# Perturbation formulae for quenched random dynamics with applications to open
systems and extreme value theory

Jason Atnip  School of Mathematics and Statistics, University of New South
Wales, Sydney, NSW 2052, Australia<EMAIL_ADDRESS>,  Gary Froyland
School of Mathematics and Statistics, University of New South Wales, Sydney,
NSW 2052, Australia<EMAIL_ADDRESS>,  Cecilia González-Tokman  School
of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072,
Australia<EMAIL_ADDRESS>and  Sandro Vaienti  Aix Marseille Université,
Université de Toulon, CNRS, CPT, 13009 Marseille, France<EMAIL_ADDRESS>mrs.fr

###### Abstract.

We consider quasi-compact linear operator cocycles
$\mathcal{L}^{n}_{\omega}:=\mathcal{L}_{\sigma^{n-1}\omega}\circ\cdots\circ\mathcal{L}_{\sigma\omega}\circ\mathcal{L}_{\omega}$
driven by an invertible ergodic process $\sigma:\Omega\to\Omega$, and their
small perturbations $\mathcal{L}_{\omega,\varepsilon}^{n}$. We prove an
abstract $\omega$-wise first-order formula for the leading Lyapunov
multipliers
$\lambda_{\omega,\varepsilon}=\lambda_{\omega}-\theta_{\omega}\Delta_{\omega,\varepsilon}+o(\Delta_{\omega,\varepsilon})$,
where $\Delta_{\omega,\varepsilon}$ quantifies the closeness of
$\mathcal{L}_{\omega,\varepsilon}$ and $\mathcal{L}_{\omega}$. We then
consider the situation where $\mathcal{L}_{\omega}^{n}$ is a transfer operator
cocycle for a random map cocycle
$T_{\omega}^{n}:=T_{\sigma^{n-1}\omega}\circ\cdots\circ T_{\sigma\omega}\circ
T_{\omega}$ and the perturbed transfer operators
$\mathcal{L}_{\omega,\varepsilon}$ are defined by the introduction of small
random holes $H_{\omega,\varepsilon}$ in $[0,1]$, creating a random open
dynamical system. We obtain a first-order perturbation formula in this
setting, which reads
$\lambda_{\omega,\varepsilon}=\lambda_{\omega}-\theta_{\omega}\mu_{\omega}(H_{\omega,\varepsilon})+o(\mu_{\omega}(H_{\omega,\varepsilon})),$
where $\mu_{\omega}$ is the unique equivariant random measure (and equilibrium
state) for the original closed random dynamics. Our new machinery is then
deployed to create a spectral approach for a quenched extreme value theory
that considers random dynamics with general ergodic invertible driving, and
random observations. An extreme value law is derived using the first-order
terms $\theta_{\omega}$. Further, in the setting of random piecewise expanding
interval maps, we establish the existence of random equilibrium states and
conditionally invariant measures for random open systems via a random
perturbative approach. Finally we prove quenched statistical limit theorems
for random equilibrium states arising from contracting potentials. We
illustrate the theory with a variety of explicit examples.

###### Contents

  1. 1 Introduction
    1. 1.1 Quenched perturbations of abstract operator cocycles and sequential/random dynamical systems
    2. 1.2 Quenched extreme value theory general sequential/random dynamics
    3. 1.3 Thermodynamic formalism for quenched random open systems
    4. 1.4 Quenched limit theorems for random dynamics with contracting potentials
    5. 1.5 Examples
  2. 2 Sequential Perturbation Theorem
  3. 3 Random Open Systems
    1. 3.1 Open Systems
    2. 3.2 Some of the Terms from Sequential Perturbation Theorem
  4. 4 Quenched perturbation theorem and escape rate asymptotics for random open systems
    1. 4.1 Escape Rate Asymptotics
  5. 5 Quenched extreme value law
    1. 5.1 Gumbel’s law in the quenched regime
    2. 5.2 The relationship between condition (1.5) and the Hüsler condition
    3. 5.3 Hitting time statistics
  6. 6 Quenched Thermodynamic Formalism for Random Open Interval Maps via Perturbation
  7. 7 Limit theorems
    1. 7.1 The Nagaev-Guivarc’h approach
    2. 7.2 The martingale approach
  8. 8 Examples
    1. 8.1 Example 1: Random maps and random holes centred on a non-random fixed point (random observations with non-random maximum)
    2. 8.2 Example 2: Random $\beta$-maps, random holes containing a non-random fixed point (random observations with non-random maximum), general geometric potential
    3. 8.3 Example 3: A fixed map with random holes containing a fixed point (random observations with a non-random maximum)
    4. 8.4 Example 4: Random maps with random holes
  9. A A version of [22] with general weights
  10. B A summary of checks that relevant results from [17] can be applied to BV
  11. C A var–$\nu_{\omega,0}(|\cdot|)$ Lasota–Yorke inequality
  12. D Proof of Claim (6) of Theorem 6.11

