon_{N}}+Q_{\sigma^{-k}\omega,\varepsilon_{N}}^{k}\varphi_{\sigma^{-k}\omega,0}\|_{\infty,\omega}$
$\displaystyle\leq$
$\displaystyle\|C_{\varepsilon_{N}}C_{2}+C_{\varphi_{0}}C_{2}\alpha(k)\|_{\infty,\omega}$
$\displaystyle\leq$ $\displaystyle
C_{2}C_{\varepsilon_{N}}+C_{2}C_{\varphi_{0}}\alpha.$

Finally, we have for $N$ sufficiently large and by ((S))

$\displaystyle g_{N,n}^{(2)}(\omega)$ $\displaystyle\leq$
$\displaystyle\frac{NC_{2}C_{3}\lambda_{\omega,0}(C_{2}C_{\varepsilon_{N}}+C_{2}C_{\varphi_{0}}\alpha)}{N\lambda_{\omega,0}\nu_{\sigma^{-n}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-n}\omega,0})}\cdot\frac{C_{3}(t_{\omega}+\xi_{\omega,N})}{N}\sum_{k=1}^{n}(t_{\sigma^{-k}\omega}+\xi_{\sigma^{-k}\omega,N}).$

Integrability of $g_{N,n}^{(2)}$ follows from the the fact that $t\in
L^{\infty}(m)$ and $|\xi_{\omega,N}|\leq W,$ for almost all $\omega.$ Moreover
by ((S)), for each $n$ we have that $g_{N,n}^{(2)}\to 0$ almost everywhere as
$N\to\infty$. By dominated convergence we have again that
$\lim_{N\to\infty}\|g_{N,n}^{(2)}-g_{n}^{(2)}\|_{1}\to 0$ for each $n$.

Referring to $Y_{\omega,\varepsilon_{N},n}^{(2)}$ in (5.26), we may now apply
Lemma 5.6 to conclude that

$\sum_{i=0}^{N-1}\frac{\theta^{\prime}_{\sigma^{i}\omega,\varepsilon_{N},n}}{\lambda_{\sigma^{i}\omega,0}\nu_{\sigma^{-n+i}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-n+i}\omega,0})}=\frac{1}{N}\sum_{i=0}^{N-1}g_{N,n}^{(2)}(\sigma^{i}\omega)\to\int_{\Omega}g_{n}^{(2)}(\omega)\
dm(\omega)=0,$

as $N\to\infty$ for each $n$ and a.e. $\omega$.

Step 5: Estimating $g^{(3)}$. We repeat a similar analysis to control the
terms in the sum (5.28) corresponding to $\theta^{\prime\prime}$. Again in
preparation for applying Lemma 5.6, recall that

$g_{N,n}^{(3)}(\omega)=\frac{N\theta^{\prime\prime}_{\omega,\varepsilon_{N},n}}{\lambda_{\omega,0}\nu_{\sigma^{-n}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-n}\omega,0})}.$

We begin developing an upper bound for $|g_{N,n}^{(3)}|$. Using (5.25),
((C2)), ((C4’)), and ((C7’)) we have for sufficiently large $N$ that

$\displaystyle|\theta_{\omega,\varepsilon_{N},n}^{\prime\prime}|$
$\displaystyle:=\nu_{\sigma\omega,0}(\mathcal{L}_{\omega,0}(\mathbbold{1}_{H_{\omega,\varepsilon_{N}}}Q^{n}_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})))$
$\displaystyle=\lambda_{\omega,0}\nu_{\omega,0}(\mathbbold{1}_{H_{\omega,\varepsilon_{N}}}Q_{\sigma^{-n}\omega,\varepsilon_{N}}^{n}\varphi_{\sigma^{-n}\omega,0})$
$\displaystyle\leq\lambda_{\omega,0}\nu_{\omega,0}(H_{\omega,\varepsilon_{N}})\|Q_{\sigma^{-n}\omega,\varepsilon_{N}}^{n}\varphi_{\sigma^{-n}\omega,0}\|_{\infty,\omega}$
$\displaystyle\leq\lambda_{\omega,0}\nu_{\omega,0}(H_{\omega,\varepsilon_{N}})C_{\varphi_{0}}\|\varphi_{\sigma^{-n}\omega,0}\|_{\mathcal{B}_{\sigma^{-n}\omega}}\alpha(n)$
$\displaystyle\leq\lambda_{\omega,0}C_{\varphi_{0}}C_{2}C_{3}\alpha(n)\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})$
$\displaystyle=\frac{\lambda_{\omega,0}C_{\varphi_{0}}C_{2}C_{3}\alpha(n)(t_{\omega}+\xi_{\omega,N})}{N}.$

Therefore, using Lemma 5.4

$|g_{N,n}^{(3)}(\omega)|\leq\frac{N\lambda_{\omega,0}C_{\varphi_{0}}C_{2}C_{3}\alpha(n)(t_{\omega}+\xi_{\omega,N})}{N\lambda_{\omega,0}\nu_{\sigma^{-n}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-n}\omega,0}))}\leq
C_{\varepsilon_{N}}C_{\varphi_{0}}C_{2}C_{3}\alpha(n)(t_{\omega}+\xi_{\omega,N})=:\tilde{g}^{(3)}_{N,n}(\omega).$

We set
$\tilde{g}^{(3)}_{n}(\omega)=C_{\varphi_{0}}C_{2}C_{3}\alpha(n)t_{\omega}$.
Integrability of $\tilde{g}_{N,n}^{(3)}$ and $\tilde{g}_{n}^{(3)}$ follows
from ((S)) and the fact $t\in L^{\infty}(m)$ and $|\xi_{\omega,N}|\leq W,$ for
almost all $\omega.$ Similarly, (recalling that $C_{\varepsilon_{N}}\to 1$ as
$N\to\infty$ by Lemma 5.4) for each $n$,
$\tilde{g}_{N,n}^{(3)}\to\tilde{g}_{n}^{(3)}$ almost everywhere as
$N\to\infty$. For the same reasons by ((S)) and dominated convergence we also
have that
$\lim_{N\to\infty}\|\tilde{g}_{N,n}^{(3)}-\tilde{g}_{n}^{(3)}\|_{1}\to 0$ for
each $n$.

Referring to $Y_{\omega,\varepsilon_{N},n}^{(3)}$ in (5.26), we may now apply
Lemma 5.6 to $\tilde{g}_{N,n}^{(3)}$ and $\tilde{g}_{n}^{(3)}$ to conclude
that

$\frac{1}{N}\sum_{i=0}^{N-1}g_{N,n}^{(3)}(\sigma^{i}\omega)\leq\frac{1}{N}\sum_{i=0}^{N-1}\tilde{g}_{N,n}^{(3)}(\sigma^{i}\omega)\to\int_{\Omega}\tilde{g}_{n}^{(3)}(\omega)\
dm(\omega)=C_{\varphi_{0}}C_{2}C_{3}\alpha(n)\int_{\Omega}t_{\omega}\
dm(\omega)$

as $N\to\infty$ for each $n$ and a.e. $\omega$.

Step 6: Finishing up. Recall from (5.3) that

$(\ref{evtexpression})=\frac{\lambda_{\omega,\varepsilon_{N}}^{N}}{\lambda_{\omega,0}^{N}}\left(\nu_{\omega,\varepsilon_{N}}(\mathbbold{1})+\nu_{\sigma^{N}\omega,0}\left(Q_{\omega,\varepsilon_{N}}^{N}(\mathbbold{1})\right)\right).$

Using ((C4’)) and Lemma 5.4 we see that
$\lim_{N\to\infty}\left(\nu_{\omega,\varepsilon_{N}}(\mathbbold{1})+\nu_{\sigma^{N}\omega,0}\left(Q_{\omega,\varepsilon_{N}}^{N}(\mathbbold{1})\right)\right)=1$
for $m$-a.e. $\omega\in\Omega$. By (5.29) and Steps 3, 4, and 5 we see that
for any $n$

$\displaystyle\lim_{N\to\infty}\exp\left(-\frac{1}{N}\sum_{i=0}^{N-1}\left(g^{(1)}_{N,n}(\sigma^{i}\omega)+g^{(2)}_{N,n}(\sigma^{i}\omega)+g^{(3)}_{N,n}(\sigma^{i}\omega)\right)\right)\exp\left(-\sum_{i=0}^{N-1}\frac{Y_{\sigma^{i}\omega,\varepsilon_{N},n}^{2}}{2(1-y)^{2}}\right)$
$\displaystyle\leq\lim_{N\to\infty}\frac{\lambda_{\omega,\varepsilon_{N}}^{N}}{\lambda_{\omega,\varepsilon_{N}}^{N}}$
$\displaystyle\leq\lim_{N\to\infty}\exp\left(-\frac{1}{N}\sum_{i=0}^{N-1}\left(g^{(1)}_{N,n}(\sigma^{i}\omega)+g^{(2)}_{N,n}(\sigma^{i}\omega)-\tilde{g}^{(3)}_{N,n}(\sigma^{i}\omega)\right)\right)\exp\left(-\sum_{i=0}^{N-1}\frac{Y_{\sigma^{i}\omega,\varepsilon_{N},n}^{2}}{2(1-y)^{2}}\right),$

where $0\leq y\leq Y_{\sigma^{i}\omega,\varepsilon_{N},n}$. We now treat the
Taylor remainder terms. From Steps 3, 4, and 5 for all $N$ sufficiently large
we have and for almost all $\omega$:

(5.36) $\displaystyle|g^{(1)}_{N,n}|$ $\displaystyle\leq$ $\displaystyle
C_{\varepsilon_{N}}(|t|_{\infty}+W),$ (5.37) $\displaystyle|g^{(2)}_{N,n}|$
$\displaystyle\leq$
$\displaystyle{C_{\varepsilon_{N}}C_{2}C_{3}(C_{2}C_{\varepsilon_{N}}+C_{2}C_{\varphi_{0}}\alpha)}\cdot\frac{C_{3}n(|t|_{\infty}+W)^{2}}{N},$
(5.38) $\displaystyle|g^{(3)}_{N,n}|$ $\displaystyle\leq$
$\displaystyle\tilde{g}^{(3)}_{N,n}:=C_{\varphi_{0}}C_{2}C_{3}\alpha(|t|_{\infty}+W).$

Further,

(5.39)
$\sum_{i=0}^{N-1}\frac{Y_{\sigma^{i}\omega,\varepsilon_{N},n}^{2}}{2(1-y)^{2}}\leq\sum_{i=0}^{N-1}\frac{Y_{\sigma^{i}\omega,\varepsilon_{N},n}^{2}}{2(1-Y_{\sigma^{i}\omega,\varepsilon_{N},n})^{2}}=\sum_{i=0}^{N-1}\frac{(G_{\sigma^{i}\omega,\varepsilon_{N},n}/N)^{2}}{2(1-(G_{\sigma^{i}\omega,\varepsilon_{N},n}/N))^{2}},$

where
$G_{\omega,\varepsilon_{N},n}:=g^{(1)}_{N,n}+g^{(2)}_{N,n}+g^{(3)}_{N,n}$.
Using the bounds (5.36)–(5.38) we see that (5.39) approaches $0$ for almost
all $\omega$ for each $n$ as $N\to\infty$. Therefore, combining the
expressions developed in Steps 3, 4, and 5 for the $N\to\infty$ limits with
(5.1) we have

$\displaystyle\exp\left(-\int_{\Omega}\theta_{\omega,0,n}t_{\omega}\
dm(\omega)\right)\leq\lim_{N\to\infty}\frac{\lambda_{\omega,\varepsilon_{N}}^{N}}{\lambda_{\omega,0}^{N}}\leq\exp\left(-\int_{\Omega}\theta_{\omega,0,n}t_{\omega}\
dm(\omega)+C_{2}C_{3}C_{\varphi_{0}}\alpha(n)\int_{\Omega}t_{\omega}\
dm(\omega)\right).$

Recalling the definitions of $\theta_{\omega,0,n}$ (5.31) and
$\theta_{\omega,0}$ (2.20) and the fact that $0\leq\theta_{\omega,0,n}\leq 1$
(by (4.11)), we may use dominated convergence to take the $n\to\infty$ limit
to obtain

$\exp\left(-\int_{\Omega}\theta_{\omega,0}t_{\omega}\
dm(\omega)\right)\leq\lim_{N\to\infty}\frac{\lambda_{\omega,\varepsilon_{N}}^{N}}{\lambda_{\omega,0}^{N}}\leq\exp\left(-\int_{\Omega}\theta_{\omega,0}t_{\omega}\
dm(\omega)\right),$

thus completing the proof that

$\lim_{N\to\infty}\nu_{\omega,0}\left(X_{\omega,N-1,\varepsilon_{N}}\right)=\lim_{N\to\infty}\frac{\lambda_{\omega,\varepsilon_{N}}^{N}}{\lambda_{\omega,0}^{N}}=\exp\left(-\int_{\Omega}t_{\omega}\theta_{\omega,0}\,dm(\omega)\right).$

To see that
$\lim_{N\to\infty}\mu_{\omega,0}\left(X_{\omega,N-1,\varepsilon_{N}}\right)$
is also equal to this value, we simply recall that (5.4) gives that

$\displaystyle\mu_{\omega,0}(X_{\omega,N-1,\varepsilon_{N}})=\frac{\lambda_{\omega,\varepsilon_{N}}^{N}}{\lambda_{\omega,0}^{N}}\left(\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})+\nu_{\sigma^{N}\omega,0}\left(Q_{\omega,\varepsilon_{N}}^{N}(\varphi_{\omega,0})\right)\right).$

Now since ((C4’)) and Lemma 5.4 together give that

$\lim_{N\to\infty}\left(\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})+\nu_{\sigma^{N}\omega,0}\left(Q_{\omega,\varepsilon_{N}}^{N}(\varphi_{\omega,0})\right)\right)=1$

for $m$-a.e. $\omega\in\Omega$, we must in fact have that

$\lim_{N\to\infty}\mu_{\omega,0}\left(X_{\omega,N-1,\varepsilon_{N}}\right)=\lim_{N\to\infty}\frac{\lambda_{\omega,\varepsilon_{N}}^{N}}{\lambda_{\omega,0}^{N}}=\exp\left(-\int_{\Omega}t_{\omega}\theta_{\omega,0}\,dm(\omega)\right),$

which completes the proof of Theorem 5.5.

∎

###  5.2. The relationship between condition (1.5) and the Hüsler condition

We now return to the discussion initiated in the Introduction to compare our
assumption (1.5) for the thresholds $z_{\omega,N}$ with the Hüsler type
condition (1.4). We show that in the more general situation considered in our
paper with random boundary level $t_{\omega}$, the limit (1.4) will follow
from the simpler assumption (1.5), provided we replace $t$ in (1.4) with the
expectation of $t_{\omega}.$

Recall our assumption for the choice of the thresholds ((S)) is
$\mu_{\omega,0}(h_{\omega}(x)>z_{\omega,N})=(t_{\omega}+\xi_{\omega,N})/N$,
where $\xi_{\omega,N}$ goes to zero almost surely when $N\rightarrow\infty$
and $\xi_{\omega,N}\leq W$ for a.e. $\omega$. It is immediate to see by
dominated convergence that:

(5.40)
$\lim_{N\rightarrow\infty}\int_{\Omega}|N\mu_{\omega,0}(h_{\omega}(x)>z_{\omega,N})-t_{\omega}|\
dm(\omega)=0.$

Applying our non-standard ergodic Lemma 5.6 with
$g_{N}(\omega):=N\mu_{\omega,0}(h_{\omega}(x)>z_{\omega,N})$ and
$g(\omega):=t_{\omega}$, one may transform the sum in (1.4) as follows:

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=0}^{N-1}N\mu_{\omega,0}(h_{\sigma^{j}\omega}(T^{j}_{\omega}(x))>z_{\sigma^{j}\omega,N})=\int_{\Omega}t_{\omega}\
dm(\omega),$

which is the condition (1.4) with $t$ replaced by the expectation of
$t_{\omega}.$

###  5.3. Hitting time statistics

It is well known that in the deterministic setting there is a close
relationship between extreme value theory and the statistics of first hitting
time, see for instance [25, 41]. We now show how our Theorem 5.5, with a
slight modification, can be interpreted in terms of a suitable definition of
(quenched) first hitting time distribution. Let us consider as in the previous
sections, a sequence of small random holes
$\mathcal{H}_{\omega,N}:=\\{H_{\sigma^{j}\omega,\varepsilon_{N}}\\}_{j\geq
0},$ and define the first random hitting time as

$\tau_{\omega,\mathcal{H}_{\omega,N}}(x)=\inf\\{k\geq 1,T^{k}_{\omega}(x)\in
H_{\sigma^{k}\omega,\varepsilon_{N}}\\}.$

We recall that the usual statistics of hitting times is written in the form
$\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}>t\right),$ for
nonnegative values of $t.$ Since the sets
$H_{\sigma^{j}\omega,\varepsilon_{N}}$ have measure tending to zero when
$N\rightarrow\infty,$ and therefore the first hitting times could eventually
grow to infinity, one needs a rescaling in order to get a meaningful limit
distribution. This is achieved in the next Proposition. In our current
setting, condition ((S)) reads:
$\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})=\frac{t_{\omega}+\xi_{\omega,N}}{N},$
with $\lim_{N\to\infty}\xi_{\omega,N}=0$ for a.e. $\omega$ and
$|\xi_{\omega,N}|\leq W$ for a.e. $\omega$ and all $N\geq 1$.

######  Proposition 5.7.

If our random open system $(\Omega,\allowbreak
m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$ satisfies the assumptions of Theorem 5.5 with the sequence
$H_{\omega,\varepsilon_{N}}$ verifying condition ((S)), then the first random
hitting time satisfies the limit, for $\omega$ $m$-a.e.

(5.41)
$\lim_{N\to\infty}\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>t_{\omega}\right)=\exp\left(-{\int_{\Omega}t_{\omega}\theta_{\omega,0}dm}\right).$

###### Proof.

For $N\geq 1$ the event,

(5.42) $\\{\tau_{\omega,\mathcal{H}_{\omega,N}}>N\\}=\\{x\in
I;T_{\omega}(x)\in
H_{\sigma\omega,\varepsilon_{N}}^{c},\dots,T^{N}_{\omega}(x)\in
H_{\sigma^{N}\omega,\varepsilon_{N}}^{c}\\}$

is also equal to

$T^{-1}_{\omega}\left(x\in I;x\in
H_{\sigma\omega,\varepsilon_{N}}^{c},T_{\sigma\omega}(x)\in
H_{\sigma^{2}\omega,\varepsilon_{N}}^{c},\dots,T_{\sigma\omega}^{N-1}(x)\in
H_{\sigma^{N}\omega,\varepsilon_{N}}^{c}\right).$

Then, by equivariance of $\mu_{0}$ we obtain the link between the statistics
of hitting time and extreme value theory:

$\displaystyle\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}>N\right)$
$\displaystyle=$ $\displaystyle\mu_{\omega,0}\left(T^{-1}_{\omega}\left(x\in
I:x\in H_{\sigma\omega,\varepsilon_{N}}^{c},T_{\sigma\omega}(x)\in
H_{\sigma^{2}\omega,\varepsilon_{N}}^{c},\dots,T_{\sigma\omega}^{N-1}(x)\in
H_{\sigma^{N}\omega,\varepsilon_{N}}^{c}\right)\right)$ $\displaystyle=$
$\displaystyle\mu_{\sigma\omega,0}\left(x\in
H_{\sigma\omega,\varepsilon_{N}}^{c},T_{\sigma\omega}(x)\in
H_{\sigma^{2}\omega,\varepsilon_{N}}^{c},\dots,T_{\sigma\omega}^{N-1}(x)\in
H_{\sigma^{N}\omega,\varepsilon_{N}}^{c}\right)$ (5.44) $\displaystyle=$
$\displaystyle\mu_{\sigma\omega,0}(X_{\sigma\omega,N-1,\varepsilon_{N}}).$

In order to rescale the eventually growing first random hitting times, we
invoke the condition ((S)); by substituting
$N=(t_{\omega}+\xi_{\omega,N})/\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})$ in
the LHS of (5.3) we have

(5.45)
$\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}>N\right)=\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>t_{\omega}+\xi_{\omega,N}\right).$

Our final preparation before applying Theorem 5.5 is to show that

(5.46)
$|\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>t_{\omega}+\xi_{\omega,N}\right)-\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>t_{\omega}\right)|\rightarrow
0,\ N\rightarrow\infty.$

Since by a standard trick, see for instance eq. 5.3.6 in [41],

$\\{\tau_{\omega,\mathcal{H}_{\omega,N}}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>t_{\omega}\\}\backslash\\{\tau_{\omega,\mathcal{H}_{\omega,N}}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>t_{\omega}+\xi_{\omega,N}\\}\subset\bigcup_{j=\left\lceil{\frac{t_{\omega}}{\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})}}\right\rceil}^{\left\lceil{\frac{t_{\omega}+\xi_{\omega,n}}{\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})}}\right\rceil}T_{\omega}^{-j}(H_{\sigma^{j}\omega,\varepsilon_{N}})$

we have by equivariance

$|\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>t_{\omega}+\xi_{\omega,N}\right)-\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>t_{\omega}\right)|\leq\sum_{j=\left\lceil{\frac{t_{\omega}}{\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})}}\right\rceil}^{\left\lceil{\frac{t_{\omega}+\xi_{\omega,n}}{\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})}}\right\rceil}\mu_{\sigma^{j}\omega,0}({H}_{\sigma^{j}\omega,N}).$

For $N$ large enough:

$\sum_{j=\left\lceil{\frac{t_{\omega}}{\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})}}\right\rceil}^{\left\lceil{\frac{t_{\omega}+\xi_{\omega,n}}{\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})}}\right\rceil}\mu_{\sigma^{j}\omega,0}({H}_{\sigma^{j}\omega,N})\leq\left\lceil{\frac{|\xi_{\omega,N}|}{\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})}}\right\rceil\frac{|t|_{\infty}+W}{N}\leq\left\lceil{\frac{|\xi_{\omega,N}|\
N}{t_{\omega}-|\xi_{\omega,N}|}}\right\rceil\frac{|t|_{\infty}+W}{N},$

which goes to zero by ((S)). Recalling $t_{\omega}>0$ for a.e. $\omega$, the
final expression above goes to zero as $N\rightarrow\infty$ for $\omega$
$m$-a.e.

Using (5.44)–(5.46) and noting that
$\lim_{N\to\infty}\mu_{\sigma\omega,0}(X_{\sigma\omega,N-1,\varepsilon_{N}})$
is nonrandom, applying Theorem 5.5 yields

(5.47)
$\lim_{N\to\infty}\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>t_{\omega}\right)=\exp\left(-\int_{\Omega}t_{\omega}\theta_{\omega,0}\
dm\right).$

∎

##  6\. Quenched Thermodynamic Formalism for Random Open Interval Maps via
Perturbation

In this section we present an explicit class of random piecewise-monotonic
interval maps for which our Theorem 4.6, Corollary 4.9, and Theorem 5.5 apply.
Using a perturbative approach, we introduce a family of small random holes
parameterised by $\varepsilon>0$ into a random closed dynamical system, and
for every small $\varepsilon$ we prove (i) the existence of a unique random
conformal measure
$\left\\{\nu_{\omega,\varepsilon}\right\\}_{\omega\in\Omega}$ with fiberwise
support in $X_{\omega,\infty,\varepsilon}$ and (ii) a unique random absolutely
continuous invariant measure
$\left\\{\mu_{\omega,\varepsilon}\right\\}_{\omega\in\Omega}$ which satisfies
an exponential decay of correlations and is the unique relative equilibirum
state for the random open system $(\Omega,\allowbreak
m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$. In addition, we prove the existence of a random absolutely
continuous (with respect to $\nu_{\omega,0}$) conditionally invariant
probability measure
$\left\\{\varrho_{\omega,\varepsilon}\right\\}_{\omega\in\Omega}$ with
fiberwise support in $[0,1]\backslash H_{\omega,\varepsilon}$.

We now suppose that the spaces $\mathcal{J}_{\omega,0}=[0,1]$ for each
$\omega\in\Omega$ and the maps $T_{\omega}:[0,1]\to[0,1]$ are surjective,
finitely-branched, piecewise monotone, nonsingular (with respect to Lebesgue),
and that there exists $C\geq 1$ such that

(E1) $\displaystyle\operatorname*{ess\sup}_{\omega}|T_{\omega}^{\prime}|\leq
C\qquad\text{ and }\qquad\operatorname*{ess\sup}_{\omega}D(T_{\omega})\leq C,$

where $D(T_{\omega}):=\sup_{y\in[0,1]}\\#T_{\omega}^{-1}(y)$. We let
$\mathcal{Z}_{\omega,0}$ denote the (finite) monotonicity partition of
$T_{\omega}$ and for each $n\geq 2$ we let $\mathcal{Z}_{\omega,0}^{(n)}$
denote the partition of monotonicity of $T_{\omega}^{n}$.

  1. (MC)

The map $\sigma:\Omega\to\Omega$ is a homeomorphism, the skew-product map
$T:\Omega\times[0,1]\to\Omega\times[0,1]$ is measurable, and $\omega\mapsto
T_{\omega}$ has countable range.

######  Remark 6.1.

Under assumption ((MC)), the family of transfer operator cocycles
$\\{\mathcal{L}_{\omega,\varepsilon}\\}_{\varepsilon\geq 0}$ satisfies the
conditions of Theorem 17 [29] ($m$-continuity and $\sigma$ a homeomorphism).
Note that condition ((MC)) implies that $T$ satisfies ((M1)) and the cocycle
generated by $\mathcal{L}_{0}$ satisfies condition ((M2)).

Let the _variation_ of $f:[0,1]\to\mathbb{R}_{+}$ on $Z\subset[0,1]$ be

$\displaystyle\text{{\rm var}}_{Z}(f)=\underset{x_{0}<\dots<x_{k},\,x_{j}\in
Z}{\sup}\sum_{j=0}^{k-1}\left\lvert f(x_{j+1})-f(x_{j})\right\rvert,$

and $\text{{\rm var}}(f):=\text{{\rm var}}_{[0,1]}(f)$. We let $\text{{\rm
BV}}=\text{{\rm BV}}([0,1])$ denote the set of functions on $[0,1]$ that have
bounded variation. Given a non-atomic and fully supported measure $\nu$ (i.e.
for any nondegenerate interval $J\subseteq[0,1]$ we have $\nu(J)>0$) we let
$\text{{\rm BV}}_{\nu}\subseteq L^{\infty}(\nu)$ be the set of (equivalence
classes of) functions of _bounded variation_ on $[0,1]$, with norm given by

$\displaystyle\|f\|_{\text{{\rm BV}}_{\nu}}:=\underset{\tilde{f}=f\ \nu\text{
a.e.}}{\inf}\text{{\rm var}}(\tilde{f})+\nu(|f|).$

If we require to emphasise that elements of $\text{{\rm BV}}_{\nu}$ are
equivalence classes, we denote these by $[f]_{\nu}$ (resp. $[f]_{1}$). Note
that if $f\in\text{{\rm BV}}$ is a function of bounded variation, then it is
always possible to choose a representative of minimal variation from the
equivalence class $[f]_{\nu}$. We define $\text{{\rm BV}}_{1}\subseteq
L^{\infty}(\text{{\rm Leb}})$ and
$\|\makebox[4.30554pt]{{$\cdot$}}\|_{\text{{\rm BV}}_{1}}$ similarly, with the
measure $\nu$ replaced with Lebesgue measure. We denote the supremum norm on
$L^{\infty}(\text{{\rm Leb}})$ by
$\|\makebox[4.30554pt]{{$\cdot$}}\|_{\infty,1}$. It follows from Rychlik [46]
that $\text{{\rm BV}}_{\nu}$ and $\text{{\rm BV}}_{1}$ are Banach spaces. The
following proposition gives the equivalence of the norms
$\|\makebox[4.30554pt]{{$\cdot$}}\|_{\text{{\rm BV}}_{\nu}}$ and
$\|\makebox[4.30554pt]{{$\cdot$}}\|_{\text{{\rm BV}}_{1}}$.

######  Proposition 6.2.

Given a fully supported and non-atomic measure $\nu$ on $[0,1]$ and
$f\in\text{{\rm BV}}$ we have that

$(1/2)\|f\|_{\text{{\rm BV}}_{1}}\leq\|f\|_{\text{{\rm BV}}_{\nu}}\leq
2\|f\|_{\text{{\rm BV}}_{1}}.$

###### Proof.

We first show that for $f\in\text{{\rm BV}}$ we have

(6.1) $\displaystyle[f]_{\nu}\cap\text{{\rm BV}}=[f]_{1}\cap\text{{\rm BV}}.$

To see this let $\tilde{f}\in[f]_{\nu}\cap\text{{\rm BV}}$. As $\nu$ is a
fully supported and non-atomic measure, we must have that the set
$\\{x:f(x)\neq\tilde{f}(x)\\}$ is countable. Thus
$\tilde{f}\in[f]_{1}\cap\text{{\rm BV}}$. As Leb is also fully supported and
non-atomic the same reasoning implies that the reverse inclusion also holds,
proving (6.1). As a direct consequence of (6.1) we have that

(6.2) $\displaystyle\underset{\tilde{f}=f\ \nu\text{ a.e.}}{\inf}\text{{\rm
var}}(\tilde{f})=\underset{\tilde{f}=f\ \text{{\rm Leb}}\text{
a.e.}}{\inf}\text{{\rm var}}(\tilde{f}).$

Since $f$ is continuous everywhere except on a set of at most countably many
points, letting $\mathcal{C}$ denote the set of intervals of continuity for
$f$, we have

(6.3) $\displaystyle\operatorname*{\text{Leb-ess}\inf}_{[0,1]}f$
$\displaystyle=\inf_{J\in\mathcal{C}}\operatorname*{\text{Leb-
ess}\inf}_{J}f=\inf_{J\in\mathcal{C}}\inf_{J}f=\inf_{J\in\mathcal{C}}\operatorname*{\nu\text{-ess}\inf}_{J}f=\operatorname*{\nu\text{-ess}\inf}_{[0,1]}f.$

Using similar reasoning we must also have

(6.4) $\displaystyle\operatorname*{\text{Leb-
ess}\sup}f=\operatorname*{\nu\text{-ess}\sup}f.$

Combining (6.2) and (6.3) we have

$\displaystyle\|f\|_{\text{{\rm BV}}_{1}}=\inf_{\tilde{f}=f\ \text{{\rm Leb}}\
a.e.}\text{{\rm var}}(\tilde{f})+\text{{\rm Leb}}(|f|)$ $\displaystyle\leq
2\inf_{\tilde{f}=f\ \text{{\rm Leb}}\ a.e.}\text{{\rm
var}}(\tilde{f})+\operatorname*{\text{Leb-ess}\inf}|f|$
$\displaystyle=2\inf_{\tilde{f}=f\ \nu\ a.e.}\text{{\rm
var}}(\tilde{f})+\operatorname*{\nu\text{-ess}\inf}|f|$ $\displaystyle\leq
2\inf_{\tilde{f}=f\ \nu\ a.e.}\text{{\rm var}}(\tilde{f})+\nu(|f|)\leq
2\|f\|_{\text{{\rm BV}}_{\nu}}.$

Similarly, using (6.2) and (6.4) we have $\|f\|_{\text{{\rm BV}}_{\nu}}\leq
2\|f\|_{\text{{\rm BV}}_{1}}$, and thus the proof is complete.

∎

Proposition 6.2 will be used later to provide (non-random) equivalence of
$\|\makebox[4.30554pt]{{$\cdot$}}\|_{\text{{\rm BV}}_{\nu_{\omega,0}}}$ and
$\|\makebox[4.30554pt]{{$\cdot$}}\|_{\text{{\rm BV}}_{1}}$ for each
$\omega\in\Omega$. We set $J_{\omega}:=|T_{\omega}^{\prime}|$ and define the
random Perron–Frobenius operator, acting on functions in $BV$

$P_{\omega}(f)(x):=\sum_{y\in T_{\omega}^{-1}(x)}\frac{f(y)}{J_{\omega}(y)}.$

The operator $P$ satisfies the well-known property that

(6.5) $\displaystyle\int_{[0,1]}P_{\omega}(f)\,d\text{{\rm
Leb}}=\int_{[0,1]}f\,d\text{{\rm Leb}}$

for $m$-a.e. $\omega\in\Omega$ and all $f\in\text{{\rm BV}}$. Recall from
Section 3 that $g_{0}=\left\\{g_{\omega,0}\right\\}_{\omega\in\Omega}$ and
that

$\mathcal{L}_{\omega,0}(f)(x):=\sum_{y\in
T_{\omega}^{-1}(x)}g_{\omega,0}(y)f(y),\quad f\in\text{{\rm BV}}.$

We assume that the weight function $g_{\omega,0}$ lies in BV for each
$\omega\in\Omega$ and satisfies

(E2) $\operatorname*{ess\sup}_{\omega}\|g_{\omega,0}\|_{\infty,1}<\infty,$

and

(E3) $\operatorname*{ess\inf}_{\omega}\inf g_{\omega,0}>0.$

Note that (E1) and (E2) together imply

(6.6)
$\displaystyle\operatorname*{ess\sup}_{\omega}\|\mathcal{L}_{\omega,0}\mathbbold{1}\|_{\infty,1}\leq\operatorname*{ess\sup}_{\omega}D(T_{\omega})\|g_{\omega,0}\|_{\infty,1}<\infty$

and

(6.7)
$\displaystyle\operatorname*{ess\sup}_{\omega}\|g_{\omega,0}J_{\omega}\|_{\infty,1}<\infty.$

We also assume a uniform covering condition222We could replace the covering
condition with the assumption of a strongly contracting potential. See [3] for
details. :

  1. (E4)

For every subinterval $J\subset[0,1]$ there is a $k=k(J)$ such that for a.e.
$\omega$ one has $T^{k}_{\omega}(J)=[0,1]$.

Concerning the open system we assume that the holes
$H_{\omega,\varepsilon}\subseteq[0,1]$ are chosen so that assumption ((A))
holds. We also assume for each $\omega\in\Omega$ and each $\varepsilon>0$ that
$H_{\omega,\varepsilon}$ is composed of a finite union of intervals such that

  1. (E5)

There is a uniform-in-$\varepsilon$ and uniform-in-$\omega$ upper bound on the
number of connected components of $H_{\omega,\varepsilon}$,

and

(E6) $\displaystyle\lim_{\varepsilon\to
0}\operatorname*{ess\sup}_{\omega}\text{{\rm Leb}}(H_{\omega,\varepsilon})=0,$

and

  1. (EX)

There exists an $\varepsilon>0$ and an open neighborhood
$\tilde{H}_{\omega,\varepsilon}\supseteq H_{\omega,\varepsilon}$ such that
$T_{\omega}(U_{\omega})\supseteq\tilde{H}_{\sigma\omega,\varepsilon}^{c}$,
where
$U_{\omega}:=\cup_{Z\in\mathcal{Z}_{\omega,0}}\overline{Z}_{\varepsilon}$ and
$\overline{Z_{\varepsilon}}$ denotes the closure of $Z_{\varepsilon}\in
A_{\omega,\varepsilon}:=\\{Z\cap\tilde{H}_{\omega,\varepsilon}^{c}:Z\in\mathcal{Z}_{\omega,0}\\}$
with $m(\left\\{\omega\in\Omega:\\#A_{\omega,\varepsilon}\geq 2\right\\})>0$.

######  Remark 6.3.

Assumption ((EX)) is satisfied for any random open system such that each map
contains at least two intervals of monotonicity, the holes are contained in
the interior of exactly one interval of monotonicity, and the image of the
complement of the hole is the full interval, i.e.
$T_{\omega}(H_{\omega,\varepsilon}^{c})=[0,1]$. In particular, ((EX)) is
satisfied if there exists a full branch outside of the hole.

The following proposition ensures that condition ((X)) holds.

######  Proposition 6.4.

The assumption ((EX)) implies ((X)).

###### Proof.

In light of Remark 3.5, to show that ((X)) is satisfied, it suffices to show
that there is some $\varepsilon>0$ such that
$X_{\omega,\infty,\varepsilon}\neq\emptyset$ for $m$-a.e. $\omega\in\Omega$.

Let $\tilde{T}_{\omega,Z}$ denote the continuous extension of $T_{\omega}$
onto $\overline{Z}_{\varepsilon}$ for each $Z_{\varepsilon}\in
A_{\omega,\varepsilon}$, and let $\tilde{X}_{\omega,\infty,\varepsilon}$
denote the survivor set for the open system consisting of the maps
$\tilde{T}_{\omega}$ and holes $\tilde{H}_{\omega,\varepsilon}$. By
Proposition 3.6 (taking
$V_{\omega}=[0,1]\backslash\tilde{H}_{\omega,\varepsilon}$ and
$U_{\omega,j}=\overline{Z}_{\varepsilon}$ for each $1\leq
j\leq\\#A_{\omega,\varepsilon}$) we see that
$\tilde{X}_{\omega,\infty,\varepsilon}$ is uncountable. Let

$\mathcal{D}_{\omega}:=\bigcup_{j\geq
0}\tilde{T}_{\omega}^{-j}(\cup_{Z_{\varepsilon}\in
A_{\sigma^{j}\omega,\varepsilon}}\overline{Z}_{\varepsilon}\backslash
Z_{\varepsilon}).$

Since the survivor set for the original (unmodified) open system
$X_{\omega,\infty,\varepsilon}\subseteq\tilde{X}_{\omega,\infty,\varepsilon}\backslash\mathcal{D}_{\omega}$,
and since $\mathcal{D}_{\omega}$ is at most countable, we must in fact have
that $X_{\omega,\infty,\varepsilon}\neq\emptyset$, thus satisfying ((X)). ∎

Further, we suppose that for $m$-a.e. $\omega\in\Omega$ and all
$\varepsilon>0$ sufficiently small

(E7) $\displaystyle T_{\omega}(\mathcal{J}_{\omega,\varepsilon})=[0,1]$

and there exists $n^{\prime}\geq 1$ and $\varepsilon_{0}>0$ such that 333Note
that the $9$ appearing in (E8) is not optimal. See [4, 2] for how this
assumption may be improved.

(E8)
$9\cdot\operatorname*{ess\sup}_{\omega}\|g_{\omega,0}^{(n^{\prime})}\|_{\infty,1}<\operatorname*{ess\inf}_{\omega}\inf_{0\leq\varepsilon\leq\varepsilon_{0}}\inf\mathcal{L}_{\omega,\varepsilon}^{n^{\prime}}\mathbbold{1},$

where

$\mathcal{L}_{\omega,\varepsilon}(f)(x):=\mathcal{L}_{\omega,0}(\mathbbold{1}_{H_{\omega,\varepsilon}^{c}}f)(x)=\sum_{y\in
T_{\omega}^{-1}(x)}g_{\omega,\varepsilon}(y)f(y),\quad f\in\text{{\rm BV}}$

and
$g_{\omega,\varepsilon}:=g_{\omega,0}\mathbbold{1}_{H_{\omega,\varepsilon}^{c}}$
as in Section 3.1.

Note that (E7) and (E3) together imply that
$\mathcal{L}_{\omega,\varepsilon}\mathbbold{1}(x)>0$ for all $x\in[0,1]$:

(6.8)
$\displaystyle\operatorname*{ess\inf}_{\omega}\inf_{\varepsilon\leq\varepsilon_{0}}\inf\mathcal{L}_{\omega,\varepsilon}\mathbbold{1}>\operatorname*{ess\inf}_{\omega}\inf
g_{\omega,0}>0,$

and since
$\|g_{\omega,\varepsilon}\|_{\infty,1}\leq\|g_{\omega,0}\|_{\infty,1}$ for all
$\varepsilon>0$, (E8) is equivalent to the following

(6.9) $\displaystyle
9\cdot\sup_{0\leq\varepsilon\leq\varepsilon_{0}}\operatorname*{ess\sup}_{\omega}\|g_{\omega,\varepsilon}^{(n^{\prime})}\|_{\infty,1}<\operatorname*{ess\inf}_{\omega}\inf_{0\leq\varepsilon\leq\varepsilon_{0}}\inf\mathcal{L}_{\omega,\varepsilon}^{n^{\prime}}\mathbbold{1}.$

######  Remark 6.5.

Note that the assumption (E7) is equivalent to there existing $N^{\prime}\geq
1$ such that for $m$-a.e. $\omega\in\Omega$ and all $\varepsilon\geq 0$
sufficiently small

$T_{\omega}^{N^{\prime}}(X_{\omega,N^{\prime}-1,\varepsilon})=[0,1].$

Indeed, since the surviving sets are forward invariant (3.4), we have that
$T_{\omega}^{N^{\prime}-1}(X_{\omega,N^{\prime}-1,\varepsilon})\subseteq
X_{\sigma^{N^{\prime}-1}\omega,0,\varepsilon}=\mathcal{J}_{\sigma^{N^{\prime}-1}\omega,\varepsilon}$,
and thus,

$[0,1]=T_{\omega}^{N^{\prime}}(X_{\omega,N^{\prime}-1,\varepsilon})\supseteq
T_{\sigma^{N^{\prime}-1}\omega}(\mathcal{J}_{\sigma^{N^{\prime}-1}\omega,\varepsilon}).$

For each $n\in\mathbb{N}$ and $\omega\in\Omega$ we let
$\mathscr{A}_{\omega,0}^{(n)}$ be the collection of all finite partitions of
$[0,1]$ such that

(6.10) $\displaystyle\text{{\rm var}}_{A_{i}}(g_{\omega,0}^{(n)})\leq
2\|g_{\omega,0}^{(n)}\|_{\infty,1}$

for each $\mathcal{A}=\left\\{A_{i}\right\\}\in\mathscr{A}_{\omega,0}^{(n)}$.
Given $\mathcal{A}\in\mathscr{A}_{\omega,0}^{(n)}$, let
$\widehat{}\mathcal{Z}_{\omega,\varepsilon}^{(n)}(\mathcal{A})$ be the
coarsest partition amongst all those finer than $\mathcal{A}$ and
$\mathcal{Z}_{\omega,0}^{(n)}$ such that all elements of
$\widehat{}\mathcal{Z}_{\omega,\varepsilon}^{(n)}(\mathcal{A})$ are either
disjoint from $X_{\omega,n-1,\varepsilon}$ or contained in
$X_{\omega,n-1,\varepsilon}$.