##  1\. Introduction

The spectral approach to studying deterministic dynamical systems $T:X\to X$
on a phase space $X$, centres on the analysis of a transfer operator
$\mathcal{L}:\mathcal{B}(X)\to\mathcal{B}(X)$, given by
$\mathcal{L}f(y)=\sum_{x\in T^{-1}y}g(x)f(x)$, for $f$ in a suitable Banach
space $\mathcal{B}(X)$ and for suitable weight functions
$g:X\to\mathbb{R}^{+}$. If the map $T$ is _covering_ and the weight function
is _contracting_ in the sense of [40] (or similarly if $\sup\log g<P(T)$ as in
[19, 18] or if $\sup\log g-\inf\log g<h_{top}(T)$ as in [31, 14]), then one
obtains the existence of an equilibrium state $\mu$, with associated conformal
measure $\nu$, with the topological pressure $P(T)$ and the density of
equilibrium state $d\mu/d\nu$ given by the logarithm of the leading (positive)
eigenvalue $\lambda$ of $\mathcal{L}$ and the corresponding positive
eigenfunction $h$, respectively. The map $T$ exhibits an exponential decay of
correlations with respect to $\nu$ and $\mu$.

Keller and Liverani [36] showed that the leading eigenvalue $\lambda$ and
eigenfunction $h$ of $\mathcal{L}$ vary continuously with respect to certain
small perturbations of $\mathcal{L}$. One example of such a perturbation is
the introduction of a small hole $H\subset X$. When trajectories of $T$ enter
$H$, the trajectory ends; this generates an _open_ dynamical system. The set
of initial conditions of trajectories that never land in $H$ is the survivor
set $X_{\infty}$. For small holes, specialising to Lasota-Yorke maps of the
interval, Liverani and Maume-Dechamps [39] apply the perturbation theory of
[36] to obtain the existence of a unique conformal measure $\nu$ and an
absolutely continuous _conditionally_ invariant measure $\mu$, with density
$h\in\text{{\rm BV}}([0,1])$. The leading eigenvalue $\lambda$ is
interpretable as an _escape rate_ , and the open system displays an
exponential decay of correlations with respect to $\nu$ and $\mu$.

To obtain finer information on the behaviour of $\lambda$ with respect to
perturbation size, particularly in the situation where the perturbation is not
smooth, such as perturbations arising from the introduction of a hole, one
requires some additional control on the perturbation. Keller and Liverani [37]
develop abstract conditions on $\mathcal{L}$ and its perturbations
$\mathcal{L}_{\varepsilon}$ to ensure good first-order behaviour with respect
to the perturbation size. Following [37], several authors [24, 27, 45, 13]
have used the Keller-Liverani [36] perturbation theory to obtain similar
first-order behaviour of the escape rate with respect to the perturbation size
for open systems in various settings.

This “linear response” of $\lambda$ is exploited in Keller [35] to develop an
elegant spectral approach to deriving an exponential extreme value law to
describe likelihoods of observing extreme values from evaluating an
observation function $h:X\to\mathbb{R}$ along orbits of $T$. In particular,
the $N\to\infty$ limiting law of

(1.1) $\nu\left(\left\\{x\in X:h(T^{j}(x))\leq
z_{N},j=0,\dots,N-1\right\\}\right),$

where the thresholds $z_{N}$ are chosen so that
$\lim_{N\to\infty}N\mu(\\{x:h(x)>z_{N}\\})\to t$ for some $t>0$. The spectral
approach of [35] also provides a relatively explicit expression for the limit
of (1.1), namely $\lim_{N\to\infty}\nu\left(\left\\{x\in X:h(T^{j}(x))\leq
z_{N},j=0,\dots,N-1\right\\}\right)=\exp(-t\theta_{0}),$ where $\theta_{0}$ is
the extremal index.

###  1.1. Quenched perturbations of abstract operator cocycles and
sequential/random dynamical systems

In the present work we begin with sequential compositions of linear operators
$\mathcal{L}_{\omega,0}^{n}:=\mathcal{L}_{\sigma^{n-1}\omega,0}\circ\cdots\circ\mathcal{L}_{\sigma\omega,0}\circ\mathcal{L}_{\omega,0}$,
where $\sigma:\Omega\to\Omega$ is an invertible map on a configuration set
$\Omega$. The driving $\sigma$ could also be an ergodic map on a probability
space $(\Omega,\mathcal{F},m)$. We then consider a family of perturbed
cocycles
$\mathcal{L}_{\omega,\varepsilon}^{n}:=\mathcal{L}_{\sigma^{n-1}\omega,\varepsilon}\circ\cdots\circ\mathcal{L}_{\sigma\omega,\varepsilon}\circ\mathcal{L}_{\omega,\varepsilon}$,
where the size of the perturbation
$\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon}$ is quantified by the
value $\Delta_{\omega,\varepsilon}$ (Definition 2.1). Our first main result is
an abstract quenched formula (Theorem 2.2) for the Lyapunov multipliers
$\lambda_{\omega,0}$ up to first order in the size of the perturbation
$\Delta_{\omega,\varepsilon}$ of the operators $\mathcal{L}_{\omega,0}$. This
quenched random formula generalises the main abstract first-order formula in
[37] stated in the case of a single deterministic operator $\mathcal{L}_{0}$.

> Theorem A: Suppose that assumptions ((P1))-((P9)) hold (see Section 2). If
> $\Delta_{\omega,\varepsilon}>0$ for all $\varepsilon>0$ then for $m$-a.e.
> $\omega\in\Omega$:
>
> $\displaystyle\lim_{\varepsilon\to
> 0}\frac{\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}}{\Delta_{\omega,\varepsilon}}=1-\sum_{k=0}^{\infty}\hat{q}_{\omega,0}^{(k)}=:\theta_{\omega,0}.$

We then introduce random compositions of maps $T_{\omega}:X\to X$, drawn from
a collection $\\{T_{\omega}\\}_{\omega\in\Omega}$. A driving map
$\sigma:\Omega\to\Omega$ on a probability space $(\Omega,\mathcal{F},m)$
creates a map cocycle $T_{\omega}^{n}:=T_{\sigma^{n-1}\omega}\circ\cdots\circ
T_{\sigma\omega}\circ T_{\omega}$. This map cocycle generates a transfer
operator cocycle
$\mathcal{L}_{\omega,0}^{n}:=\mathcal{L}_{\sigma^{n-1}\omega,0}\circ\cdots\circ\mathcal{L}_{\sigma\omega,0}\circ\mathcal{L}_{\omega,0}$,
where $\mathcal{L}_{\omega,0}$ is the transfer operator for the map
$T_{\omega}$. At each iterate, a random hole $H_{\omega}\subset X$ is
introduced.

The existence of a random quenched equilibrium state, conformal measure,
escape rates, and exponential decay of correlations is established in [2] for
relatively large holes, generalising the large-hole constructions of [39] for
a single deterministic map $T$ to the random setting with general driving. In
contrast, our focus here is establishing a random quenched analogue of the
results of [39], [37], and [35] discussed above. To this end, we let
$\mathcal{L}_{\omega,\varepsilon}$ be the transfer operator for the open map
$T_{\omega}$ with a hole $H_{\omega,\varepsilon}$ introduced in $X$, namely
$\mathcal{L}_{\omega,\varepsilon}(f)=\mathcal{L}_{\omega}(\mathbbold{1}_{X\setminus
H_{\omega,\varepsilon}}f)$. Our second collection of main results is a
quenched formula for the derivative of $\lambda_{\omega,\varepsilon}$ with
respect to the sample invariant probability measure of the hole
$\mu_{\omega,0}(H_{\omega,\varepsilon})$ (Theorem 4.6), as well as a quenched
formula for the derivative of the escape rate
$R_{\varepsilon}(\mu_{\omega,0})$ with respect to the sample invariant
probability measure of the hole $\mu_{\omega,0}(H_{\omega,\varepsilon})$
(Corollary 4.9).