######  Remark 6.6.

Note that if $\text{{\rm var}}_{Z}(g_{\omega,0})\leq
2\|g_{\omega,0}^{(n)}\|_{\infty,1}$ for each
$Z\in\mathcal{Z}_{\omega,0}^{(n)}$ then we can take the partition
$\mathcal{A}=\mathcal{Z}_{\omega,0}^{(n)}$. Furthermore, the $2$ above can be
replaced by some $\hat{}\alpha\geq 0$ (depending on $g_{\omega,0}$) following
the techniques of [4, 2].

Define the subcollection

(6.11)
$\displaystyle\mathcal{Z}_{\omega,*,\varepsilon}^{(n)}:=\\{Z\in\widehat{}\mathcal{Z}_{\omega,\varepsilon}^{(n)}(\mathcal{A}):Z\subseteq
X_{\omega,n-1,\varepsilon}\\}.$

Recalling that
$g_{\omega,\varepsilon}^{(n)}:=g_{\omega,0}^{(n)}\mathbbold{1}_{X_{\omega,n-1,\varepsilon}}$,
(6.10) implies that

(6.12) $\displaystyle\text{{\rm var}}_{Z}(g_{\omega,\varepsilon}^{(n)})\leq
2\|g_{\omega,0}^{(n)}\|_{\infty,1}$

for each $Z\in\mathcal{Z}_{\omega,*,\varepsilon}^{(n)}$. We assume the
following covering condition for the open system

  1. (E9)

There exists $k_{o}(n^{\prime})\in\mathbb{N}$ such that for $m$-a.e.
$\omega\in\Omega$, all $\varepsilon>0$ sufficiently small, and all
$Z\in\mathcal{Z}_{\omega,*,\varepsilon}^{(n^{\prime})}$ we have
$T_{\omega}^{k_{o}(n^{\prime})}(Z)=[0,1]$, where $n^{\prime}$ is the number
coming from (E8).

######  Remark 6.7.

Note that the uniform open covering time assumption ((E9)) clearly holds if
((E4)) holds and if there are only finitely many maps $T_{\omega}$. In Remark
C.2 we present an alternative assumption to ((E9)).

The following lemma extends several results in [22] from the specific weight
$g_{\omega,0}=1/|T^{\prime}_{\omega}|$ to general weights satisfying the
conditions just outlined.

######  Lemma 6.8.

Assume that a family of random piecewise-monotonic interval maps
$\\{T_{\omega}\\}$ satisfies (E2), (E3), and ((E4)), as well as (E8) and
((MC)) for $\varepsilon=0$. Then ((C1’)) and the $\varepsilon=0$ parts of
((C2)), ((C3)), ((C4’)), ((C5’)), and ((C7’)) as well as ((CCM)) hold.
Further, $\nu_{\omega,0}$ is fully supported, condition ((C4’)) holds with
$C_{f}=K$, for some $K<\infty$, and with $\alpha(N)=\gamma^{N}$ for some
$\gamma<1$.

###### Proof.

See Appendix A. ∎

In what follows we consider transfer operators acting on the Banach spaces
$\mathcal{B}_{\omega}=\text{{\rm BV}}_{\nu_{\omega,0}}$ for a.e.
$\omega\in\Omega$. The norm we will use is
$\|\cdot\|_{\mathcal{B}_{\omega}}=\|\cdot\|_{\text{{\rm BV}}}:=\text{{\rm
var}}(\cdot)+\nu_{\omega,0}(|\cdot|)$. As $\nu_{\omega,0}$ is fully supported
and non-atomic (Lemma 6.8), Proposition 6.2 implies that

(6.13) $(1/2)\|f\|_{\text{{\rm BV}}_{1}}\leq\|f\|_{\mathcal{B}_{\omega}}\leq
2\|f\|_{\text{{\rm BV}}_{1}}$

for $m$-a.e. $\omega\in\Omega$ and $f\in\text{{\rm BV}}$. Furthermore,
applying (6.13) twice, we see that

(6.14)
$(1/4)\|f\|_{\mathcal{B}_{\omega}}\leq\|f\|_{\mathcal{B}_{\sigma^{n}\omega}}\leq
4\|f\|_{\mathcal{B}_{\omega}}$

for $m$-a.e. $\omega\in\Omega$ and all $n\in\mathbb{Z}$. It follows from the
proof of Proposition 6.2 that

(6.15) $\|f\|_{\infty,\omega}=\|f\|_{\infty,1}$

for all $f\in\text{{\rm BV}}$ and $m$-a.e. $\omega\in\Omega$, where
$\|\makebox[4.30554pt]{{$\cdot$}}\|_{\infty,\omega}$ denotes the supremum norm
with respect to $\nu_{\omega,0}$. From (6.13) we see that ((B)) is clearly
satisfied.

From Lemma 6.8 we have that
$\lambda_{\omega,0}:=\nu_{\sigma\omega,0}(\mathcal{L}_{\omega,0}\mathbbold{1})$
and thus we may update (6.8) to get

(6.16)
$\displaystyle\operatorname*{ess\inf}_{\omega}\lambda_{\omega,0}^{n^{\prime}}\geq\operatorname*{ess\inf}_{\omega}\inf_{\varepsilon\leq\varepsilon_{0}}\inf\mathcal{L}_{\omega,\varepsilon}^{n^{\prime}}\mathbbold{1}\geq\operatorname*{ess\inf}_{\omega}\inf
g_{\omega,0}^{(n^{\prime})}>0.$

Note that since the conditions ((B)), ((X)), and ((CCM)) have been verified
and we have assumed ((A)) and ((MC)), we see that
$(\Omega,m,\sigma,[0,1],T,\text{{\rm
BV}},\mathcal{L}_{0},\varphi_{0},H_{\varepsilon})$ forms a random open system
as defined in Section 3.1 for all $\varepsilon>0$ sufficiently small. We now
use hyperbolicity of the $\varepsilon=0$ transfer operator cocycle to
guarantee that we have hyperbolic cocycles for small $\varepsilon>0$, which
will yield ((C2)), ((C3)), ((C4’)), ((C5’)), ((C6)), and ((C7’)) for small
positive $\varepsilon$.

######  Lemma 6.9.

Assume that the conditions (E1)–((E9)) hold for the random open system
$(\Omega,\allowbreak m,\allowbreak\sigma,\allowbreak[0,\allowbreak
1],\allowbreak T,\allowbreak\text{{\rm
BV}}_{1},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$. Then for sufficiently small $\varepsilon>0$, conditions
((C2)), ((C3)), ((C4’)), ((C5’)), ((C6)), and ((C7’)) hold. Furthermore, the
functionals $\nu_{\omega,\varepsilon}\in\text{{\rm BV}}_{1}^{*}$ can be
identified with non-atomic Borel measures.

###### Proof.

For each $\omega$ and $\varepsilon>0$ we define
$\hat{}\mathcal{L}_{\omega,\varepsilon}:=\lambda_{\omega,0}^{-1}\mathcal{L}_{\omega,\varepsilon}$;
note that $\hat{}\mathcal{L}_{\omega,0}=\tilde{}\mathcal{L}_{\omega,0}$. Our
strategy is to apply Theorem 4.8 [17], to conclude that for small
$\varepsilon$ the cocycles $\\{\hat{}\mathcal{L}_{\omega,\varepsilon}\\}$ are
uniformly hyperbolic when considered as cocycles on the Banach space
$(\text{{\rm BV}}_{1},\|\cdot\|_{\text{{\rm BV}}_{1}})$. Because of (6.13), we
will conclude the existence of a uniformly hyperbolic splitting in
$\|\cdot\|_{\mathcal{B}_{\omega}}$ for a.e. $\omega$.

First, we note that Theorem 4.8 [17] assumes that the Banach space on which
the transfer operator cocycle acts is separable. A careful check of the proof
of Theorem 4.8 [17] shows that it holds for the Banach space $(\text{{\rm
BV}}_{1},\|\cdot\|_{\text{{\rm BV}}_{1}})$ under the alternative condition
((MC)) (see Appendix B). To apply Theorem A [17] we require, in our notation,
that:

  1. (1)

$\hat{}\mathcal{L}_{\omega,0}$ is a hyperbolic transfer operator cocycle on
$\text{{\rm BV}}_{1}$ with norm $\|\cdot\|_{\text{{\rm BV}}_{1}}$ and a one-
dimensional leading Oseledets space (see Definition 3.1 [17]), and slow and
fast growth rates $0<\gamma<\Gamma$, respectively. We will construct $\gamma$
and $\Gamma$ shortly.

  2. (2)

The family of cocycles
$\\{\hat{}\mathcal{L}_{\omega,\varepsilon}\\}_{0\leq\varepsilon\leq\varepsilon_{0}}$
satisfy a uniform Lasota–Yorke inequality

$\|\hat{}\mathcal{L}^{k}_{\omega,\varepsilon}f\|_{\text{{\rm BV}}_{1}}\leq
A\alpha^{k}\|f\|_{\text{{\rm BV}}_{1}}+B^{k}\|f\|_{1}$

for a.e. $\omega$ and $0\leq\varepsilon\leq\varepsilon_{0}$, where
$\alpha\leq\gamma<\Gamma\leq B$.

  3. (3)

$\lim_{\varepsilon\to
0}\operatorname*{ess\sup}_{\omega}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\hat{}\mathcal{L}_{\omega,0}-\hat{}\mathcal{L}_{\omega,\varepsilon}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}=0$,
where
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\makebox[4.30554pt]{{$\cdot$}}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
is the $\text{{\rm BV}}-L^{1}(\text{{\rm Leb}})$ triple norm.

By Lemma 6.8 we obtain a unique measurable family of equivariant functions
$\\{\varphi_{\omega,0}\\}$ satisfying ((C7’)) and ((C5’)) for $\varepsilon=0$.
We have the equivariant splitting $\mathrm{span}\\{\varphi_{\omega,0}\\}\oplus
V_{\omega}$, where $V_{\omega}=\\{f\in\text{{\rm
BV}}_{1}:\nu_{\omega,0}(f)=0\\}$. We claim that this splitting is hyperbolic
in the sense of Definition 3.1 [17]; this will yield item (1) above. To show
this, we verify conditions (H1)–(H3) in [17]. In our setting, Condition (H1)
[17] requires the norm of the projection onto the top space spanned by
$\varphi_{\omega,0}$, along the annihilator of $\nu_{\omega,0}$, to be
uniformly bounded in $\omega$. This is true because this projection acting on
$f\in\text{{\rm BV}}_{1}$ is $\nu_{\omega,0}(f)\varphi_{\omega,0}$ and
therefore

$\|\nu_{\omega,0}(f)\varphi_{\omega,0}\|_{\text{{\rm
BV}}_{1}}\leq\operatorname*{ess\sup}_{\omega}\|\varphi_{\omega,0}\|_{\text{{\rm
BV}}_{1}}\cdot\nu_{\omega,0}(f)\leq
2\operatorname*{ess\sup}_{\omega}\|\varphi_{\omega,0}\|_{\text{{\rm
BV}}_{1}}\cdot\|f\|_{\text{{\rm BV}}_{1}},$

using ((C5’)) and equivalence of $\|\cdot\|_{\text{{\rm BV}}_{1}}$ and
$\|\cdot\|_{\mathcal{B}_{\omega}}$ (6.13). Next, we define

$\alpha^{n^{\prime}}:=\frac{9\operatorname*{ess\sup}_{\omega}\|g_{\omega,0}^{(n^{\prime})}\|_{\infty,1}}{\operatorname*{ess\inf}_{\omega}\inf_{\varepsilon\geq
0}\inf\mathcal{L}^{n^{\prime}}_{\omega,0}\mathbbold{1}}<1,$

which is possible by (E8). Condition (H2) requires
$\|\hat{}\mathcal{L}^{n}_{\omega,0}\varphi_{\omega,0}\|_{\text{{\rm
BV}}_{1}}\geq C\Gamma^{n}\|\varphi_{\omega,0}\|_{\text{{\rm BV}}_{1}}$ for
some $C>0$, $\Gamma>0$, all $n$ and a.e. $\omega$. By ((C7’)) one has

$\|\hat{}\mathcal{L}^{n}_{\omega,0}\varphi_{\omega,0}\|_{\text{{\rm
BV}}_{1}}=\|\varphi_{\sigma^{n}\omega,0}\|_{\text{{\rm
BV}}_{1}}\geq\operatorname*{ess\inf}_{\omega}\inf\varphi_{\sigma^{n}\omega,0}>0,$

and thus we obtain (H2) with $C=\Gamma=1$. Condition (H3) requires
$\|\hat{}\mathcal{L}^{n}_{\omega,0}|_{V_{\omega}}\|_{\text{{\rm BV}}_{1}}\leq
K\gamma^{n}$ for some $K<\infty$, $\alpha\leq\gamma<1$, all $n$ and a.e.
$\omega$. This is provided by the $\varepsilon=0$ part of ((C4’))—specifically
the stronger exponential version guaranteed by Lemma 6.8—and the equivalence
of $\|\cdot\|_{\text{{\rm BV}}_{1}}$ and
$\|\cdot\|_{\mathcal{B}_{\sigma^{n}\omega}}$.

For item (2) we begin with the Lasota–Yorke inequality for $\text{{\rm
var}}(\cdot)$ and $\nu_{\omega,0}(|\cdot|)$ provided by the final line of the
proof of Lemma C.1 (equation (C.7)). Dividing through by
$\lambda_{\omega,0}^{n^{\prime}}$ we obtain

$\displaystyle\text{{\rm
var}}(\hat{}\mathcal{L}_{\omega,\varepsilon}^{n^{\prime}}f)$
$\displaystyle\leq\frac{9\|g_{\omega,\varepsilon}^{({n^{\prime}})}\|_{\infty,1}}{\lambda_{\omega,0}^{n^{\prime}}}\text{{\rm
var}}(f)+\frac{8\|g_{\omega,\varepsilon}^{(n^{\prime})}\|_{\infty,1}}{\lambda_{\omega,0}^{n^{\prime}}\min_{Z\in\mathcal{Z}_{\omega,*,\varepsilon}^{(n^{\prime})}(\mathcal{A})}\nu_{\omega,0}(Z)}\nu_{\omega,0}(|f|)$
(6.17) $\displaystyle\leq\alpha^{n^{\prime}}\text{{\rm
var}}(f)+\frac{\alpha^{n^{\prime}}}{\min_{Z\in\mathcal{Z}_{\omega,*,\varepsilon}^{(n^{\prime})}(\mathcal{A})}\nu_{\omega,0}(Z)}\nu_{\omega,0}(|f|),$

noting that
$\operatorname*{ess\inf}_{\omega}\min_{Z\in\mathcal{Z}_{\omega,*,\varepsilon}^{(n^{\prime})}(\mathcal{A})}\nu_{\omega,0}(Z)>0$
since the uniform open covering assumption ((E9)) together with (E3) and
(6.16) imply that and equivariance of the backward adjoint cocycle together
imply that for $Z\in\mathcal{Z}_{\omega,*,\varepsilon}^{(n^{\prime})}$ we have

(6.18)
$\displaystyle\nu_{\omega,0}(Z)=\nu_{\sigma^{k_{o}(n^{\prime})}\omega,0}\left(\left(\lambda_{\omega,0}^{k_{o}(n^{\prime})}\right)^{-1}\mathcal{L}_{\omega,0}^{k_{o}(n^{\prime})}\mathbbold{1}_{Z}\right)\geq\frac{\inf
g_{\omega,0}^{k_{o}(n^{\prime})}}{\lambda_{\omega,0}^{k_{o}(n^{\prime})}}>0.$

As the holes $H_{\omega,\varepsilon}$ are composed of finite unions of
disjoint intervals, assumption (E6) implies that the radii of each of these
intervals must go to zero as $\varepsilon\to 0$. Thus, using (E6) together
with the fact that $\nu_{\omega,0}$ is fully supported and non-atomic, we see
that ((C6)) must hold.

We construct a uniform Lasota–Yorke inequality for all $n$ in the usual way by
using blocks of length $jn^{\prime}$; we write this as

(6.19) $\text{{\rm var}}(\hat{}\mathcal{L}_{\omega,\varepsilon}^{n}f)\leq
A_{1}\alpha^{n}\text{{\rm var}}(f)+A_{2}^{n}\nu_{\omega,0}(|f|)$

for some $A_{2}>\alpha$. We now wish to convert this to an inequality

(6.20) $\text{{\rm var}}(\hat{}\mathcal{L}_{\omega,\varepsilon}^{n}f)\leq
A_{1}^{\prime}\alpha^{n}\text{{\rm var}}(f)+(A_{2}^{\prime})^{n}\|f\|_{1}.$

In light of (E2), (6.16), (E1), and (6.5), we see that there is a constant $B$
so that for $m$-a.e. $\omega\in\Omega$ we have

$\displaystyle\|\hat{}\mathcal{L}_{\omega,\varepsilon}^{n^{\prime}}f\|_{1}$
$\displaystyle=\left(\lambda_{\omega,0}^{n^{\prime}}\right)^{-1}\int_{[0,1]}\left|\sum_{y\in
T_{\omega}^{-n^{\prime}}x}g_{\omega,0}^{(n^{\prime})}(y)\hat{X}_{\omega,n^{\prime}-1,\varepsilon}(y)f(y)\right|d\text{{\rm
Leb}}(x)$
$\displaystyle\leq\frac{\|g_{\omega,0}^{(n^{\prime})}J_{\omega}^{(n^{\prime})}\|_{\infty,1}}{\lambda_{\omega,0}^{n^{\prime}}}\int_{[0,1]}\left|\sum_{y\in
T_{\omega}^{-n^{\prime}}(x)}\frac{f(y)}{J_{\omega}^{(n^{\prime})}(y)}\right|d\text{{\rm
Leb}}(x)$
$\displaystyle=\frac{\|g_{\omega,0}^{(n^{\prime})}J_{\omega}^{(n^{\prime})}\|_{\infty,1}}{\lambda_{\omega,0}^{n^{\prime}}}\int_{[0,1]}\left|P_{\omega}^{n^{\prime}}(f)\right|d\text{{\rm
Leb}}(x)$
$\displaystyle\leq\frac{\|g_{\omega,0}^{(n^{\prime})}J_{\omega}^{(n^{\prime})}\|_{\infty,1}}{\lambda_{\omega,0}^{n^{\prime}}}\|f\|_{1}\leq
B^{n^{\prime}}\|f\|_{1}.$

Using the non-atomicicty of $\nu_{\omega,0}$ from ((CCM)) (shown in (6.8)) and
the fact that $\text{{\rm var}}(|f|)\leq\text{{\rm var}}(f)$, we may apply
Lemma 5.2 [10] to $\nu_{\omega,0}$ to obtain that for each $\zeta>0$, there is
a $B_{\zeta}<\infty$ such that $\nu_{\omega,0}(|f|)\leq\zeta\text{{\rm
var}}(f)+B_{\zeta}\|f\|_{1}$. Now using (6.19) we see that

$\displaystyle\|\hat{\mathcal{L}}_{\omega,\varepsilon}^{n}f\|_{\text{{\rm
BV}}_{1}}$ $\displaystyle=\text{{\rm
var}}(\hat{\mathcal{L}}_{\omega,\varepsilon}^{n}f)+\|\hat{\mathcal{L}}_{\omega,\varepsilon}^{n}(f)\|_{1}$
$\displaystyle\leq A_{1}\alpha^{n}\text{{\rm
var}}(f)+A_{2}^{n}\nu_{\omega,0}(|f|)+B^{n}\|f\|_{1}$
$\displaystyle\leq(A_{1}\alpha^{n}+A_{2}^{n}\zeta)\text{{\rm
var}}(f)+(B^{n}+A_{2}^{n}B_{\zeta})\|f\|_{1}$
$\displaystyle\leq(A_{1}\alpha^{n}+A_{2}^{n}\zeta)\|f\|_{\text{{\rm
BV}}_{1}}+(B^{n}+A_{2}^{n}(B_{\zeta}-\zeta)-A_{1}\alpha^{n}))\|f\|_{1}.$

Selecting $\zeta$ sufficiently small and $n^{\prime\prime}$ sufficiently large
so that $C(\alpha^{n^{\prime\prime}}+\zeta)<1$ we again (by proceeding in
blocks of $n^{\prime\prime}$) arrive at a uniform Lasota–Yorke inequality of
the form (6.20) for all $n\geq 0$.