> Corollary A: If $(\Omega,\allowbreak
> m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
> T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
> H_{\varepsilon})$ is a random open system (see Section 3.1) with
> $\mu_{\omega,0}(H_{\omega,\varepsilon})>0$ for all $\varepsilon>0$ and
> ((C1))–((C8)) hold (see Section 4), then for $m$-a.e. $\omega\in\Omega$:
>
> (1.2) $\displaystyle\lim_{\varepsilon\to
> 0}\frac{1-\lambda_{\omega,\varepsilon}/\lambda_{\omega,0}}{\mu_{\omega,0}(H_{\omega,\varepsilon})}=1-\sum_{k=0}^{\infty}\hat{q}_{\omega,0}^{(k)}$
>
> and
>
> $\displaystyle
> R_{\varepsilon}(\mu_{\omega,0})=\int_{\Omega}\log\lambda_{\omega,0}-\log\lambda_{\omega,\varepsilon}\,dm(\omega).$
>
> In addition, if (4.12) holds and the $\mu_{\omega,0}$-measures of the random
> holes scale with $\varepsilon$ according to (4.13), then for $m$-a.e.
> $\omega\in\Omega$
>
> $\lim_{\varepsilon\to
> 0}\frac{R_{\varepsilon}(\mu_{\omega,0})}{\mu_{\omega,0}(H_{\omega,\varepsilon})}=\int_{\Omega}\left(1-\sum_{k=0}^{\infty}\hat{q}_{\omega,0}^{(k)}\right)\,dm(\omega).$

###  1.2. Quenched extreme value theory general sequential/random dynamics

To generalize to the random setting the rescaled distribution of the maxima
given by (1.1), we now consider the (random) real-valued random observables
$h_{\omega}$ defined on the phase space $X$ and construct a process
$h_{\sigma^{j}\omega}\circ T^{j}_{\omega}$. We are interested in determining
the limiting law of

(1.3) $\mu_{\omega,0}\left(\left\\{x\in
X:h_{\sigma^{j}\omega}(T^{j}_{\omega}(x))\leq
z_{\sigma^{j}\omega,N},j=0,\dots,N-1\right\\}\right),$

where $\\{z_{\sigma^{j}\omega,N}\\}_{0\leq j\leq N-1}$ is a collection of
real-valued thresholds. The sample probability measure $\mu_{\omega,0}$ enjoys
the equivariance property $T^{*}_{\omega}\mu_{\omega,0}=\mu_{\sigma\omega,0},$
however the process $h_{\sigma^{j}\omega}\circ T^{j}_{\omega}$ is not
stationary in the probabilistic sense, which makes the theory slightly more
difficult.

The first approach to non-stationary extreme value theory (EVT) was given
under convenient assumptions, by Hüsler in [33, 34]. He was able to recover
the usual extremal behaviour seen for i.i.d. or stationary sequences under
Leadbetter’s conditions [38], namely (i) guaranteed mixing properties for the
probability measure governing the process and (ii) that the exceedances should
appear scattered through the time period under consideration. Hüsler’s results
can not be applied in the dynamical systems setting because his uniform bounds
on the control of the exceedances are not satisfied for deterministic, random,
or sequential compositions of maps. The first contribution dealing explicitly
with extreme value theory for random and sequential systems is the paper [28];
see also [26] for an application to point processes. These works were an
adaptation of Leadbetter’s conditions and Hüsler’s approach: let us call them
the probabilistic approach to extreme value theory, to distinguish it from the
spectral and perturbative approach used in the current paper.