For item (3) we note that

$\displaystyle{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\hat{}\mathcal{L}_{\omega,0}-\hat{}\mathcal{L}_{\omega,\varepsilon}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
$\displaystyle:=\sup_{\|f\|_{\text{{\rm
BV}}_{1}}=1}\|(\hat{}\mathcal{L}_{\omega,0}-\hat{}\mathcal{L}_{\omega,\varepsilon})f\|_{1}=\sup_{\|f\|_{\text{{\rm
BV}}_{1}}=1}\|\hat{}\mathcal{L}_{\omega,0}(f\mathbbold{1}_{H_{\omega,\varepsilon}})\|_{1}\leq\|\hat{}\mathcal{L}_{\omega,0}(\mathbbold{1}_{H_{\omega,\varepsilon}})\|_{1}$
$\displaystyle=\lambda_{\omega,0}^{-1}\int_{[0,1]}\left|\sum_{y\in
T_{\omega}^{-1}(x)}g_{\omega,0}(y)\mathbbold{1}_{H_{\omega,\varepsilon}}(y)\right|d\text{{\rm
Leb}}(x)$
$\displaystyle\leq\frac{\|g_{\omega,0}J_{\omega}\|_{\infty,1}}{\lambda_{\omega,0}}\int_{[0,1]}\left|\sum_{y\in
T_{\omega}^{-1}(x)}\frac{\mathbbold{1}_{H_{\omega,\varepsilon}}(y)}{J_{\omega}(y)}\right|d\text{{\rm
Leb}}(x)$
$\displaystyle=\frac{\|g_{\omega,0}J_{\omega}\|_{\infty,1}}{\lambda_{\omega,0}}\int_{[0,1]}\left|P_{\omega}(\mathbbold{1}_{H_{\omega,\varepsilon}})\right|d\text{{\rm
Leb}}(x)$
$\displaystyle\leq\operatorname*{ess\sup}_{\omega}\frac{\|g_{\omega,0}J_{\omega}\|_{\infty,1}}{\lambda_{\omega,0}}\cdot\operatorname*{ess\sup}_{\omega}\mathrm{Leb}(H_{\omega,\varepsilon}).$

Because (E2), (E1) and (6.16) imply
$\operatorname*{ess\sup}_{\omega}\nicefrac{{\|g_{\omega,0}J_{\omega}\|_{\infty,1}}}{{\lambda_{\omega,0}}}<\infty$,
and since (E6) implies that $\lim_{\varepsilon\to
0}\operatorname*{ess\sup}_{\omega}\mathrm{Leb}(H_{\omega,\varepsilon})=0$, we
obtain item (3).

We may now apply Theorem 4.8 [17] to conclude that given $\delta>0$ there is
an $\varepsilon_{0}>0$ such that for all $\varepsilon\leq\varepsilon_{0}$ the
cocycle generated by $\hat{}\mathcal{L}_{\varepsilon}$ is hyperbolic, with

  1. (i)

the existence of an equivariant family
$\hat{\varphi}_{\omega,\varepsilon}\in\text{{\rm BV}}$ with
$\operatorname*{ess\sup}_{\omega}\|\hat{\varphi}_{\omega,\varepsilon}-\varphi_{\omega,0}\|_{\text{{\rm
BV}}_{1}}<\delta$,

  2. (ii)

existence of corresponding Lyapunov multipliers
$\hat{\lambda}_{\omega,\varepsilon}$ satisfying
$|\hat{\lambda}_{\omega,\varepsilon}-1|<\delta$,

  3. (iii)

operators $\hat{Q}_{\omega,\varepsilon}$ satisfying
$\|(\hat{Q}_{\omega,\varepsilon})^{n}\|_{\text{{\rm BV}}_{1}}\leq
K^{\prime}(\gamma+\delta)^{n}$, where $\gamma$ is the decay rate for
$Q_{\omega,0}$ from the proof of Lemma 6.8.

To obtain an $\hat{}\mathcal{L}_{\omega,\varepsilon}^{*}$-equivariant family
of linear functionals $\hat{\nu}_{\omega,\varepsilon}\in\text{{\rm
BV}}_{1}^{*}$ we apply Corollary 2.5 [23]. Using the one-dimensionality of the
leading Oseledets space for the forward cocycle, this result shows that the
leading Oseledets space for the backward adjoint cocycle is also one-
dimensional. This leading Oseledets space is spanned by some
$\hat{\nu}_{\omega,\varepsilon}\in\text{{\rm BV}}_{1}^{*}$, satisfying
$\hat{\nu}_{\sigma\omega,\varepsilon}(\hat{}\mathcal{L}_{\omega,\varepsilon}(f))=\hat{}\vartheta_{\omega,\varepsilon}\hat{\nu}_{\omega,\varepsilon}(f)$,
for Lyapunov multipliers $\hat{}\vartheta_{\omega,\varepsilon}$. By Lemma 2.6
[23], we may scale the $\hat{\nu}_{\omega,\varepsilon}$ so that
$\hat{\nu}_{\omega,\varepsilon}(\hat{\varphi}_{\omega,\varepsilon})=1$ for
a.e. $\omega$. We show that in fact
$\hat{}\vartheta_{\omega,\varepsilon}=\hat{\lambda}_{\omega,\varepsilon}$
$m$-a.e. Indeed,

$1=\hat{\nu}_{\sigma\omega,\varepsilon}(\hat{\varphi}_{\sigma\omega,\varepsilon})=\hat{\nu}_{\sigma\omega,\varepsilon}(\hat{}\mathcal{L}_{\omega,\varepsilon}\hat{\varphi}_{\omega,\varepsilon}/\hat{\lambda}_{\omega,\varepsilon})=(\hat{}\vartheta_{\omega,\varepsilon}/\hat{\lambda}_{\omega,\varepsilon})\hat{\nu}_{\omega,\varepsilon}(\hat{\varphi}_{\omega,\varepsilon})=\hat{}\vartheta_{\omega,\varepsilon}/\hat{\lambda}_{\omega,\varepsilon}.$

Note that
$\mathcal{L}_{\omega,\varepsilon}=\lambda_{\omega,0}\hat{}\mathcal{L}_{\omega,\varepsilon}$,
and we now define
$\varphi_{\omega,\varepsilon},\lambda_{\omega,\varepsilon},Q_{\omega,\varepsilon}$,
and $\nu_{\omega,\varepsilon}$ by the following:

$\displaystyle\varphi_{\omega,\varepsilon}$
$\displaystyle:=\frac{1}{\nu_{\omega,0}(\hat{\varphi}_{\omega,\varepsilon})}\cdot\hat{\varphi}_{\omega,\varepsilon},$
$\displaystyle\nu_{\omega,\varepsilon}(f)$
$\displaystyle:=\nu_{\omega,0}(\hat{\varphi}_{\omega,\varepsilon})\hat{\nu}_{\omega,\varepsilon}(f),$
$\displaystyle\lambda_{\omega,\varepsilon}$
$\displaystyle:=\lambda_{\omega,0}\frac{\nu_{\sigma\omega,0}(\hat{\varphi}_{\sigma\omega,\varepsilon})}{\nu_{\omega,0}(\hat{\varphi}_{\omega,\varepsilon})}\hat{}\lambda_{\omega,\varepsilon},$
$\displaystyle Q_{\omega,\varepsilon}(f)$
$\displaystyle:=\frac{\nu_{\omega,0}(\hat{\varphi}_{\omega,\varepsilon})}{\nu_{\sigma\omega,0}(\hat{\varphi}_{\sigma\omega,\varepsilon})}\hat{Q}_{\omega,\varepsilon}(f)$

Clearly all of the properties of ((C2)) and ((C3)) are now satisfied except
for the log-integrability of $\lambda_{\omega,\varepsilon}$ in ((C2)). To
demonstrate this last point, we note that by uniform hyperbolicity of the
perturbed cocycles, $\lambda^{\prime}_{\omega,\varepsilon}$ are uniformly
bounded below and are therefore log-integrable. Since

$\displaystyle|1-\nu_{\omega,0}(\hat{\varphi}_{\omega,\varepsilon})|=|\nu_{\omega,0}(\varphi_{\omega,0})-\nu_{\omega,0}(\hat{\varphi}_{\omega,\varepsilon})|\leq\|\hat{\varphi}_{\omega,\varepsilon}-\varphi_{\omega,0}\|_{\text{{\rm
BV}}_{1}}<\delta,$

the $\lambda_{\omega,\varepsilon}$ are uniformly small perturbations of the
$\hat{\lambda}_{\omega,\varepsilon}$, and since $\lambda_{\omega,0}$ is
$\log$-integrable by (E2) and (6.16), we must therefore have that the log
integrability condition on $\lambda_{\omega,\varepsilon}$ in ((C2)) is
satisfied. Point ((i)) above, combined with the $\varepsilon=0$ part of
((C5’)) (resp. ((C7’))) and the uniform estimate for
$|1-\nu_{\omega,0}(\hat{\varphi}_{\omega,\varepsilon})|$, immediately yields
the $\varepsilon>0$ part of ((C5’)) (resp. ((C7’))). Point ((iii)) above
combined with the same estimates also ensures that the norm of
$\|Q_{\omega,\varepsilon}^{n}\|_{\infty,1}$ decays exponentially fast,
uniformly in $\omega$ and $\varepsilon$, satisfying the stronger exponential
version of ((C4’)). In fact point ((iii)) implies the stronger statement that
$\|Q_{\omega,\varepsilon}^{n}\|_{\text{{\rm BV}}_{1}}$ decays exponentially
fast, uniformly in $\omega$ and $\varepsilon$.

Finally we show that $\nu_{\omega,\varepsilon}:C^{0}([0,1])\to\mathbb{C}$ is a
positive linear functional with $\nu_{\omega,\varepsilon}(f)\in\mathbb{R}$ if
$f$ is real. From this fact it will follow by Riesz-Markov (e.g. Theorem
A.3.11 [49]) that $\nu_{\omega,\varepsilon}$ can be identified with a real
finite Borel measure on $[0,1]$. By linearity we may consider the two cases:
(i) $f=\varphi_{\omega,\varepsilon}>0$ (the generator of the leading Oseledets
space) and (ii) $f\in F_{\omega,\varepsilon}$, where $F_{\omega,\varepsilon}$
is the Oseledets space complementary to
$\mathrm{span\\{\varphi_{\omega,\varepsilon}\\}}$. In case (i)
$\nu_{\omega,\varepsilon}(\varphi_{\omega,\varepsilon})=1>0$. In case (ii),
Lemma 2.6 [23] implies $\nu_{\omega,\varepsilon}(f)=0$. In summary we see that
$\nu_{\omega,\varepsilon}$ is positive. ∎

######  Remark 6.10.

As we have just shown that assumptions ((C1’)), ((C2)), ((C3)), ((C4’)),
((C5’)), ((C6)), ((C7’)) (Lemmas 6.8 and 6.9), we see that Proposition 4.4
holds as well as Theorem 4.6 for the random open system
$(\Omega,m,\sigma,[0,1],T,\text{{\rm
BV}},\mathcal{L}_{0},\nu_{0},\varphi_{0},H_{\varepsilon})$ under the
additional assumption of ((C8)). In light of Remark 5.3, if we assume ((S)) in
addition to ((C8)) then both Corollary 4.9 and Theorem 5.5 apply.

The following theorem is the main result of this section and elaborates on the
dynamical significance of the perturbed objects produced in Lemma 6.9.

######  Theorem 6.11.

Suppose
$(\Omega,m,\sigma,[0,1],T,\mathcal{B},\mathcal{L}_{0},\nu_{0},\varphi_{0},H_{\varepsilon})$
is a random open system and that the assumptions of Lemma 6.9 hold. Then there
exists $\varepsilon_{0}>0$ sufficiently small such that for every
$0\leq\varepsilon<\varepsilon_{0}$ we have the following:

  1. (1)

There exists a unique random probability measure
$\zeta_{\varepsilon}=\left\\{\zeta_{\omega,\varepsilon}\right\\}_{\omega\in\Omega}$
on $[0,1]$ such that, for $\varepsilon>0$, $\zeta_{\omega,\varepsilon}$ is
supported in $X_{\omega,\infty,\varepsilon}$ and

$\displaystyle\zeta_{\sigma\omega,\varepsilon}(\mathcal{L}_{\omega,\varepsilon}f)=\rho_{\omega,\varepsilon}\zeta_{\omega,\varepsilon}(f),$

for $m$-a.e. $\omega\in\Omega$ and each $f\in\text{{\rm BV}}$, where

$\displaystyle\rho_{\omega,\varepsilon}:=\zeta_{\sigma\omega,\varepsilon}(\mathcal{L}_{\omega,\varepsilon}\mathbbold{1}).$

Furthermore, for $\varepsilon=0$ we have $\zeta_{\omega,0}=\nu_{\omega,0}$ and
$\rho_{\omega,0}=\lambda_{\omega,0}$ and for $\varepsilon>0$ we have that
$C_{1}^{-1}\leq\rho_{\omega,\varepsilon}\leq C_{1}$ for $m$-a.e.
$\omega\in\Omega$.

  2. (2)

There exists a measurable function
$h_{\varepsilon}:\Omega\times[0,1]\to(0,\infty)$ such that
$\zeta_{\omega,\varepsilon}(h_{\omega,\varepsilon})=1$ and

$\displaystyle\mathcal{L}_{\omega,\varepsilon}h_{\omega,\varepsilon}=\rho_{\omega,\varepsilon}h_{\sigma\omega,\varepsilon}$

for $m$-a.e. $\omega\in\Omega$. Moreover, $h_{\varepsilon}$ is unique modulo
$\zeta_{\varepsilon}$, and there exists $C\geq 1$ such that $C^{-1}\leq
h_{\omega,\varepsilon}\leq C$ for $m$-a.e. $\omega\in\Omega$. Furthermore, for
$m$-a.e. $\omega\in\Omega$ we have that
$\rho_{\omega,\varepsilon}\to\rho_{\omega,0}=\lambda_{\omega,0}$ and
$h_{\omega,\varepsilon}\to h_{\omega,0}=\varphi_{\omega,0}$ (in
$\mathcal{B}_{\omega}$) as $\varepsilon\to 0$, where $\varphi_{\omega,0}$ and
$\lambda_{\omega,0}$ are defined in Lemma 6.8.

  3. (3)

The random measure
$\mu_{\varepsilon}=\left\\{\mu_{\omega,\varepsilon}:=h_{\omega,\varepsilon}\zeta_{\omega,\varepsilon}\right\\}_{\omega\in\Omega}$
is a $T$-invariant and ergodic random probability measure whose fiberwise
support, for $\varepsilon>0$, is contained in $X_{\omega,\infty,\varepsilon}$.
Furthermore, $\mu_{\varepsilon}$ is the unique relative equilibrium state,
i.e.

$\displaystyle\int_{\Omega}\log\rho_{\omega,\varepsilon}\ dm(\omega)$
$\displaystyle=:\mathcal{E}P_{\varepsilon}(\log
g_{0})=h_{\mu_{\varepsilon}}(T)+\int_{\mathcal{J}_{0}}\log
g_{0}\,d{\mu_{\varepsilon}}=\sup_{\eta_{\varepsilon}\in\mathcal{P}_{T,m}^{H_{\varepsilon}}(\mathcal{J}_{0})}h_{\eta_{\varepsilon}}(T)+\int_{\mathcal{J}_{0}}\log
g_{0}\,d\eta_{\varepsilon},$

where $h_{\eta_{\varepsilon}}(T)$ is the entropy of the measure
$\eta_{\varepsilon}$, $\mathcal{E}P_{\varepsilon}(\log g_{0})$ is the expected
pressure of the weight function
$g_{0}=\left\\{g_{\omega,0}\right\\}_{\omega\in\Omega}$, and
$\mathcal{P}_{T,m}^{H_{\varepsilon}}(\mathcal{J}_{0})$ denotes the collection
of $T$-invariant random probability measures $\eta_{\varepsilon}$ on
$\mathcal{J}_{0}$ whose disintegration
$\\{\eta_{\omega,\varepsilon}\\}_{\omega\in\Omega}$ satisfies
$\eta_{\omega,\varepsilon}(H_{\omega,\varepsilon})=0$ for $m$-a.e.
$\omega\in\Omega$. Furthermore, $\lim_{\varepsilon\to
0}\mathcal{E}P_{\varepsilon}(\log g_{0})=\mathcal{E}P_{0}(\log
g_{0}):=\int_{\Omega}\log\lambda_{\omega,0}\,dm.$

  4. (4)

For $\varepsilon>0$, let
$\varrho_{\varepsilon}=\left\\{\varrho_{\omega,\varepsilon}\right\\}_{\omega\in\Omega}$
be the random probability measure with fiberwise support in $[0,1]\backslash
H_{\omega,\varepsilon}$ whose disintegrations are given by

$\displaystyle\varrho_{\omega,\varepsilon}(f):=\frac{\nu_{\omega,0}\left(\mathbbold{1}_{H_{\omega,\varepsilon}^{c}}h_{\omega,\varepsilon}f\right)}{\nu_{\omega,0}\left(\mathbbold{1}_{H_{\omega,\varepsilon}^{c}}h_{\omega,\varepsilon}\right)}$

for all $f\in\text{{\rm BV}}$. $\varrho_{\omega,\varepsilon}$ is the unique
random conditionally invariant probability measure that is absolutely
continuous (with respect to
$\left\\{\nu_{\omega,0}\right\\}_{\omega\in\Omega}$) with density of bounded
variation.

  5. (5)

For each $f\in\text{{\rm BV}}$ there exists $D>0$ and
$\kappa_{\varepsilon}\in(0,1)$ such that for $m$-a.e. $\omega\in\Omega$ and
all $n\in\mathbb{N}$ we have

$\displaystyle\|\left(\rho_{\omega,\varepsilon}^{n}\right)^{-1}\mathcal{L}_{\omega,\varepsilon}^{n}f-\zeta_{\omega,\varepsilon}(f)h_{\sigma^{n}\omega,\varepsilon}\|_{\mathcal{B}_{\sigma^{n}\omega}}\leq
D\|f\|_{\mathcal{B}_{\omega}}\kappa_{\varepsilon}^{n}.$

Furthermore, for all $A\in\mathscr{B}$ and $f\in\text{{\rm BV}}$ we have

$\displaystyle\left\lvert\nu_{\omega,0}\left(T_{\omega}^{-n}(A)\,\rvert\,X_{\omega,n,\varepsilon}\right)-\varrho_{\sigma^{n}\omega,\varepsilon}(A)\right\rvert\leq
D\kappa_{\varepsilon}^{n},$

and

$\displaystyle\left\lvert\frac{\varrho_{\omega,\varepsilon}\left(f\rvert_{X_{\omega,n,\varepsilon}}\right)}{\varrho_{\omega,\varepsilon}(X_{\omega,n,\varepsilon})}-\mu_{\omega,\varepsilon}(f)\right\rvert\leq
D\|f\|_{\mathcal{B}_{\omega}}\kappa_{\varepsilon}^{n}.$

In addition, we have $\lim_{\varepsilon\to 0}\kappa_{\varepsilon}=\kappa_{0}$,
where $\kappa_{0}$ is defined in Lemma 6.8.

  6. (6)

There exists $C>0$ such that for every $f,h\in\text{{\rm BV}}$, every
$n\in\mathbb{N}$ sufficiently large, and for $m$-a.e. $\omega\in\Omega$ we
have

$\displaystyle\left\lvert\mu_{\omega,\varepsilon}\left(\left(f\circ
T_{\omega}^{n}\right)h\right)-\mu_{\sigma^{n}\omega,\varepsilon}(f)\mu_{\omega,\varepsilon}(h)\right\rvert\leq
C\|f\|_{\infty,\omega}\|h\|_{\mathcal{B}_{\omega}}\kappa_{\varepsilon}^{n}.$

###### Proof.

First we note that the claims of items (1) – (3) and (5) – (6) above for
$\varepsilon=0$ follow immediately from Lemma 6.8. Now we are left to prove
each of the claims for $\varepsilon>0$.

Claims (1) – (3) follow from Lemma 6.9 with the scaling:

$\displaystyle h_{\omega,\varepsilon}$
$\displaystyle:=\nu_{\omega,\varepsilon}(\varphi_{\omega,\varepsilon})\varphi_{\omega,\varepsilon},$
$\displaystyle\zeta_{\omega,\varepsilon}(f)$
$\displaystyle:=\frac{\nu_{\omega,\varepsilon}(f)}{\nu_{\omega,\varepsilon}(\mathbbold{1})},$
$\displaystyle\rho_{\omega,\varepsilon}$
$\displaystyle:=\frac{\nu_{\omega,\varepsilon}(\mathbbold{1})}{\nu_{\sigma\omega,\varepsilon}(\mathbbold{1})}\lambda_{\omega,\varepsilon}.$

The uniform boundedness on $\rho_{\omega,\varepsilon}$ and
$h_{\omega,\varepsilon}$ follows from the uniform boundedness on
$\lambda_{\omega,\varepsilon}$ and $\varphi_{\omega,\varepsilon}$ coming from
Lemma 6.9. The fact that $\mu_{\varepsilon}$ is the unique relative
equilibrium state follows similarly to the proof of Theorem 2.23 in [4] (see
also Remark 2.24, Lemma 12.2 and Lemma 12.3). The claim that
$\operatorname{supp}(\zeta_{\omega,\varepsilon})\subseteq
X_{\omega,\infty,\varepsilon}$ follows similarly to Lemma 11.11 of [2]. Noting
that $\|f\|_{\infty,\omega}=\|f\|_{\infty,1}$ for $f\in\text{{\rm BV}}$ by the
proof of Proposition 6.2, we now proof Claims (4) – (6).

Claim (4) follows from Lemma 3.5 [2] and the uniqueness of the density
$h_{\omega,\varepsilon}\in\text{{\rm BV}}$.

The first item of Claim (5) follows from Lemma 6.9 and the remaining items are
proved similarly to Corollary 12.8 [2].

Claim (6) follows from Claim (5) and is proven in Appendix D. ∎

######  Remark 6.12.

If one considers a two-sided (bi-infinite) sequential analogue of random open
systems, then because Theorem 4.8 [17] also applies to two-sided sequential
systems, one could prove similar results to items (1), (2), (4), (5), and (6)
of Theorem 6.11.

##  7\. Limit theorems

In this section we prove a few limit theorems for the closed systems discussed
in Section 6 $(\Omega,m,\sigma,[0,1],T,\text{{\rm
BV}},\mathcal{L}_{0},\nu_{0},\varphi_{0})$. We will in fact show that such
systems are admissible in the sense of [21]. This will allow us to adapt to
our setting the spectral approach à la Nagaev-Guivarc’h developed in [21] and
get a quenched central limit theorem, a quenched large deviation theorem and a
quenched local central limit theorem. We will also present an alternative
approach based on martingale techniques [22, 1, 20], which produces an almost
sure invariance principle (ASIP) for the random measure
$\mu_{0}=\left\\{\mu_{\omega,0}\right\\}_{\omega\in\Omega}$. Moreover, the
ASIP implies that $\mu_{0}$ satisfies the central limit theorem as well as the
law of the iterated logarithm. The martingale approach will also give an upper
bound for any (large) deviation from the expected value and a Borel-Cantelli
dynamical lemma. At the moment we could not extend the previous limit theorems
to the open systems investigated in Section 6. There are a few reasons for
that which concern the Banach space $\mathcal{B}_{\omega,\varepsilon}$
associated to those systems and defined by the norm:
$||\cdot||_{\mathcal{B}_{\omega,\varepsilon}}=\text{var}(\cdot)+\zeta_{\omega,\varepsilon}(|\cdot|).$
First of all, we do not know if the random cocycle
$\mathcal{R}_{\varepsilon}=(\Omega,m,\sigma,\mathcal{B}_{\omega,\varepsilon},\tilde{\mathcal{L}}_{\omega,\varepsilon})$
is quasi-compact which is an essential requirement for admissibility. Second,
the results of Theorem 6.11 are not particularly compatible with the Banach
spaces $\mathcal{B}_{\omega,\varepsilon}$, $\varepsilon>0$, since the
inequalities of items (5) and (6) are in terms of the norms
$\|\makebox[4.30554pt]{{$\cdot$}}\|_{\infty,\omega}$ and
$\|\makebox[4.30554pt]{{$\cdot$}}\|_{\mathcal{B}_{\omega}}$ which are defined
modulo $\nu_{\omega,0}$ and the $\mathcal{B}_{\omega,\varepsilon}$ norm is
defined via $\zeta_{\omega,\varepsilon}$, a measure which is supported on a
$\nu_{\omega,0}$-null set.

###  7.1. The Nagaev-Guivarc’h approach

The paper [21] developed a general scheme to adapt the Nagaev-Guivarc’h
approach to random quenched dynamical systems, allowing one to prove limit
theorems by exploring the connection between a twisted operator cocycle and
the distribution of the Birkhoff sums. The results in [21] were confined to
the geometric potential $|\det(DT_{\omega})|^{-1}$ and the associated
conformal measure, Lebesgue measure. We now show how to extend those results
to the systems verifying the assumptions stated in Section 6 and the results
of Theorem 6.11, whenever $\varepsilon=0,$ that is we will consider random
closed systems for a larger class of potentials.

The starting point is to replace the linear operator $\mathcal{L}_{\omega,0}$
associated to the geometric potential and the (conformal) Lebesgue measure
introduced in [21], with our operator $\mathcal{L}_{\omega,0}$ and the
associated conformal measures $\nu_{\omega,0}.$ In particular, if we work with
the normalized operator
$\tilde{\mathcal{L}}_{\omega,0}:=\lambda_{\omega,0}^{-1}\mathcal{L}_{\omega,0}$
the results in [21] are reproducible almost verbatim with a few precautions
which we are going to explain. As before, let $\mathcal{B}_{\omega}$ be the
Banach space defined by
$||\cdot||_{\mathcal{B}_{\omega}}=\text{var}(\cdot)+\nu_{\omega,0}(|\cdot|),$
where the variation is defined using equivalence classes mod-$\nu_{\omega,0}.$
In order to apply the theory in [21] we must show that our random cocycle is
admissible. This reduces to check two sets of properties which were listed in
[21] respectively as conditions (V1) to (V9) and conditions (C0) up to (C4).
The first set of conditions reproduces the classical properties of the total
variation of a function and its relationship with the $L^{1}(\text{{\rm
Leb}})$ norm. We should emphasize that in our case the variation is defined
using equivalence classes mod-$\nu_{\omega,0}.$ It is easy to check that
properties (V1, V2, V3, V5, V8, V9) hold with respect to this variation. In
particular (V3) asserts that for any $f\in\mathcal{B}_{\omega}$ we have
$||f||_{L^{\infty}(\nu_{\omega,0})}\leq||f||_{\mathcal{B}_{\omega}};$ we will
refer to it in the following just as the (V3) property. Notation: recall that
given an element $f\in\mathcal{B}_{\omega}$, the $L^{\infty}$ norm of $f$ with
respect to the measure $\nu_{\omega,0}$ is denoted by $||f||_{\infty,\omega}$
in the rest of this section.

Property (V7) is not used in the current paper; (V6) is a general density
embedding result proved in Hofbauer and Keller (Lemma 5 [32]). We elaborate on
property (V4). To obtain (V4) in our situation one needs to prove that the
unit ball of $\mathcal{B}_{\omega}$ is compactly injected into
$L^{1}(\nu_{\omega,0})$. As we will see, this is used to get the quasi-
compactness of the random cocycle. The result follows easily by adapting
Proposition 2.3.4 in [11] to our conformal measure $\nu_{\omega,0},$ which is
fully supported and non-atomic. We now rename the other set of properties
(C0)–(C4) in [21] as ($\mathcal{C}$0) to ($\mathcal{C}$4) to distinguish them
from our (C) properties stated earlier in Sections 4 and 5.

  * •

Assumption ($\mathcal{C}$0) coincides with our condition ((MC)).

  * •

Condition ($\mathcal{C}$1) requires us to prove in our case that

(7.1) $||\tilde{\mathcal{L}}_{\omega,0}f||_{\mathcal{B}_{\sigma\omega}}\leq
K||f||_{\mathcal{B}_{\omega}},$

for every $f\in\mathcal{B}_{\omega}$ and for $m$-a.e. $\omega$, with
$\omega$-independent $K$.

  * •

Condition ($\mathcal{C}$2) asks that there exists $N\in\mathbb{N}$ and
measurable $\tilde{\alpha}^{N},\tilde{\beta}^{N}:\Omega\rightarrow(0,\infty),$
with $\int_{\Omega}\log\tilde{\alpha}^{N}(\omega)dm(\omega)<0,$ such that for
every $f\in\mathcal{B}_{\omega}$ and $m$-a.e. $\omega\in\Omega,$

(7.2)
$||\tilde{\mathcal{L}}^{N}_{\omega,0}f||_{\mathcal{B}_{\sigma^{N}\omega}}\leq\tilde{\alpha}^{N}(\omega)||f||_{\mathcal{B}_{\omega}}+\tilde{\beta}^{N}(\omega)||f||_{L^{1}(\nu_{\omega,0})}.$

  * •

Condition ($\mathcal{C}$3) is the content of the first display equation in
item 5 in the statement of Theorem 6.11.

  * •

Condition ($\mathcal{C}$4) is only used to obtain Lemma 2.11 in [21]. There
are three results in that Lemma which we now compare with our situation. The
third result is the decay of correlations stated in item 6 of Theorem 6.11.
The second result is the almost-sure strictly positive lower bound for the
density $\varphi_{\omega,0}$ stated in item 2 of Theorem 6.11. The first
result requires that
$\operatorname*{ess\sup}_{\omega\in\Omega}||\varphi_{\omega,0}||_{\mathcal{B}_{\omega}}<\infty.$
This follows by checking that the proof of Proposition 1 in [22] works in our
current setting with the obvious modifications; we note that Proposition 1
[22] only assumes conditions ($\mathcal{C}$1) and ($\mathcal{C}$3).

We are thus left with showing conditions ($\mathcal{C}$1) and ($\mathcal{C}$2)
in our setting. We will get both at the same time as a consequence of the
following argument, which consists in adapting to our current situation the
final part of Lemma C.1. Our starting point will be the inequality (C.6).
Since we are working with the normalized operator cocycle
$\tilde{\mathcal{L}}_{\omega,0}^{N}$, we have to divide the expression in
(C.6) by $\lambda^{n^{\prime}}_{\omega,0};$ moreover we have to replace the
measure $\nu_{\omega,0}$ with the equivalent conformal probability measure
$\nu_{\omega,0}.$ Therefore (C.6) now becomes

(7.3) $(\lambda_{\omega,0}^{n})^{-1}\text{{\rm
var}}\left(\mathbbold{1}_{T_{\omega}^{n}(Z)}\left((fg_{\omega,0}^{(n)})\circ
T_{\omega,Z}^{-n}\right)\right)\leq
9\frac{\|g_{\omega,0}^{(n)}\|_{\infty,\omega}}{\lambda^{n}_{\omega,0}}\text{var}_{Z}(f)+\frac{8\|g_{\omega,0}^{(n)}\|_{\infty,\omega}}{\lambda^{n}_{\omega,0}\nu_{\omega,0}(Z)}\nu_{\omega,0}(|f|_{Z}),$

where $Z$ is an element of $\mathcal{Z}^{(n)}_{\omega,0}.$ Since each element
of the partition $\mathcal{Z}^{(n)}_{\omega,0}$ has nonempty interior and the
measure $\nu_{\omega,0}$ charges open intervals, if we set

$\overline{\nu}^{(n)}_{\omega,0}:=\min_{Z\in\mathcal{Z}_{\omega,0}^{(n)}}\nu_{\omega,0}(Z)>0,$

and we take the sum over the $Z\in\mathcal{Z}^{(n)}_{\omega,0}$ we finally get

(7.4) $\text{var}(\tilde{\mathcal{L}}^{n}_{\omega,0}f)\leq
9\frac{\|g_{\omega,0}^{(n)}\|_{\infty,\omega}}{\lambda^{n}_{\omega,0}}\text{var}(f)+8\frac{\|g_{\omega,0}^{(n)}\|_{\infty,\omega}\nu_{\omega,0}(|f|)}{\lambda^{n}_{\omega,0}\overline{\nu}_{\omega,0}^{(n)}}.$

Notice that condition ($\mathcal{C}$2) requires that $\tilde{\alpha}^{N}<1$,
which in our case becomes

(7.5)
$9\frac{\operatorname*{ess\sup}_{\omega}\|g_{\omega,0}^{(n)}\|_{\infty,\omega}}{\operatorname*{ess\inf}_{\omega}\lambda^{n}_{\omega,0}}<1,$

or equivalently

$9\frac{\operatorname*{ess\sup}_{\omega}\|g_{\omega,0}^{(n)}\|_{\infty,\omega}}{\operatorname*{ess\inf}_{\omega}\inf\mathcal{L}^{n}_{\omega,0}\mathbbold{1}}<1,$

which is guaranteed by (E8). We now move on to check condition
($\mathcal{C}$1). From (7.4), setting $n=1$ we see that a sufficient condition
for ($\mathcal{C}$1) is

(7.6)
$\frac{\operatorname*{ess\sup}_{\omega}\|g_{\omega,0}^{(1)}\|_{\infty,\omega}}{\operatorname*{ess\inf}_{\omega}\lambda^{1}_{\omega,0}\overline{\nu}_{\omega,0}}<\infty.$

or equivalently

(7.7)
$\frac{\operatorname*{ess\sup}_{\omega}\|g_{\omega,0}^{(1)}\|_{\infty,\omega}}{\operatorname*{ess\inf}_{\omega}\inf\mathcal{L}_{\omega,0}\mathbbold{1}\overline{\nu}_{\omega,0}}<\infty$

Condition (7.7) is in principle checkable ($n=1$) and we could assume it as a
part of the admissibility condition for our random cocycle
$\mathcal{R}=(\Omega,m,\sigma,\mathcal{B}_{\omega},\tilde{\mathcal{L}}_{\omega,0}).$

The admissibility conditions stated in [21], in particular ($\mathcal{C}$2)
and ($\mathcal{C}$3), were sufficient to prove the quasi-compactness of the
random cocycle introduced in [21], but they rely on another assumption, which
was a part of the Banach space construction, namely that $\text{{\rm BV}}_{1}$
was compactly injected into $L^{1}(\text{Leb}).$ We saw above that the same
result holds for our Banach space $\mathcal{B}_{\omega}$ and our measure
$\nu_{\omega,0}.$ In order to prove quasi-compactness following Lemma 2.1 in
[21], we use condition ($\mathcal{C}$2) with the almost-sure bound
$\tilde{\alpha}^{N}(\omega)<1.$ We must additionally prove that the top
Lyapunov exponent $\Lambda(\mathcal{R})$ defined by the limit
$\Lambda(\mathcal{R})=\lim_{n\rightarrow\infty}\frac{1}{n}\log||\tilde{\mathcal{L}}_{\omega,0}^{n}||_{\mathcal{B}_{\omega}}$
is not less than zero. This will follow by applying item 5 and the bound on
the density $\varphi_{\omega,0}$ in item 2, both in Theorem 6.11. Since the
density $\varphi_{\omega,0}$ is for $m$-a.e. $\omega\in\Omega$ bounded from
below by the constant $C^{-1},$ we have that
$\limsup_{n\rightarrow\infty}\frac{1}{n}\log\|\varphi_{\sigma^{n}\omega,0}\|_{\mathcal{B}_{\sigma^{n}\omega}}\geq\limsup_{n\rightarrow\infty}\frac{1}{n}\log
C^{-1}=0$. Now we continue as in the proof of Theorem 3.2 in [21]. Let $N$ be
the integer for which $\tilde{\alpha}^{N}(\omega)<1$ $\omega$ a.s. We then
consider the cocycle $\mathcal{R}_{N}$ generated by the map
$\omega\rightarrow\mathcal{L}_{\omega,0}^{N}.$ It is easy to verify that
$\Lambda(\mathcal{R}_{N})=N\Lambda(\mathcal{R})$ and the index of compactness
$\kappa$ (see section 2.1 in [21] for the definition) of the two cocycles also
satisfies $\kappa(\mathcal{R}_{N})=N\kappa(\mathcal{R}).$ Because
$\int\log\tilde{\alpha}^{N}(\omega)\ dm<0\leq\Lambda(\mathcal{R}_{N})$, Lemma
2.1 in [21] guarantees that
$\kappa(\mathcal{R}_{N})\leq\int\log\tilde{\alpha}^{N}(\omega)\ dm$, proving
quasicompactness of $\mathcal{R}.$

As we said at the beginning of this section, we could now follow almost
verbatim the proofs in [21] by using our operators $\mathcal{L}_{\omega,0}$
and by replacing the Lebesgue measure with the conformal measures
$\nu_{\omega,0}.$ We briefly sketch the main steps of the approach; we first
define the observable $v$ as a measurable map
$v:\Omega\times[0,1]\rightarrow\mathbb{R}$ with the additional properties that
$||v(\omega,x)||_{L^{1}(\nu_{0})}<\infty$ and
$\text{ess}\sup_{\omega}||v_{\omega}||_{\infty,\omega}<\infty,$ where we set
$v_{\omega}(x)=v(\omega,x).$ Moreover we assume $v_{\omega}$ is fibrewise
centered: $\int v_{\omega}(x)d\mu_{\omega,0}(x)=0,$ for $m$-a.e.
$\omega\in\Omega.$ We then define the twisted (normalized) transfer operator
cocycle as:

$\tilde{\mathcal{L}}_{\omega,0}^{\theta}(f)=\tilde{\mathcal{L}}_{\omega,0}(e^{\theta
v_{\omega}(\cdot)}f),\ f\in\mathcal{B}_{\omega}.$

The link with the limit theorems is provided by the following equality which
follows easily by the duality property of the operator:

$\int\tilde{\mathcal{L}}_{\omega,0}^{\theta,n}(f)d\nu_{\sigma^{n}\omega,0}=\int
e^{\theta S_{n}v_{\omega}(\cdot)}fd\nu_{\omega,0},$

where
$\tilde{\mathcal{L}}_{\omega,0}^{\theta,n}=\tilde{\mathcal{L}}_{\sigma^{n-1}\omega,0}^{\theta}\circ\cdots\circ\tilde{\mathcal{L}}_{\omega,0}^{\theta}.$
The adaptation of Theorem 3.12 in [21] to our case, shows that the twisted
operator is quasi-compact (in $\mathcal{B}_{\omega}$) for $\theta$ close to
zero. Moreover, by denoting with
$\Lambda(\theta)=\lim_{n\rightarrow\infty}\frac{1}{n}\log||\tilde{\mathcal{L}}_{\omega,0}^{\theta,n}||_{\mathcal{B}_{\omega}}$
the top Lyapunov exponent of the cocycle, the map
$\theta\rightarrow\Lambda(\theta)$ is of class $C^{2}$ and strictly convex in
a neighborhood of zero. We have also the analog of Lemma 4.3 in [21], linking
the asymptotic behavior of characteristic functions associated to Birkhoff
sums with $\Lambda(\theta):$

$\lim_{n\rightarrow\infty}\frac{1}{n}\log\left|\int e^{\theta
S_{n}v_{\omega}(x)}d\mu_{\omega,0}\right|=\Lambda(\theta).$

We are now ready to collect our results on a few limit theorems; we first
define the variance $\Sigma^{2}=\int v_{\omega}(x)^{2}\ d\mu_{\omega,0}\
dm+2\sum_{n=1}^{\infty}\int v_{\omega}(x)v(T_{\omega}^{n}(x))\
d\mu_{\omega,0}\ dm.$ We also define the aperiodicity condition by asking that
for $m$-a.e. $\omega\in\Omega$ and for every compact interval
$J\subset\mathbb{R}\setminus\\{0\\},$ there exists $C(\omega)>0$ and
$\rho\in(0,1)$ such that
$||\tilde{\mathcal{L}}_{\omega,0}^{it,n}||_{\mathcal{B}_{\omega}}\leq
C(\omega)\rho^{n},$ for $t\in J$ and $n\geq 0.$

######  Theorem 7.1.

Suppose that our random cocycle
$\mathcal{R}=(\Omega,m,\sigma,\mathcal{B}_{\omega},\tilde{\mathcal{L}}_{\omega,0})$
is admissible and take the centered observable $v$ verifying
$\text{ess}\sup_{\omega}||v_{\omega}||_{\infty,\omega}<\infty.$ Then:

  * •

(Large deviations). There exists $\varkappa_{0}>0$ and a non-random function
$c:(-\varkappa_{0},\varkappa_{0})\rightarrow\mathbb{R}$ which is nonnegative,
continuous, strictly convex, vanishing only at $0$ and such that

$\lim_{n\rightarrow\infty}\frac{1}{n}\log\mu_{\omega,0}(S_{n}v_{\omega}(\cdot)>n\varkappa)=-c(\varkappa),\
\text{for}\ 0<\varkappa<\varkappa_{0},\ \text{and}\ m-\text{a.e.}\
\omega\in\Omega.$

  * •

(Central limit theorem). Assume that $\Sigma^{2}>0.$ Then for every bounded
and continuous function $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ and $m$-a.e.
$\omega\in\Omega$, we have

$\lim_{n\rightarrow\infty}\int\varphi\left(\frac{S_{n}v_{\omega}(x)}{\sqrt{n}}\right)d\mu_{\omega,0}=\int\varphi\
d\mathcal{N}(0,\Sigma^{2}).$

  * •

(Local central limit theorem). Suppose the aperiodicity condition holds. Then
for $m$-a.e. $\omega\in\Omega$ and every bounded interval
$J\subset\mathbb{R},$ we have