As in the deterministic case, in order to avoid a degenerate limit
distribution, one should conveniently choose the thresholds
$z_{\sigma^{j}\omega,N}.$ Hüsler proved convergence to the Gumbel’s law if for
some $0<t<\infty$ we have convergence of the sum

(1.4)
$\sum_{j=0}^{N-1}\mu_{\omega,0}\left(h_{\sigma^{j}\omega}(T^{j}_{\omega}(x))>z_{\sigma^{j}\omega,N}\right)\rightarrow
t.$

for $m$-a.e. $\omega.$ In our current framework we will additionally allow the
positive number $t$ to be any positive random variable in $L^{\infty}(m)$. The
nonstationary theory developed in [28] for quenched random processes, has the
further restrictions that the observation function is fixed
($\omega$-independent), and the thresholds $z_{N}$ (like the scaling $t$) are
just real numbers, and requires the obvious restricted equivalent of (1.4). In
our framework the observation function $h_{\omega}$, the scaling $t_{\omega}$,
and the thresholds $z_{\omega,N}$ may all be random (but need not be). We
generalise and simplify the requirement (1.4) to

(1.5)
$N\mu_{\omega,0}\left(h_{\omega}(x)>z_{\omega,N}\right)=t_{\omega}+\xi_{\omega,N},$

where the scaling $t$ may be a random variable $t\in L^{\infty}(m)$ and the
“errors” $\xi_{\omega,N}$ satisfy (i)
$\lim_{N\rightarrow\infty}\xi_{\omega,N}=0$ a.e., and (ii)
$|\xi_{\omega,N}|\leq W<\infty$ for a.e. $\omega$ and all sufficiently large
$N$.

We provide a more detailed discussion of the relationship between the
conditions (1.4) and (1.5) at the end of section 5.

In summary, we derive a spectral approach for a quenched random extreme value
law, where the dynamics $T_{\omega}$ is random, the observation functions
$h_{\omega}$ can be random, the thresholds controlling what is an extreme
value can be random, and the scalings of the likelihoods of observing extreme
values can be random, all controlled by general invertible ergodic driving.
Moreover, we obtain a formula for the explicit form of Gumbel law for the
extreme value distribution. This leads to our main extreme value theory result
(stated in detail later as Theorem 5.5):

> Theorem B: For a random open system $(\Omega,\allowbreak
> m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
> T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
> H_{\varepsilon})$ (see Section 3.1), assuming ((C1’)), ((C2)), ((C3)),
> ((C4’)), ((C5’)), ((C7’)), ((C8)), and ((S)) (see Sections 4 and 5), for
> $m$-a.e. $\omega\in\Omega$ one has
>
> $\displaystyle\lim_{N\to\infty}\nu_{\omega,0}\left(x\in
> X:h_{\sigma^{j}\omega}(T^{j}_{\omega}(x))\leq
> z_{\sigma^{j}\omega,N},j=0,\dots,N-1\right)$ $\displaystyle\qquad=\hbox
> to0.0pt{$\displaystyle\lim_{N\to\infty}\mu_{\omega,0}\left(x\in
> X:h_{\sigma^{j}\omega}(T^{j}_{\omega}(x))\leq
> z_{\sigma^{j}\omega,N},j=0,\dots,N-1\right)$\hss}$
> $\displaystyle\qquad=\exp\left(-\int_{\Omega}t_{\omega}\theta_{\omega,0}\
> dm(\omega)\right),$
>
> where $\nu_{\omega,0}$ and $\mu_{\omega,0}$ are the random conformal measure
> and the random invariant measure, respectively, for our random dynamics,
> $t_{\omega}$ is a random scaling function, and $\theta_{\omega,0}$ is an
> $\omega$-local extremal index.

This result generalises the spectral approach to extreme value theory in [35],
for a single map $T$, single observation function $h$, single scaling, and
single sequence of thresholds.

Given a family of random holes
$\mathcal{H}_{\omega,N}:=\\{H_{\sigma^{j}\omega,\varepsilon_{N}}\\}_{j\geq
0},$ one can define the first (random) hitting time to a hole, starting at
initial condition $x$ and random configuration $\omega$:

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

When this family of holes shrink with increasing $N$ according to Condition
((S)) (see Section 5.3), Theorem B provides a description of the statistics of
random hitting times, scaled by the measure of the holes (see Theorem 5.7).