$\lim_{n\rightarrow\infty}\sup_{s\in\mathbb{R}}\left|\Sigma\sqrt{n}\mu_{\omega,0}(s+S_{n}v_{\omega}(\cdot)\in
J)-\frac{1}{\sqrt{2\pi}}e^{-\frac{s^{2}}{2n\Sigma^{2}}}|J|\right|=0.$

###  7.2. The martingale approach

The deviation result quoted above allows to control, asymptotically,
deviations of order $\varkappa,$ for $\varkappa$ in a sufficiently small
bounded interval around $0.$ We now show how to extend that result to any
$\varkappa$ by getting an exponential bound on the deviation of the
distribution function instead of an asymptotic expansion; in particular our
bound shows that deviation probability stays small for finite $n$,which is a
typical concentration property. We now derive this result since it will
provide us with the martingale that is used to obtain the ASIP. Recall that
the equivariant measure $\mu_{\omega,0}$ is equivalent to $\nu_{\omega,0}.$ We
consider again the fibrewise centered observable $v$ from the previous
section, and we wish to estimate

$\mu_{\omega,0}\left(\left|\frac{1}{n}\sum_{k=0}^{n-1}v_{\sigma^{k}\omega}\circ
T^{k}_{\omega}\right|>\varkappa\right).$

We will use the following result (Azuma, [7]): Let
$\\{M_{i}\\}_{i\in\mathbb{N}}$ be a sequence of martingale differences. If
there is $a>0$ such that $||M_{i}||_{\infty}<a$ for all $i,$ then we have for
all $b\in\mathbb{R}:$

$\mu_{\omega,0}\left(\sum_{i=1}^{n}M_{i}\geq nb\right)\leq
e^{-n\frac{b^{2}}{2a^{2}}}.$

If we denote $\mathcal{F}^{k}_{\omega}:=(T^{k}_{\omega})^{-1}(\mathcal{F}),$
we can easily prove that for a measurable map
$\varphi:[0,1]\rightarrow\mathbb{{R}},$ we have (the expectations
$\mathbb{E}_{\omega}$ will be taken with respect to $\mu_{\omega,0}$):

(7.8) $\mathbb{E}_{\omega}(\varphi\circ
T^{l}_{\o}|\mathcal{F}^{n}_{\omega})=\left(\frac{\tilde{\mathcal{L}}_{\sigma^{l}\omega,0}^{n-l}(\lambda_{\sigma^{l}\omega,0}\varphi)}{\lambda_{\sigma^{n}\omega,0}}\right)\circ
T_{\omega}^{n}$

We now set:

$M_{n}:=v_{\sigma^{n}\omega}+G_{n}-G_{n+1}\circ T_{\sigma^{n}\omega},$

with $G_{0}=0$ and

(7.9)
$G_{n+1}=\frac{\tilde{\mathcal{L}}_{\sigma^{n}\omega,0}(v_{\sigma^{n}\omega}\lambda_{\sigma^{n}\omega,0}+G_{n}\lambda_{\sigma^{n}\omega,0})}{\lambda_{\sigma^{k+1}\omega,0}}$

It is easy to check that

$\mathbb{E}_{\omega}(M_{n}\circ T_{\omega}^{n}|\mathcal{F}^{n+1}_{\omega})=0,$

which means that the sequence $(M_{n}\circ T_{\omega}^{n})$ is a reversed
martingale difference with respect to the filtration
$\mathcal{F}^{n}_{\omega}.$ By iterating (7.9) we get

(7.10)
$G_{n}=\frac{1}{\lambda_{\sigma^{n}\omega,0}}\sum_{j=0}^{n-1}\tilde{\mathcal{L}}^{(n-j)}_{\sigma^{j}\omega,0}(v_{\sigma^{j}\omega}\lambda_{\sigma^{j}\omega,0}),$

and by a telescopic trick we have

$\sum_{k=0}^{n-1}M_{k}\circ
T_{\omega}^{k}=\sum_{k=0}^{n-1}v_{\sigma^{k}\omega}T_{\omega}^{n}-G_{n}\circ
T_{\omega}^{n}.$

Suppose for the moment we could bound $G_{n}$ uniformly in $n$, but not
necessarily in $\omega$, by $||G_{n}||_{\infty,\omega}\leq C_{1}(\omega).$
Since by assumption there exists a constant $C_{2}$ such that
$\text{ess}\sup_{\omega}||v_{\omega}||_{\infty,\omega}\leq C_{2},$ we have
$||M_{n}||_{\infty,\omega}\leq C_{2}+2C_{1}({\omega}),$ and by Azuma:

$\mu_{\omega,0}\left(\left|\frac{1}{n}\sum_{k=0}^{n-1}M_{k}\circ
T^{k}_{\omega}\right|>\frac{\varkappa}{2}\right)\leq
2\exp\left\\{-\frac{\varkappa^{2}}{8(C_{2}+2C_{1}({\omega}))^{2}}n\right\\}.$

Therefore

$\displaystyle\mu_{\omega,0}\left(\left|\frac{1}{n}\sum_{k=0}^{n-1}v_{\sigma^{k}\omega}\circ
T^{k}_{\omega}\right|>\varkappa\right)$
$\displaystyle\leq\mu_{\omega,0}\left(\left|\frac{1}{n}\sum_{k=0}^{n-1}M_{k}\circ
T^{k}_{\omega}\right|+\frac{1}{n}C_{1}(\omega)>\varkappa\right)$
$\displaystyle\leq\mu_{\omega,0}\left(\left|\frac{1}{n}\sum_{k=0}^{n-1}M_{k}\circ
T^{k}_{\omega}\right|>\frac{\varkappa}{2}\right)$ $\displaystyle\leq
2\exp\left\\{-\frac{\varkappa^{2}}{8(C_{2}+2C_{1}({\omega}))^{2}}n\right\\},$

provided $n>n_{0},$ where $n_{0}$ verifies
$\frac{1}{n}_{0}C_{1}(\omega)\leq\frac{\varkappa}{2}.$
In order to estimate $C_{1},$ we proceed in the following manner. We have

(7.11)
$||G_{n}||_{\infty}\leq\frac{1}{\lambda_{\sigma^{n}\omega,0}}\sum_{j=0}^{n-1}||\tilde{\mathcal{L}}^{(n-j)}_{\sigma^{j}\omega,0}(v_{\sigma^{j}\omega}\lambda_{\sigma^{j}\omega,0})||_{\infty,\omega}.$

The multiplier $\lambda_{\omega,0}$ is bounded from above and from below
$\omega$ a.s. respectively by, say, $U$ and $1/U$ by conditions
$(\mathcal{C}1)$ and item 1 Theorem 6.11, respectively. Then we use item 5 of
Theorem 6.11 and property (V3) to bound the term into the sum. In conclusion
we get:

$||G_{n}||_{\infty,\omega}\leq
U^{2}DC_{2}\sum_{j=0}^{\infty}\kappa^{n-j}:=C_{1}.$

We summarize this result in the following

######  Theorem 7.2.

(Large deviations bound)
Suppose that our random cocycle
$\mathcal{R}=(\Omega,m,\sigma,\mathcal{B}_{\omega},\tilde{\mathcal{L}}_{\omega,0})$
is admissible and take the fibrewise centered observable $v$ satisfying
$\text{ess}\sup_{\omega}||v_{\omega}||_{\infty,\omega}=C_{2}<\infty.$ Then
there exists $C_{1}>0$ such that for $m$-a.e. $\omega\in\Omega$ and for any
$\varkappa>0:$

$\mu_{\omega,0}\left(\left|\frac{1}{n}\sum_{k=0}^{n-1}v_{\sigma^{k}\omega}\circ
T^{k}_{\omega}\right|>\varkappa\right)\leq
2\exp\left\\{-\frac{\varkappa^{2}}{8(C_{2}+2C_{1})^{2}}n\right\\},$

provided $n>n_{0},$ where $n_{0}$ is any integer number satisfying
$\frac{1}{n_{0}}C_{1}\leq\frac{\varkappa}{2}.$

We point out that whenever our random cocycle
$\mathcal{R}=(\Omega,m,\sigma,\mathcal{B}_{\omega},\tilde{\mathcal{L}}_{\omega,0})$
is admissible, we satisfy all the assumptions in the paper [22], which allows
us to get the almost sure invariance principle. We should simply replace the
(conformal) Lebesgue measure with our random conformal measure
$\nu_{\omega,0},$ and use our operators $\tilde{\mathcal{L}}_{\omega,0},$ as
we already did in Section 7.1 for the Nagaev-Guivarc’h approach. It is
worthwhile to observe that [22] relies on the construction of a suitable
martingale, which in our case is precisely the reversed martingale difference
$M_{n}\circ T^{n}_{\omega}$ obtained above in the proof of the large deviation
bound. We denote with $\Sigma^{2}_{n}$ the variance
$\Sigma^{2}_{n}=\mathbb{E}_{\omega}\left(\sum_{k=0}^{\infty}v_{\sigma^{k}\omega}\circ
T_{\omega}^{k}\right)^{2},$ where $v_{\omega}$ is a centered observable, and
with $\Sigma^{2}$ the quantity introduced above, in the statement of the
central limit theorem. We have the following:

######  Theorem 7.3.

(Almost sure invariance principle)

Suppose that our random cocycle
$\mathcal{R}=(\Omega,m,\sigma,\mathcal{B}_{\omega},\tilde{\mathcal{L}}_{\omega,0})$
is admissible and consider a centered observable $v_{\omega}.$ Then
$\lim_{n\rightarrow\infty}\frac{1}{n}\Sigma^{2}_{n}=\Sigma^{2}.$ Moreover one
of the following cases hold:

(i) $\Sigma=0,$ and this is equivalent to the existence of $\varphi\in
L^{2}(\mu_{0})$

$v=\varphi-\varphi\circ T,$

where $T$ is the induced skew product map and $\mu$ its invariant measure, see
Definition 3.1.

(ii) $\Sigma^{2}>0$ and in this case for $m$ a.e. $\omega\in\Omega,$
$\forall\varrho>\frac{5}{4}$, by enlarging the probability space
$(X,\mathcal{F},\mu_{\omega,0})$ if necessary, it is possible to find a
sequence $(B_{k})_{k}$ of independent centered Gaussian random variables such
that

$\sup_{1\leq k\leq n}\left|\sum_{i=1}^{k}(v_{\sigma^{i}\omega}\circ
T_{\omega}^{i})-\sum_{i=1}^{k}B_{i}\right|=O(n^{1/4}\log^{\varrho}(n)),\
\mu_{\omega,0}\ \text{a.s.}.$

We show in Section 5.47 that the distribution of the first hitting random time
in a decreasing sequence of holes follows an exponential law, see Proposition
5.7. We now prove a recurrence result by giving a quantitative estimate of the
asymptotic number of entry times in a descending sequence of balls. This is
known as the shrinking target problem.

######  Proposition 7.4.

Suppose that our random cocycle
$\mathcal{R}=(\Omega,m,\sigma,\mathcal{B}_{\omega},\tilde{\mathcal{L}}_{\omega,0})$
is admissible. Take $B_{\xi_{\sigma^{k}\omega}}(p)$ as a descending sequence
of balls centered at the point $p$ and with radius $\xi_{\sigma^{k}\omega},$
such that

$E_{\omega,n}:=\sum_{k=0}^{n-1}\mu_{\sigma^{k}_{\omega},0}(B_{\xi_{\sigma^{k}\omega}}(p))\rightarrow\infty\quad\mbox{
for $m$-a.e.\ $\omega$}.$

Then for $m$-a.e. $\omega$ and $\mu_{\omega,0}$-almost all $x,$
$T^{k}_{\omega}(x)\in B_{\xi_{\sigma^{k}\omega}}(p)$ for infinitely many $k$
and

$\frac{1}{E_{\omega,n}}\\#\\{k\geq 0,T^{k}_{\omega}(x)\in
B_{\xi_{\sigma^{k}\omega}}(p)\\}\rightarrow 1.$

The proposition is a simple consequence of the following Borel-Cantelli like
property, whenever we put the observable $v$ in the theorem below as
$v_{\sigma^{k}\omega}=\mathbbold{1}_{B_{\xi_{\sigma^{k}\omega}}(p)}.$

######  Theorem 7.5.

Suppose that our random cocycle
$\mathcal{R}=(\Omega,m,\sigma,\mathcal{B}_{\omega},\tilde{\mathcal{L}}_{\omega,0})$
is admissible. Take a nonnegative observable $v,$ satisfying
$\text{ess}\sup_{\omega}||v_{\omega}||_{\mathcal{B}_{\omega}}\leq M.$ Suppose
that for $m$ a.e. $\omega\in\Omega$

$E_{\omega,n}:=\sum_{k=0}^{n-1}\int
v_{\sigma^{k}\omega}(x)d\mu_{\sigma^{k}_{\omega},0}(x)\rightarrow\infty,\
n\rightarrow\infty.$

Then for $m$ a.e. $\omega\in\Omega$

$\lim_{n\rightarrow\infty}\frac{1}{E_{\omega,n}}\sum_{k=0}^{n-1}v_{\sigma^{k}\omega}(T^{k}_{\omega}x)=1,$

for $\mu_{\omega,0}$-almost all $x$.

###### Proof.

Let us write

$S_{\omega}:=\int\left(\sum_{m<k\leq
n}\left(v_{\sigma^{k}\omega}(T^{k}_{\omega}y)-\int
v_{\sigma^{k}\omega}d\mu_{\sigma^{k}_{\omega},0}\right)\right)^{2}d\mu_{\omega,0}(y).$

If we could prove that $S_{\omega}\leq\text{constant}\sum_{m<k\leq n}\int
v_{\sigma^{k}\omega}d\mu_{\sigma^{k}\omega,0},$ then the result will follow by
applying Sprindzuk theorem [47]444We recall here the Sprindzuk theorem. Let
$(\Omega,\mathcal{B},\mu_{0})$ be a probability space and let $(f_{k})_{k}$ be
a sequence of nonnegative and measurable functions on $\Omega.$ Moreover, let
$(g_{k})_{k}$ and $(h_{k})_{k}$ be bounded sequences of real numbers such that
$0\leq g_{k}\leq h_{k}.$ Assume that there exists $C>0$ such that
$\int\left(\sum_{m<k\leq n}(f_{k}(x)-g_{k})\right)^{2}d\mu_{0}(x)\leq
C\sum_{m<k\leq n}h_{k}$ for $m<n.$ Then, for every $\varepsilon>0$ we have
$\sum_{1\leq k\leq n}f_{k}(x)=\sum_{1\leq k\leq
n}g_{k}+O(\Theta^{1/2}(n)\log^{3/2+\varepsilon}\Theta(n)),$ for $\mu$ a.e.
$x\in\Omega,$ where $\Theta(n)=\sum_{1\leq k\leq n}h_{k}.$ , where we identify
$\int v_{\sigma^{k}\omega}d\mu_{\sigma^{k}\omega,0}$ with the functions
$g_{k}$ and $h_{k}$ in the previous footnote. So we have

$\displaystyle S_{\omega}$ $\displaystyle=\int\left[\sum_{m<k\leq
n}v_{\sigma^{k}\omega}(T^{k}_{\omega}y)\right]^{2}d\mu_{\omega,0}(y)-\left[\sum_{m<k\leq
n}\int v_{\sigma^{k}\omega}d\mu_{\sigma^{k}_{\omega},0}\right]^{2}$
$\displaystyle=\sum_{m<k\leq n}\int
v_{\sigma^{k}\omega}(T^{k}_{\omega}y)^{2}d\mu_{\omega,0}+2\sum_{m<i<j\leq
n}\int
v_{\sigma^{i}\omega}(T^{i}_{\omega}y)v_{\sigma^{j}\omega}(T^{j}_{\omega}y)d\mu_{\omega,0}$
$\displaystyle\qquad-\sum_{m<k\leq n}\left[\int
v_{\sigma^{k}\omega}d\mu_{\sigma^{k}\omega,0}\right]^{2}-2\sum_{m<i<j\leq
n}\int v_{\sigma^{i}\omega}d\mu_{\sigma^{i}\omega,0}\int
v_{\sigma^{j}\omega}d\mu_{\sigma^{j}\omega,0}.$

We now discard the third negative piece and bound the first by Hölder
inequality as

$\sum_{m<k\leq n}\int
v_{\sigma^{k}\omega}(T^{k}_{\omega}y)^{2}d\mu_{\omega,0}\leq\sum_{m<k\leq
n}M\int v_{\sigma^{k}\omega}d\mu_{\sigma^{k}\omega,0}.$

Then for the remaining two pieces we use the decay of correlations given in
Theorem 2.21 of our paper [4], where the observables are taken in
$\mathcal{B}_{\omega}$ and in $L^{1}(\mu_{\omega,0}).$ We can apply it to our
case since the two observables coincide with $v_{\omega}$, and the measures
$\mu_{\omega,0}$ and $\nu_{\omega,0}$ are equivalent. We point out that this
result improves the decay bound given in item (6) of Theorem 6.11 where the
presence of holes for $\varepsilon>0$ forced us to use
$L^{\infty}(\nu_{\omega,\varepsilon})$ instead of
$L^{1}(\nu_{\omega,\varepsilon}).$ Hence:

$\left|\int
v_{\sigma^{i}\omega}(T^{i}_{\omega}y)v_{\sigma^{j}\omega}(T^{j}_{\omega}y)d\mu_{\omega,0}-\int
v_{\sigma^{i}\omega}d\mu_{\sigma^{i}\omega,0}\int
v_{\sigma^{j}\omega}d\mu_{\sigma^{j}\omega,0}\right|\leq CM\kappa^{j-i}\int
v_{\sigma^{j}\omega}d\mu_{\sigma^{j}\omega,0}.$

In conclusion, we get

$\displaystyle S_{\omega}$ $\displaystyle\leq\sum_{m<k\leq n}M\int
v_{\sigma^{k}\omega}d\mu_{\sigma^{k}\omega,0}+CM\sum_{m<i<j\leq
n}\kappa^{j-i}\int v_{\sigma^{j}\omega}d\mu_{\sigma^{j}\omega,0}$
$\displaystyle\leq M\left(\frac{C}{1-\kappa}+1\right)\sum_{m<k\leq n}\int
v_{\sigma^{k}\omega}d\mu_{\sigma^{k}\omega,0},$

which satisfies Sprindzuk. ∎

##  8\. Examples

In this section we present explicit examples of random dynamical systems to
illustrate the quenched extremal index formula in Theorem 5.5. In all cases,
Theorem 6.11 can be applied to guarantee the existence of all objects therein
for the perturbed random open system (in brief, the thermodynamic formalism
for the closed random system implies a thermodynamic formalism for the
perturbed random open system). In Examples 1–3 we derive explicit expressions
for the quenched extreme value laws.

Our examples are piecewise-monotonic interval map cocycles whose transfer
operators act on $\mathcal{B}_{\omega}=\text{{\rm BV}}:=\text{{\rm
BV}}([0,1])$ for a.e. $\omega$. The norm we will use is
$\|\cdot\|_{\mathcal{B}_{\omega}}=\|\cdot\|_{\text{{\rm BV}}}:=\text{{\rm
var}}(\cdot)+\nu_{\omega,0}(|\cdot|)$. In Section 6 we noted that ((B)) is
automatically satisfied. Lemma 6.8 shows that ((CCM)) holds under assumptions
(E2), (E3), ((E4)), and (E8). To obtain a random open system, in addition to
((B)) and ((CCM)) we need ((MC)) (implying ((M1)) and ((M2))), ((A)), and
((X)). Therefore, in each example we verify conditions ((A)), ((X)) (Section
3.1) and ((MC)) and (E1)–((E9)) (Section 6); and where required ((S)).

###  8.1. Example 1: Random maps and random holes centred on a non-random
fixed point (random observations with non-random maximum)

We show that nontrivial quenched extremal indices occur even for holes centred
on a non-random fixed point $x_{0}$. To define a random (open) map, let
$(\Omega,m)$ be a complete probability space, and
$\sigma:(\Omega,m)\to(\Omega,m)$ be an ergodic, invertible, probability-
preserving transformation. For example, one could consider an irrational
rotation on the circle. Let $\Omega=\cup_{j=1}^{\infty}\Omega_{j}$ be a
countable partition of $\Omega$ into measurable sets on which $\omega\mapsto
T_{\omega}$ is constant. This ensures that ((MC)) is satisfied. For each
$\omega\in\Omega$ let $\mathcal{J}_{\omega,0}=[0,1]$ and let
$T_{\omega}:\mathcal{J}_{\omega,0}\to\mathcal{J}_{\sigma\omega,0}$ be random,
with all maps fixing $x_{0}\in[0,1]$. The observation functions
$h_{\omega}:\mathcal{J}_{\omega,0}\to\mathbb{R}$ have a unique maximum at
$x_{0}$ for a.e. $\omega$.

To make all of the objects and calculations as simple as possible and
illustrate some of the underlying mechanisms for nontrivial extremal indices
we use the following specific family of maps $\\{T_{\omega}\\}$:

(8.1)
$T_{\omega}(x)=\left\\{\begin{array}[]{ll}1-2x/(1-1/s_{\omega}),&\hbox{$0\leq
x\leq(1-1/s_{\omega})/2$;}\\\
s_{\omega}x-(s_{\omega}-1)/2,&\hbox{$(1-1/s_{\omega})/2\leq
x\leq(1+1/s_{\omega})/2$;}\\\
1-(2x-(1+1/s_{\omega}))/(1-1/s_{\omega})&\hbox{$(1+1/s_{\omega})/2\leq x\leq
1$,}\end{array}\right.$

where $1<s\leq s_{\omega}\leq S<\infty$ and $s\rvert_{\Omega_{j}}$ is constant
for each $j\geq 1$; thus ((M1)) holds. These maps have three full linear
branches, and therefore Lebesgue measure is preserved by all maps
$T_{\omega}$; i.e. $\mu_{\omega,0}=\mathrm{Leb}$ for a.e. $\omega$. The
central branch has slope $s_{\omega}$ and passes through the fixed point
$x_{0}=1/2$, which lies at the centre of the central branch; see Figure 1.

Figure 1. Graph of map $T_{\omega}$, with $s_{\omega}=2$.

A specific random driving could be $\sigma:S^{1}\to S^{1}$ given by
$\sigma(\omega)=\omega+\alpha$ for some $\alpha\notin\mathbb{Q}$ and
$s_{\omega}=s^{0}+s^{1}\cdot\omega$ for $1<s^{0}<\infty$ and $0<s^{1}<\infty$,
but only the ergodicity of $\sigma$ will be important for us. Setting
$g_{\omega,0}=1/|T^{\prime}_{\omega}|$ it is immediate that (E1), (E2), (E3),
and ((E4)) hold.

We select a measurable family of observation functions
$h:\Omega\times[0,1]\to\mathbb{R}$. For a.e. $\omega\in\Omega$, $h_{\omega}$
is $C^{1}$, has a unique maximum at $x_{0}=1/2$, and is a locally even
function about $x_{0}$. For small $\varepsilon_{N}$ we will then have that
$H_{\omega,\varepsilon_{N}}=\\{x\in[0,1]:h_{\omega}(x)>z_{\omega,N}\\}$ is a
small neighbourhood of $x_{0}$, satisfying ((E5)) and ((A)). In light of
Remark 6.3 and Proposition 6.4 we see that ((X)) holds. The corresponding
cocycle of open operators $\mathcal{L}_{\omega,\varepsilon}$ satisfy ((MC)).
Given an essentially bounded function $t$, the $z_{\omega,N}$ are chosen to
satisfy ((S)). Because the $h_{\omega}$ are $C^{1}$ with unique maxima and
$\mu_{\omega,0}$ is Lebesgue for all $\omega$ we can make such a choice of
$z_{\omega,N}$ with555In the situation where $z_{\omega,N}$ is constant a.e.
for each $N\geq 1$, in general we cannot satisfy ((S)) for a given fixed
scaling function $t$. In the case where $t$ is also constant a.e. if
$h_{\omega}$ is random but sufficiently close to a non-random observation $h$
for each $\omega$, we can satisfy ((S)) by incorporating the error term
$\xi_{\omega,N}$. This situation corresponds to a decreasing family of holes
whose length fluctuates slightly in $\omega$ with the fluctation decreasing to
zero sufficiently fast. A similar situation can occur if the scaling function
is a general measurable function. These situations motivate the use of the
error term $\xi_{\omega,N}$ in condition ((S)). $\xi_{\omega,N}\equiv 0$.
Furthermore, as $\mu_{\omega,0}$ is Lebesgue we have that ((S)) implies that
assumption (E6) is automatically satisfied. With the above choices, (E7) is
clearly satisfied for $n^{\prime}=1$. To show condition (E8) with
$n^{\prime}=1$ we note that
$\|g_{\omega,0}\|_{\infty}=(1-1/s_{\omega})/2<(1-1/S)/2$, while
$\inf\mathcal{L}_{\omega,\varepsilon}\mathbbold{1}\geq
2\cdot(1-1/s_{\omega})/2\geq 1-1/s$. Therefore we require $(1-1/S)/2<1-1/s$,
which by elementary rearrangement is always true for $0<s<S<\infty$. Because
there is a full branch outside the branch with the hole, ((E9)) is satisfied
with $n^{\prime}=1$ and $k_{o}(n^{\prime})=1$.

At this point we have checked all of the hypotheses of Theorem 6.11 and we
obtain that for sufficiently small holes as defined above, the corresponding
random open dynamical system has a quenched thermodynamic formalism and
quenched decay of correlations. We note that this result does not require
((S)); the holes $H_{\varepsilon_{N}}$ do not need to scale in any particular
way with $N$, they simply need to be sufficiently small. In order to next
apply Theorem 5.5 we do require ((S)) and additionally ((C8)).

To finish the example, we verify ((C8)). We claim that for each fixed $k>0$,
one has $\hat{q}_{\omega,\varepsilon_{N}}^{(k)}=0$ for sufficiently
sufficiently large $N$ (sufficiently small $\varepsilon_{N}$). If this were
not the case, there must be a positive $\mu_{\omega,0}$-measure set of points
that (i) lie in $H_{\sigma^{-k}\omega,\varepsilon_{N}}$, (ii) land outside the
sequence of holes
$H_{\sigma^{-(k-1)}\omega,\varepsilon_{N}},\ldots,H_{\sigma^{-1}\omega,\varepsilon_{N}}$
for the next $k-1$ iterations, and then (iii) land in
$H_{\omega,\varepsilon_{N}}$ on the $k^{\mathrm{th}}$ iteration. In this
example, because all $H_{\omega,\varepsilon_{N}}$ are neighbourhoods of
$x_{0}$ of diameter smaller than $(|t|_{\infty}+\gamma_{N})/N$, and the maps
are locally expanding about $x_{0}$, for fixed $k$ one can find a large enough
$N$ so that it is impossible to leave a small neighbourhood of $x_{0}$ and
return after $k$ iterations. For $k=0$, by definition

(8.2)
$\hat{q}_{\omega,\varepsilon_{N}}^{(0)}=\frac{\mu_{\sigma^{-1}\omega,0}(T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}}\cap
H_{\sigma^{-1}\omega,\varepsilon_{N}})}{\mu_{\sigma^{-1}\omega,0}(T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}})}=\frac{\mu_{\sigma^{-1}\omega,0}(T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}}\cap
H_{\sigma^{-1}\omega,\varepsilon_{N}})}{\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})}.$

There are two cases to consider:

Case 1: $H_{\sigma^{-1}\omega,\varepsilon_{N}}$ is larger than the local
preimage of $H_{\omega,\varepsilon_{N}}$; that is,
$T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}}\cap
H_{\omega,\varepsilon_{N}}\subset H_{\sigma^{-1}\omega,\varepsilon_{N}}$.
Because of the linearity of the branch containing $x_{0}$, one has
$\hat{q}_{\omega,\varepsilon_{N}}^{(0)}=1/T^{\prime}_{\sigma^{-1}\omega}(x_{0})$.

Case 2: $H_{\sigma^{-1}\omega,\varepsilon_{N}}$ is smaller than the local
preimage of $H_{\omega,\varepsilon_{N}}$; that is,
$T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}}\cap
H_{\omega,\varepsilon_{N}}\supset H_{\sigma^{-1}\omega,\varepsilon_{N}}$. By
(8.2) and ((S)) we then have

$\frac{t_{\sigma^{-1}\omega}-\gamma_{N}}{t_{\omega}+\gamma_{N}}\leq\hat{q}_{\omega,\varepsilon_{N}}^{(0)}\leq\frac{t_{\sigma^{-1}\omega}+\gamma_{N}}{t_{\omega}-\gamma_{N}}$

and thus for such an $\omega$,
$\lim_{N\to\infty}\hat{q}_{\omega,\varepsilon_{N}}^{(0)}=t_{\sigma^{-1}\omega}/t_{\omega}$.

Thus, combining the two cases,

$\hat{q}^{(0)}_{\omega,0}:=\lim_{N\to\infty}\hat{q}^{(0)}_{\omega,\varepsilon_{N}}=\min\left\\{\frac{t_{\sigma^{-1}\omega}}{t_{\omega}},\frac{1}{|T^{\prime}_{\sigma^{-1}\omega}(x_{0})|}\right\\}$

exists for a.e. $\omega$, verifying $\eqref{C8}$.

Recalling that
$\theta_{\omega,0}=1-\sum_{k=0}^{\infty}\hat{q}^{(k)}_{\omega,0}=1-\hat{q}^{(0)}_{\omega,0}$,
we may now apply Theorem 5.5 to obtain the quenched extreme value law:

(8.3)
$\displaystyle\lim_{N\to\infty}\nu_{\omega,0}\left(X_{\omega,N-1,\varepsilon_{N}}\right)$
$\displaystyle=$
$\displaystyle\exp\left(-\int_{\Omega}t_{\omega}\theta_{\omega,0}\,dm(\omega)\right)$
$\displaystyle=$
$\displaystyle\exp\left(-\int_{\Omega}t_{\omega}\left(1-\min\left\\{\frac{t_{\sigma^{-1}\omega}}{t_{\omega}},\frac{1}{|T^{\prime}_{\sigma^{-1}\omega}(x_{0})|}\right\\}\right)\,dm(\omega)\right)$

This formula is a generalisation of the formula in Remark 8 [35], where we may
create nontrivial laws from either the random dynamics $T_{\omega}$, or the
random scalings $t_{\omega}$, or both. The following two special cases
consider these mechanisms separately.

  1. (1)

Random maps, non-random scaling ($t_{\omega}$ takes a constant value $t>0$):
Since $|T_{\omega}^{\prime}|>1$, in this case (8.3) becomes

(8.4)
$\lim_{N\to\infty}\nu_{\omega,0}\left(X_{\omega,N-1,\varepsilon_{N}}\right)=\exp\left(-t\left[1-\int_{\Omega}\frac{1}{|T_{\omega}^{\prime}(x_{0})|}\
dm(\omega)\right]\right),$

and we see that we can interpret
$\theta=1-\int_{\Omega}\frac{1}{|T_{\omega}^{\prime}(x_{0})|}\ dm(\omega)$ as
an extremal index.

  2. (2)

Fixed map, random scaling: Suppose $T_{\omega}\equiv T$; then we may replace
$T_{\omega}^{\prime}(x_{0})$ with $T^{\prime}(x_{0})$ in (8.3), and we see we
the extremal index depends on the choice of random scalings $t_{\omega}$
alone; of course the thresholds $z_{\omega,N}$ depend on the chosen
$t_{\omega}$.

Similar results could be obtained with the $T_{\omega}$ possessing nonlinear
branches. The arguments above can also be extended to the case where $x_{0}$
is a periodic point of prime period $p$ for all maps; the formula (8.3) now
includes $(T_{\omega}^{p})^{\prime}(x_{0})$ and
$t_{\omega},t_{\sigma^{1}\omega},\ldots,t_{\sigma^{-(p-1)}\omega}$. If the
scaling $t_{\omega}=t$ is non-random, one would simply replace
$T^{\prime}_{\omega}(x_{0})$ in (8.4) with $(T_{\omega}^{p})^{\prime}(x_{0})$.

We recall that for deterministic $T$ (including non-uniformly hyperbolic
maps), the extremal index enjoys a dichotomy, in the sense that it is equal to
$1$ when a single hole $H_{\varepsilon_{N}}$ shrinks to an aperiodic point,
and it is strictly smaller than $1$ when the hole shrinks to a periodic point.
In the latter case the extremal index can be expressed in terms of the period;
see [41] for a general account of the above facts. Example 1 shows that in a
simple random setting there are many more ways to obtain nontrivial
exponential limit laws, e.g. by random scaling, or by the existence of
periodic orbits for only a positive measure set of $\omega$.

###  8.2. Example 2: Random $\beta$-maps, random holes containing a non-random
fixed point (random observations with non-random maximum), general geometric
potential

We show how a nontrivial extremal index can arise from random $\beta$-maps
where statistics are generated by an equilibrium state of a general geometric
potential. Consider the “no short branches” random $\beta$-map example of
Section 13.2 [4], where $\beta_{\omega}\in\\{2\\}\cup\bigcup_{2\leq k\leq
K}[k+\delta,k+1]$ for a.e. $\omega$, and some $\delta>0$ and $K<\infty$. The
measurability of $\omega\mapsto\beta_{\omega}$ yields ((M1)). We use the
weight $g_{\omega}=1/|T^{\prime}_{\omega}|^{r}$, $r\geq 0$. To obtain ((MC))
the base dynamics is driven by an ergodic homeomorphism $\sigma$ on a Borel
subset $\Omega$ of a separable, complete metric space, assuming that
$\omega\mapsto\mathcal{L}_{\omega,0}$ has countable range, as in Example 8.1.
Our random observation function $h:\Omega\times[0,1]\to\mathbb{R}$ is
measurable and for a.e. $\omega$, $h_{\omega}$ is $C^{1}$ with a unique
maximum at $x=0$. We select a measurable scaling function $t_{\omega}$; either
or both of $h_{\omega}$ and $t_{\omega}$ could be $\omega$-independent. By
assigning thresholds $z_{\omega,N}$ (which could also be $\omega$-independent)
we obtain a decreasing family of holes $H_{\omega,\varepsilon_{N}}$, which are
decreasing intervals with left endpoint at $x=0$.

Clearly (E2) and (E3) hold and Lemma 13.5 [4] provides ((E4)). Conditions
((E5)) and (E6), and ((A)) are immediate, as is (E7) with $n^{\prime}=1$.
Condition ((EX)) holds because there is at least one full branch outside the
branch with the hole. Regarding (E8),
$\|g_{\omega,\varepsilon}\|_{\infty}\leq\beta_{\omega}^{-r}$ and arguing as in
the previous example, since
$\inf\mathcal{L}_{\omega,\varepsilon}\mathbbold{1}\geq\lfloor\beta_{\omega}\rfloor/\beta_{\omega}^{r}$,
we see (E8) holds with $n^{\prime}=1$ because $\beta_{\omega}\geq 2$. Because
there is at least one full branch outside the branch with the hole, ((E9))
holds with $n^{\prime}=1$ and $k_{o}(n^{\prime})=1$. We have now checked all
of the hypotheses of Theorem 6.11. By this theorem, for sufficiently small
holes $H_{\varepsilon}$, the corresponding random open dynamical system has a
quenched thermodynamic formalism and quenched decay of correlations. As in the
previous example, we emphasise that this result does not require ((S)).

Because $\varphi_{\omega,0}$ is uniformly bounded above and below, as long as
$z_{\omega,N}$ is random, we may adjust $z_{\omega,N}$ to obtain ((S)). We now
verify ((C8)) for our holes, which are of the form
$H_{\omega,\varepsilon_{N}}=[0,r_{\omega,N}]$, in preparation to apply Theorem
5.5. The same arguments from Case 1 and Case 2 of the previous example apply.
Case 2 is unchanged. In Case 1 we have

$\hat{q}^{(0)}_{\omega,\varepsilon_{N}}=\frac{\mu_{\sigma^{-1}\omega,0}(T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}}\cap
H_{\sigma^{-1}\omega,\varepsilon_{N}})}{\mu_{\sigma^{-1}\omega,0}(T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}})}.$

Because $T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}}$ is a finite
union of left-closed intervals, using the fact that
$\varphi_{\sigma^{-1}\omega,0}\in BV$, and therefore has left- and right-hand
limits everywhere, we may redefine $\varphi_{\sigma^{-1}\omega,0}(y)$ at the
finite collection of points $y\in T^{-1}_{\sigma^{-1}\omega}(0)$ so that

$\lim_{N\to\infty}\frac{\mu_{\sigma^{-1}\omega,0}(T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}}\cap
H_{\sigma^{-1}\omega,\varepsilon_{N}})}{\mu_{\sigma^{-1}\omega,0}(T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}})}=\frac{\varphi_{\sigma^{-1}\omega,0}(0)}{\sum_{y\in
T^{-1}_{\sigma^{-1}\omega}(0)}\varphi_{\sigma^{-1}\omega,0}(y)},$

recalling that in Case 1,
$T^{-1}_{\sigma^{-1}\omega}H_{\omega,\varepsilon_{N}}\subset
H_{\sigma^{-1}\omega,\varepsilon_{N}}$. Taking the minimum of Cases 1 and 2 as
in Example 1, we see that
$\hat{q}_{\omega,0}^{(0)}=\lim_{N\to\infty}\hat{q}_{\omega,\varepsilon_{N}}^{(0)}$
exists, which verifies ((C8)). Finally,

(8.5)
$\lim_{N\to\infty}\nu_{\omega,0}\left(X_{\omega,N-1,\varepsilon_{N}}\right)=\exp\left(-\int_{\Omega}t_{\omega}\left(1-\min\left\\{\frac{t_{\sigma^{-1}\omega}}{t_{\omega}},\frac{\varphi_{\sigma^{-1}\omega,0}(0)}{\sum_{y\in
T^{-1}_{\sigma^{-1}\omega}(0)}\varphi_{\sigma^{-1}\omega,0}(y)}\right\\}\right)\,dm(\omega)\right)$

###  8.3. Example 3: A fixed map with random holes containing a fixed point
(random observations with a non-random maximum)

Let us now consider more closely the case of a fixed map and holes moving
randomly around a fixed point $z$, a situation previously considered in [8] in
the annealed framework.

We take a piecewise uniformly expanding map $T$ of the unit interval $I$ of
class $C^{2}$, and such that $T$ is surjective on the domains of local
injectivity. Moreover $T$ preserves a mixing measure $\mu$ equivalent to the
Lebesgue measure $\text{Leb},$ with strictly positive density $\rho$. We
moreover assume that $T$ and $\rho$ are continuous at a fixed point $x_{0}$.
The observation functions $h_{\omega}$ have a common maximum at $x_{0}$,
leading to holes $H_{\omega,\varepsilon_{N}}$ that are closed intervals
containing the point $x_{0}$ for any $N$. In Example 1 the holes were centred
on $x_{0}$ but could vary dramatically in diameter between $\omega$-fibers. In
this example the holes need not be centred on $x_{0}$ but must become more
identical as they shrink. Specifically, we assume

(8.6) $\sup_{k}\frac{\text{Leb}(H_{\omega,\varepsilon_{N}}\Delta
H_{\sigma^{-k}\omega,\varepsilon_{N}})}{\text{Leb}(H_{\omega,\varepsilon_{N}})}\rightarrow
0,\ N\rightarrow\infty,$

where the use of Lebesgue is for simplicity.

Since the sample measures $\mu_{\omega,0}$ coincide with $\mu,$ we may easily
verify condition ((S)) by choosing conveniently the size of the hole
$H_{\omega,\varepsilon_{N}}$. Moreover, by choosing the holes to be contained
entirely within exactly one interval of monotonicity, we see that ((EX))
holds, and thus ((X)) holds via Remark 6.3 and Proposition 6.4. Since the
weights $g_{\omega}$ are nonrandom and equal to $1/|T^{\prime}|,$ conditions
(E1) to (E7) clearly hold. Since $T$ is continuous in $x_{0},$, the holes will
belong to one branch of $T.$ Therefore if $D(T)$ will denote the number of
branches of $T$ and $\lambda_{m}:=\min_{I}|T^{\prime}|,$
$\lambda_{M}:=\max_{I}|T^{\prime}|,$ it will be enough to have
$D(T)-1>\frac{\lambda_{M}}{\lambda_{m}}$ in order to satisfy ((E9)) with
$n^{\prime}=1.$ Moreover, still keeping $n^{\prime}=1$ and since we have
finitely many branches, condition ((E9a)) in Remark C.2 is satisfied with
$k_{o}(n^{\prime})=1$. We may now apply Theorem 6.11 to obtain a quenched
thermodynamical formalism for our fixed map with sufficiently small random
holes.