> Corollary B: For a random open system $(\Omega,\allowbreak
> m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
> T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
> H_{\varepsilon})$ (see Section 3.1), assume ((C1’)), ((C2)), ((C3)),
> ((C4’)), ((C5’)), ((C7’)), ((C8)), and ((S)) (see Sections 4 and 5). For
> $m$-a.e. $\omega\in\Omega$ one has
>
> (1.6)
> $\lim_{N\to\infty}\mu_{\omega,0}\left(\tau_{\omega,\mathcal{H}_{\omega,N}}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>t_{\omega}\right)=\exp\left(-{\int_{\Omega}t_{\omega}\theta_{\omega,0}\
> dm(\omega)}\right).$

###  1.3. Thermodynamic formalism for quenched random open systems

By assuming some additional uniformity in $\omega$ on the maps $T_{\omega}$ we
use a recent random perturbative result [17] to obtain a complete quenched
thermodynamic formalism. The following existence result extends Theorem C of
[39], which concerned inserting a single small hole into the phase space of a
single deterministic map $T$, to the situation of random map cocycles with
small random holes with the random process controlled by general invertible
ergodic driving $\sigma$.

> Theorem C: Suppose that (E1)–((E9)) (see Section 6) hold for the random open
> system $(\Omega,\allowbreak m,\allowbreak\sigma,\allowbreak[0,\allowbreak
> 1],\allowbreak T,\allowbreak\text{{\rm
> BV}},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
> H_{\varepsilon})$ (see Section 3.1). Then for each $\varepsilon>0$
> sufficiently small there exists a unique random $T$-invariant probability
> measure
> $\mu_{\varepsilon}=\left\\{\mu_{\omega,\varepsilon}\right\\}_{\omega\in\Omega}$
> with $\operatorname{supp}(\mu_{\omega,\varepsilon})\subseteq
> X_{\omega,\infty,\varepsilon}$. Furthermore, $\mu_{\varepsilon}$ is the
> unique relative equilibrium state for the random open system and satisfies a
> forward and backward exponential decay of correlations. In addition, there
> exists a random absolutely continuous (with respect to
> $\left\\{\nu_{\omega,0}\right\\}_{\omega\in\Omega}$) conditionally invariant
> probability measure
> $\varrho_{\varepsilon}=\left\\{\varrho_{\omega,\varepsilon}\right\\}_{\omega\in\Omega}$
> with
> $\operatorname{supp}(\varrho_{\omega,\varepsilon})\subseteq[0,1]\backslash
> H_{\omega,\varepsilon}$ and density function
> $h_{\omega,\varepsilon}\in\text{{\rm BV}}([0,1])$.

For a more explicit statement of Theorem C as well as the relevant assumptions
and definitions see Section 6 and Theorem 6.11.

###  1.4. Quenched limit theorems for random dynamics with contracting
potentials

In Section 7 we prove some quenched limit theorems for closed random dynamics.
These limit results are new for more general potentials and their associated
equilibrium states. We use two approaches. The first is based on the
perturbative technique developed in [21], which generalises the Nagaev-
Guivarc’h method to random cocycles. This technique establishes a relation
between a suitable twisted operator cocycle and the distribution of the random
Birkhoff sums. As a consequence, it is possible to get quenched versions of
the large deviation principle, the central limit theorem, and the local
central limit theorem. The second approach invokes the martingale techniques
previously used in the quenched random setting in [22]. We obtain the almost
sure invariance principle (ASIP) for the equivariant measure $\mu_{\omega,0},$
which also implies the central limit theorem and the law of iterated
logarithms, a general bound for large deviations and a dynamical Borel-
Cantelli lemma.