Whenever the point $x_{0}$ is not periodic, one obtains that all the
$\hat{q}^{(k)}_{\omega,\varepsilon_{N}}$ are zero by repeating the argument
given in Example 1 for $k\geq 0$ and using the continuity of $T$ at $x_{0}$.
We now take $x_{0}$ as a periodic point of minimal period $p$ and we assume
that $T^{p}$ and $\rho$ are continuous at $x_{0}$. We now begin to evaluate

(8.7)
$\hat{q}_{\omega,\varepsilon_{N}}^{(p-1)}=\frac{\mu(T^{-p}H_{\omega,\varepsilon_{N}}\cap
H_{\sigma^{-p}\omega,\varepsilon_{N}})}{\mu(H_{\omega,\varepsilon_{N}})}.$

Since $T^{p}$ is continuous and expanding in the neighborhood of $x_{0}$, by
taking $N$ large enough, the set $T^{-p}H_{\omega,\varepsilon_{N}}\cap
H_{\sigma^{-p}\omega,\varepsilon_{N}}$ has only one connected component.
Denote the local branch of $T$ through $x_{0}$ by $T_{x_{0}}$. Therefore by a
local change of variable we have for the upper bound of the numerator

$\displaystyle\mu(T^{-p}H_{\omega,\varepsilon_{N}}\cap
H_{\sigma^{-p}\omega,\varepsilon_{N}})$ $\displaystyle=$
$\displaystyle\int_{T^{p}(T^{-p}H_{\omega,\varepsilon_{N}}\cap
H_{\sigma^{-p}\omega,\varepsilon_{N}})}\rho(T^{-p}_{x_{0}}y)|DT^{p}(T^{-p}_{x_{0}}y)|^{-1}d\text{Leb}(y)$
$\displaystyle\leq$
$\displaystyle\sup_{H_{\omega,\varepsilon_{N}}}|DT^{p}|^{-1}\sup_{H_{\omega,\varepsilon_{N}}}\rho\
\text{Leb}(H_{\omega,\varepsilon_{N}}).$

For the lower bound of the numerator, since $T^{p}$ is locally expanding,
$T^{p}(H_{\sigma^{-p}\omega,\varepsilon_{N}})\supset
H_{\sigma^{-p}\omega,\varepsilon_{N}}$, and by (8.3)

$\displaystyle\mu(T^{-p}H_{\omega,\varepsilon_{N}}\cap
H_{\sigma^{-p}\omega,\varepsilon_{N}})\geq\int_{H_{\omega,\varepsilon_{N}}\cap
H_{\sigma^{-p}\omega,\varepsilon_{N}}}\rho(T^{-p}_{x_{0}}y)|DT^{p}(T^{-p}_{x_{0}}y))|^{-1}d\text{Leb}(y)$
$\displaystyle\quad\geq\inf_{H_{\omega,\varepsilon_{N}}}|DT^{p}|^{-1}\inf_{H_{\omega,\varepsilon_{N}}}\rho\
\text{Leb}(H_{\omega,\varepsilon_{N}})-\int_{H_{\omega,\varepsilon_{N}}\setminus
H_{\sigma^{-p}\omega,\varepsilon_{N}}}\rho(T^{-p}_{x_{0}}y)|DT^{p}(T^{-p}_{x_{0}}y)|^{-1}d\text{Leb}(y)$

We can bound the second negative term as

$\int_{H_{\omega,\varepsilon_{N}}\setminus
H_{\sigma^{-p}\omega,\varepsilon_{N}}}\rho(T^{-p}_{x_{0}}y)|DT^{p}(T^{-p}_{x_{0}}y)|^{-1}(y)d\text{Leb}(y)\leq\sup_{H_{\omega,\varepsilon_{N}}}\rho\
\sup_{H_{\omega,\varepsilon_{N}}}|DT^{p}|^{-1}\
\text{Leb}(H_{\omega,\varepsilon_{N}}\Delta
H_{\sigma^{-p}\omega,\varepsilon_{N}}).$

We are now in a position to verify the existence of the limit
$\lim_{\varepsilon_{N}\to 0}\hat{q}_{\omega,\varepsilon_{N}}^{(p-1)}$. Using
the above bounds we have

$\frac{\inf_{H_{\omega,\varepsilon_{N}}}|DT^{p}|^{-1}\inf_{H_{\omega,\varepsilon_{N}}}\rho\
\text{Leb}(H_{\omega,\varepsilon_{N}})}{\sup_{H_{\omega,\varepsilon_{N}}}\rho\text{Leb}(H_{\omega,\varepsilon_{N}})}-\frac{\
\sup_{H_{\omega,\varepsilon_{N}}}|DT^{p}|^{-1}\
\text{Leb}(H_{\omega,\varepsilon_{N}}\Delta
H_{\sigma^{-p}\omega,\varepsilon_{N}})}{\text{Leb}(H_{\omega,\varepsilon_{N}})}$
$\leq\hat{q}_{\omega,\varepsilon_{N}}^{(p-1)}\leq\frac{\sup_{H_{\omega,\varepsilon_{N}}}|DT^{p}|^{-1}\sup_{H_{\omega,\varepsilon_{N}}}\rho\
\text{Leb}(H_{\omega,\varepsilon_{N}})}{\inf_{H_{\omega,\varepsilon_{N}}}\rho\text{Leb}(H_{\omega,\varepsilon_{N}})}.$

Since the map $T^{p}$ and the density $\rho$ are continuous at ${x_{0}},$ and
by using the assumption (8.6), we will finally obtain the $\omega$-independent
value

$\theta_{\omega,0}=1-\frac{1}{|DT^{p}({x_{0}})|}.$

Theorem 5.5 may now be applied to obtain the quenched extreme value law.

###  8.4. Example 4: Random maps with random holes

We saw in Examples 1 and 3 that an extremal index less than one could be
obtained for observables reaching their maximum around a point which was
periodic for all the random maps, or, for a fixed map, when the holes shrink
around the periodic point. We now produce an example where periodicity is not
responsible for getting an extremal index less than one. This example is the
quenched version of the annealed cases investigated in sections 4.1.2 and
4.2.1 of the paper [15].

Let $\Omega=\\{0,\dots,l-1\\}^{\mathbb{Z}},$ with $\sigma$ the bilateral shift
map, and $m$ an invariant ergodic measure. To each letter $j=0,\dots,l-1$ we
associate a point $v(j)$ in the unit circle $S^{1}$ and we consider the well-
known observable in the extreme value theory literature:

$h_{\omega}(x)=-\log|x-v(\omega_{0})|,\ x\in S$

where $\omega_{0}$ denotes the $0$-th coordinate of $\omega\in\Omega.$

In this case the hole $H_{\omega,\varepsilon_{N}}$ will be the ball
$B(v(\omega_{0}),e^{-z_{N}(\omega)}),$ of center $v(\omega_{0})$ and radius
$e^{-z_{N}(\omega)};$. For each $\omega\in\Omega$ we associate a map
$T_{\omega_{0}},$ where $T_{0},\dots,T_{l-1}$ are maps of the circle which we
will take as $\beta$-maps of the form666The reason for this choice is that, in
order to compute the quantities $\hat{q}_{\omega,0}^{(k)}$, we have to follow
the itinerary of the points $v(i)$ under the composition of the maps $T_{i}$
and compare with their predecessors. As it will be clear in the computation
below for $k=0$ and $1,$ the task will be relatively easy and generalisable to
any $k>0$, if all the maps are at least $C^{1}$ which in particular means that
the image of the point $T_{i}(v_{j}),i,j=0,\dots,l-1$ is not a discontinuous
point of the $T_{i},i=0,\dots,l-1,$. It will also be true that all the maps
$T_{i},0=1,\dots,l-1$ are differentiable with bounded derivative in small
neighbourhoods of any $v_{i},i=0,\dots,l-1$. If these conditions are relaxed,
it could be that the limit defining the $\hat{q}_{\omega,0}^{(k)}$ for some
$k$ is not immediately computable, we defer to section 3.3 and Proposition 3.4
in [6] for a detailed discussion of the computation of the extremal index in
presence of discontinuities. Another advantage of our choice is that, as in
Example 1, all the sample measure $\mu_{\omega,0}$ are equivalent to Lebesgue,
$\text{Leb}.$ $T_{i}(x)=\beta_{i}x+r_{i}\pmod{1}$, with
$\beta_{i}\in\mathbb{N}$, $\beta_{i}\geq 3$, and $0\leq r_{i}<1$. Since the
range of $\omega\mapsto\mathcal{L}_{\omega,0}$ is finite and the shift is a
homeomorphism on $\Omega$ with respect to the usual metrics for $\Omega$,
assumption ((MC)) is verified. Since the potential is equal to
$1/|T_{\omega}^{\prime}|,$ conditions (E1) to (E7) clearly hold. Condition
(E8) holds as in Example 1, which uses similar piecewise linear expanding
maps. Condition ((E9)) is a consequence of the fact that we have finitely many
maps each of which is full branches; it will be therefore enough to invoke
((E9a)) with $n^{\prime}=k_{o}(n^{\prime})=1.$ As we have chosen
$\beta_{i}\geq 3$ we have that ((EX)) holds and thus ((X)) follows via
Proposition 6.4.

At this point we may apply Theorem 6.11 to obtain a quenched thermodynamic
formalism for sufficiently small holes.

As a more concrete example, we will consider an alphabet of four letters
$\mathcal{A}:=\\{0,1,2,3\\}$ and we set the associations

$i\rightarrow v_{i}:=v(i),\ i=0,1,2,3,$

where the points $v_{i}\in(0,1)$, are chosen on the unit interval according to
the following prescriptions:

(8.9) $T_{1}(v_{1})=T_{2}(v_{2})=v_{0};\ T_{0}(v_{0})=v_{3};\ T_{3}(v_{i})\neq
v_{3},i=0,1,2,3.$

Since the sample measures coincide with the Lebesgue measure, condition ((S))
reduces to $2e^{-z_{N}(\omega)}=\frac{t_{\omega}+\xi_{\omega,N}}{N}$ and we
can solve for $z_{N}(\omega)$ by setting $\xi_{\omega,N}\equiv 0$. The
prescription (8.9) clearly avoids any sort of periodicity when we take the
first iteration of the random maps needed to compute
$\hat{q}_{\omega,0}^{(0)};$ nevertheless we will show that
$\hat{q}_{\omega,0}^{(0)}>0$ and this is sufficient to conclude that
$\theta_{\omega,0}$ is smaller than $1$. Of course we need to prove that the
limits defining all the other $\hat{q}_{\omega,0}^{(k)}$ for any $k>0$ exist.
We will show it for $k=1$ and the same arguments could be generalized to any
$k>1.$

Using $T$-invariance of Lebesgue, we have (for simplicity we write $z_{N}$
instead of $z_{N}(\omega)$):

$\hat{q}_{\omega,0}^{(0)}=\lim_{N\rightarrow\infty}\frac{\text{Leb}\left(T^{-1}_{\sigma^{-1}\omega}B(v(\omega_{0}),e^{-z_{N}})\cap
B(v((\sigma^{-1}\omega)_{0}),e^{-z_{N}})\right)}{\text{Leb}(B(v(\omega_{0}),e^{-z_{N}}))},$

provided the limit exists, which we are going to establish. Consider first a
point $\omega$ in the cylinder $[\omega_{-1}=0,\omega_{0}=3]$; the quantity we
have to compute is therefore

(8.10)
$\hat{q}_{\omega,0}^{(0)}=\frac{\text{Leb}\left(T^{-1}_{0}B(v_{3},e^{-z_{N}})\cap
B(v_{0},e^{-z_{N}})\right)}{\text{Leb}(B(v_{3},e^{-z_{N}}))}$

If $N$ is large enough, by the prescription (8.9), $T_{0}(v_{0})=v_{3},$ we
see that the local preimage of $B(v_{3},e^{-z_{N}})$ under $T_{0}^{-1}$ will
be strictly included into $B(v_{0},e^{-z_{N}})$ and its length will be
contracted by a factor $\beta_{0}^{-1}.$ Therefore the ratio (8.10) will be
simply $\beta_{0}^{-1}.$ The same happens with the cylinders
$[\omega_{-1}=1,\omega_{0}=0]$ and $[\omega_{-1}=2,\omega_{0}=0],$ producing
respectively the quantities $\beta_{1}^{-1}$ and $\beta_{2}^{-1}.$ For all the
other cylinders the ratio will be zero since the sets entering the numerator
of (8.10) will be disjoint for $N$ large enough. In conclusion we have

$\int_{\Omega}t_{\omega}\hat{q}_{\omega,0}^{(0)}dm=\int_{[\omega_{-1}=0,\omega_{0}=3]}t_{\omega}\beta_{0}^{-1}dm+\int_{[\omega_{-1}=1,\omega_{0}=0]}t_{\omega}\beta_{1}^{-1}dm+\int_{[\omega_{-1}=2,\omega_{0}=0]}t_{\omega}\beta_{2}^{-1}dm.$

If we now take $t_{\omega}=t$, $\omega$-independent, and we choose $m$ as the
Bernoulli measure with equal weights $1/4$, the preceding expression assumes
the simpler form

$\int_{\Omega}t\hat{q}_{\omega,0}^{(0)}dm=\frac{t}{16}(\beta_{0}^{-1}+\beta_{1}^{-1}+\beta_{2}^{-1}).$

To compute $\hat{q}_{\omega,0}^{(1)}$ we have to split the integral over
cylinders of length three. As a concrete example let us consider the cylinder
$[\omega_{-2}=i,\omega_{-1}=j,\omega_{0}=k],$ where $i,j,k\in\mathcal{A}.$ By
using the preceding notations, we have to control the set

$T^{-1}_{i}T^{-1}_{j}B(v_{k},e^{-z_{N}})\cap
T^{-1}_{i}B^{c}(v_{j},e^{-z_{N}})\cap B(v_{i},e^{-z_{N}})=$
$T^{-1}_{i}\left(T^{-1}_{j}B(v_{k},e^{-z_{N}})\cap
B^{c}(v_{j},e^{-z_{N}})\right)\cap B(v_{i},e^{-z_{N}}).$

Let us first consider the intersection $T^{-1}_{j}B(v_{k},e^{-z_{N}})\cap
B^{c}(v_{j},e^{-z_{N}}).$ Call $u$ one of the preimages of $T^{-1}_{j}(v_{k})$
and $T^{-1}_{j,u}$ the inverse branch giving $T^{-1}_{j,u}(v_{k})=u$. If
$u=v_{j}$ then the intersection $T^{-1}_{j,u}B(v_{k},e^{-z_{N}})\cap
B^{c}(v_{j},e^{-z_{N}})$ is empty. Otherwise, by taking $N$ large enough, the
set $T^{-1}_{j,u}B(v_{k},e^{-z_{N}})$ will be completely included in
$B^{c}(v_{j},e^{-z_{N}})$ and moreover we have, by the linearity of the maps,
$T^{-1}_{j,u}B(v_{k},e^{-z_{N}})=B(u,\beta_{j}^{-1}e^{-z_{N}}).$ We are
therefore left with the computation of
$T^{-1}_{i}B(u,\beta_{j}^{-1}e^{-z_{N}})\cap B(v_{i},e^{-z_{N}}).$ If
$u\in\mathcal{A}$ we proceed as in the computation of
$\hat{q}_{\omega,0}^{(0)}$ by using the prescriptions (8.9); otherwise such an
intersection will be empty for $N$ large enough. For instance, if, in the
example we are considering, we take $k=0,j=2,i=0$ and we suppose that among
the $\beta_{2}$ preimages of $v_{0}$ there is, besides $v_{0},$ $v_{3}$ too,
namely $T_{2}(v_{3})=v_{0},$ and no other element of $\mathcal{A},$ 777This
condition does not intervene to compute $\hat{q}_{\omega,0}^{(0)}$ and show
that it was strictly less than one. then we get the contribution for
$\hat{q}_{\omega,0}^{(1)}:$

$\int_{[\omega_{-2}=0,\omega_{-1}=2,\omega_{0}=0]}t_{\omega}\beta_{2}^{-1}\beta_{0}^{-1}dm.$

From the above it follows that $\theta_{\omega,0}$ exists for a.e. $\omega$,
and that we may apply Theorem 5.5.

It is not difficult to give an example where all the
$\hat{q}_{\omega,0}^{(k)}$ can be explicitly computed. Let us take our beta
maps $T_{i}(x)=\beta_{i}x+r_{i}$-$\text{mod}\ 1$ in the particular case where
all the $r_{i}$ are equal to the irrational number $r.$ Then take a sequence
of random balls $B(v((\sigma^{k}\omega)_{0}),e^{-z_{N}}),k\geq 0$ with the
centers $v((\sigma^{k}\omega)_{0}),k\geq 0$ which are rational numbers. From
what we discussed above, it follows that a necessary condition to get a
$\hat{q}_{\omega,0}^{(k)}\neq 0$ is that the center
$v((\sigma^{-(k+1)}\omega)_{0})$ will be sent to the center $v(\omega_{0}).$
Let $z$ one of this rational centers; the iterate $T^{n}_{\omega}(z)$ has the
form
$T^{n}_{\omega}(z)=\beta_{\omega_{n-1}}\cdots\beta_{\omega_{0}}z+k_{n}r,$-mod$1,$
where $k_{n}$ is an integer number. Therefore such an iterate will be never a
rational number which shows that all the $\hat{q}_{\omega,0}^{(k)}=0$ for any
$k\geq 0$ and $\omega,$ and therefore $\theta_{\omega,0}=1.$

## Acknowledgments

The authors thank Harry Crimmins for helpful discussions. JA, GF, and CG-T
thank the Centro de Giorgi in Pisa and CIRM in Luminy for their support and
hospitality. JA is supported by the ARC Discovery projects DP180101223 and
DP210100357, and thanks the School of Mathematics and Physics at the
University of Queensland for their hospitality. GF, CG-T, and SV are partially
supported by the ARC Discovery Project DP180101223. The research of SV was
supported by the project Dynamics and Information Research Institute within
the agreement between UniCredit Bank and Scuola Normale Superiore di Pisa and
by the Laboratoire International Associé LIA LYSM, of the French CNRS and
INdAM (Italy).

##  Appendix A A version of [22] with general weights

In this appendix we outline how to extend relevant results from [22] from the
Perron–Frobenius weight $g_{\omega}=1/|T_{\omega}^{\prime}|$ to the general
class of weights $g_{\omega}$ in Section 6. To begin, we note that there is a
unique measurable equivariant family of functions
$\\{\varphi_{\omega,0}\\}_{\omega\in\Omega}$ guaranteed by Theorem 2.19 [4]
(called $q_{\omega}$ there). We wish to obtain uniform control on the
essential infimum and essential supremum of $\varphi_{\omega,0}$ for a
suitable class of maps.

In [22] we work with the space $\text{{\rm BV}}=\\{h\in L^{1}(\text{{\rm
Leb}}):\text{{\rm var}}(h)<\infty\\}$ and use the norm $\|\cdot\|_{\text{{\rm
BV}}}=\text{{\rm var}}(\cdot)+\|\cdot\|_{1}$. Here we have a measurable family
of random conformal probability measures $\nu_{\omega,0}$ (guaranteed by
Theorem 2.19 [4]) and we work with the random spaces $\text{{\rm
BV}}_{\omega}=\\{h\in L^{1}(\nu_{\omega,0}):\text{{\rm var}}(h)<\infty\\}$ and
the random norms $\|\cdot\|_{\text{{\rm BV}}_{\omega}}=\text{{\rm
var}}(\cdot)+\|\cdot\|_{L^{1}(\nu_{\omega,0})}$. We also work with the
normalised transfer operator
$\tilde{\mathcal{L}}_{\omega,0}(f):=\mathcal{L}_{\omega,0}(f)/\nu_{\omega,0}(\mathcal{L}_{\omega,0}\mathbbold{1})$;
throughout Appendix A we simply denote $\mathcal{L}_{\omega,0}$ and
$\tilde{\mathcal{L}}_{\omega,0}$ by $\mathcal{L}_{\omega}$ and
$\tilde{\mathcal{L}}_{\omega}$ as we only deal with the closed random system.
Similarly we denote $g_{\omega,0}$, $\nu_{\omega,0}$, and $\varphi_{\omega,0}$
by $g_{\omega}$, $\nu_{\omega}$, and $\varphi_{\omega}$ respectively. All of
the variation axioms (V1)–(V8) in [22] hold with the obvious replacements.

###### Proof of Lemma 6.8.

C1’: Since
$\operatorname*{ess\inf}_{\omega}\inf\mathcal{L}_{\omega,0}\mathbbold{1}\geq\operatorname*{ess\inf}_{\omega}\inf
g_{\omega}$, (E3) and (6.6) together imply that ((C1’)) holds.

C7’ ($\varepsilon=0$): We show
$\operatorname*{ess\inf}_{\omega}\inf\varphi_{\omega}>0$; this will give us
((C7’)). The statement
$\operatorname*{ess\inf}_{\omega}\inf\varphi_{\omega}>0$ is a generalised
version of Lemmas 1 and 5 [22] and we follow the strategy in [22]. The result
follows from Lemma A.2 below, which in turn depends on (A.3) and Lemma A.1.

C4’ ($\varepsilon=0$): It will be sufficient to show that there is a
$K<\infty$ and $0<\gamma<1$ such that for all $f\in\text{{\rm BV}}$ with
$\nu_{\omega}(f)=0$ and a.e. $\omega$, one has

(A.1) $\|\tilde{\mathcal{L}}_{\omega}^{n}f\|_{\text{{\rm
BV}}_{\sigma^{n}\omega}}\leq K\gamma^{n}\|f\|_{\text{{\rm BV}}_{\omega}}\mbox{
for all $n\geq 0$}.$

This is a generalised version of Lemma 2 [22], which has an identical proof,
making the replacements outlined in the proof of Lemma A.1 below, and using
Lemmas A.1–A.5 and (A.3). We also use the non-random equivalence (6.13) of the
$\text{{\rm BV}}_{\omega}$ norm to the usual BV norm.

C2 ($\varepsilon=0$), C3 ($\varepsilon=0$), C5’ ($\varepsilon=0$): We wish to