###  1.5. Examples

We conclude with several explicit examples. We start in Example 1 with the
weight $1/T^{\prime}_{\omega}$ for a family of random maps, random scalings,
and random observations $h_{\omega}$ with a common extremum location in phase
space, which is a common fixed point of the $T_{\omega}$. The special cases of
a fixed map $T$ on the one hand, and a fixed scaling $t$ on the other, are
also considered. The same calculations can be extended to observation
functions with common extrema on a periodic orbit common to all $T_{\omega}$.
Next in Example 2 we consider the more difficult case where orbits are
distributed according to equilibrium states of a general geometric weight
$|DT_{\omega}|^{-r}$, using random $\beta$-maps and random observation
functions $h_{\omega}$ with a common extremum at $x=0$. Example 3 investigates
a fixed map $T$ with random observation functions $h_{\omega}$ with extrema in
a shrinking neighbourhood of a fixed point of $T$, where the neighbourhood
lacks the symmetry of Example 1. In the last example, Example 4, we again
consider random maps $T_{\omega}$ with random observations $h_{\omega}$, but
now the maxima of the observations are not related to fixed or periodic points
of $T_{\omega}$.

Though we apply our results to the setting of random interval maps our results
apply equally well to other random settings including random subshifts [9,
42], random distance expanding maps [43], random polynomial systems [12],
random transcendental maps [44]. In addition, Theorem A (as well as (1.2) of
Corollary A) applies to sequential and semi-group settings including [5, 48].
See Remark 6.12 for a sequential version of Theorem C.

##  2\. Sequential Perturbation Theorem

In this section we prove a general perturbation result for sequential
operators acting on sequential Banach spaces. In particular, we will not
require any measurability or notion of randomness in this section.

Suppose that $\Omega$ is a set and that the map $\sigma:\Omega\to\Omega$ is
invertible. Furthermore, we suppose that there is a family of (fiberwise)
normed vector spaces
$\left\\{\mathcal{B}_{\omega},\|\makebox[4.30554pt]{{$\cdot$}}\|_{\mathcal{B}_{\omega}}\right\\}_{\omega\in\Omega}$
and dual spaces
$\left\\{\mathcal{B}_{\omega}^{*},\|\makebox[4.30554pt]{{$\cdot$}}\|_{\mathcal{B}_{\omega}^{*}}\right\\}_{\omega\in\Omega}$
such that for each $\omega\in\Omega$ and each
$0\leq\varepsilon\leq\varepsilon_{0}$ there are operators
$\mathcal{L}_{\omega,\varepsilon}:\mathcal{B}_{\omega}\to\mathcal{B}_{\sigma\omega}$
such that the following hold.

  1. (P1)

There exists a function $C_{1}:\Omega\to\mathbb{R}_{+}$ such that for
$f\in\mathcal{B}_{\omega}$ we have

$\displaystyle\sup_{\varepsilon\geq
0}\|\mathcal{L}_{\omega,\varepsilon}(f)\|_{\mathcal{B}_{\sigma\omega}}\leq
C_{1}(\omega)\|f\|_{\mathcal{B}_{\omega}}.$

  2. (P2)

For each $\omega\in\Omega$ and $\varepsilon\geq 0$ there is a functional
$\nu_{\omega,\varepsilon}\in\mathcal{B}_{\omega}^{*}$, the dual space of
$\mathcal{B}_{\omega}$,
$\lambda_{\omega,\varepsilon}\in\mathbb{C}\backslash\left\\{0\right\\}$, and
$\varphi_{\omega,\varepsilon}\in\mathcal{B}_{\omega}$ such that

$\displaystyle\mathcal{L}_{\omega,\varepsilon}(\varphi_{\omega,\varepsilon})=\lambda_{\omega,\varepsilon}\varphi_{\sigma\omega,\varepsilon}\quad\text{
and
}\quad\nu_{\sigma\omega,\varepsilon}(\mathcal{L}_{\omega,\varepsilon}(f))=\lambda_{\omega,\varepsilon}\nu_{\omega,\varepsilon}(f)$

for all $f\in\mathcal{B}_{\omega}$. Furthermore we assume that

$\nu_{\omega,0}(\varphi_{\omega,\varepsilon})=1.$

  3. (P3)

There is an operator
$Q_{\omega,\varepsilon}:\mathcal{B}_{\omega}\to\mathcal{B}_{\sigma\omega}$
such that for each $f\in\mathcal{B}_{\omega}$ we have

$\displaystyle\lambda_{\omega,\varepsilon}^{-1}\mathcal{L}_{\omega,\varepsilon}(f)=\nu_{\omega,\varepsilon}(f)\cdot\varphi_{\sigma\omega,\varepsilon}+Q_{\omega,\varepsilon}(f).$

Furthermore, we have

$\displaystyle
Q_{\omega,\varepsilon}(\varphi_{\omega,\varepsilon})=0\quad\text{ and
}\quad\nu_{\sigma\omega,\varepsilon}(Q_{\omega,\varepsilon}(f))=0.$

Note that assumptions ((P2)) and ((P3)) together imply that

$\displaystyle\nu_{\omega,\varepsilon}(\varphi_{\omega,\varepsilon})=1.$

  4. (P4)

There exists a function $C_{2}:\Omega\to\mathbb{R}_{+}$ such that

$\displaystyle\sup_{\varepsilon\geq
0}\|\varphi_{\omega,\varepsilon}\|_{\mathcal{B}_{\omega}}=C_{2}(\omega)<\infty.$

For each $\omega\in\Omega$ and $\varepsilon\geq 0$ we define the quantities

(2.1)
$\displaystyle\Delta_{\omega,\varepsilon}:=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\varphi_{\omega,0})\right)$

and

(2.2)
$\displaystyle\eta_{\omega,\varepsilon}:=\|\nu_{\sigma\omega,0}(\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})\|_{\mathcal{B}_{\omega}^{*}}.$

  5. (P5)

For each $\omega\in\Omega$ we have

$\displaystyle\lim_{\varepsilon\to 0}\eta_{\omega,\varepsilon}=0.$

  6. (P6)

For each $\omega\in\Omega$ such that $\Delta_{\omega,\varepsilon}>0$ for every
$\varepsilon>0$, we have that there exists a function
$C_{3}:\Omega\to\mathbb{R}_{+}$ such that

$\displaystyle\limsup_{\varepsilon\to
0}\frac{\eta_{\omega,\varepsilon}}{\Delta_{\omega,\varepsilon}}:=C_{3}(\omega)<\infty.$

Given $\omega\in\Omega$, if there is $\varepsilon_{0}>0$ such that for each
$\varepsilon\leq\varepsilon_{0}$ we have that $\Delta_{\omega,\varepsilon}=0$
then we also have that $\eta_{\omega,\varepsilon}=0$ for each
$\varepsilon\leq\varepsilon_{0}$.

  7. (P7)

For each $\omega\in\Omega$ we have

$\displaystyle\lim_{\varepsilon\to
0}\nu_{\omega,\varepsilon}(\varphi_{\omega,0})=1.$

  8. (P8)

For each $\omega\in\Omega$ with $\Delta_{\omega,\varepsilon}>0$ for all
$\varepsilon>0$ we have

$\displaystyle\lim_{n\to\infty}\limsup_{\varepsilon\to
0}\Delta_{\omega,\varepsilon}^{-1}\nu_{\sigma\omega,0}\left(\left(\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon}\right)\left(Q_{\sigma^{-n}\omega,\varepsilon}^{n}\varphi_{\sigma^{-n}\omega,0}\right)\right)=0.$

  9. (P9)

For each $\omega\in\Omega$ with $\Delta_{\omega,\varepsilon}>0$ for all
$\varepsilon>0$ we have the limit

$\displaystyle q_{\omega,0}^{(k)}:=\lim_{\varepsilon\to
0}\frac{\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\mathcal{L}_{\sigma^{-(k+1)}\omega,0}-\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)}{\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\varphi_{\omega,0})\right)}$

exists for each $k\geq 0$.

Consider the identity

$\displaystyle\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}$
$\displaystyle=\lambda_{\omega,0}\nu_{\omega,0}(\varphi_{\omega,\varepsilon})-\nu_{\sigma\omega,0}(\lambda_{\omega,\varepsilon}\varphi_{\sigma\omega,\varepsilon})$
$\displaystyle=\nu_{\sigma\omega,0}(\mathcal{L}_{\omega,0}(\varphi_{\omega,\varepsilon}))-\nu_{\sigma\omega,0}(\mathcal{L}_{\omega,\varepsilon}(\varphi_{\omega,\varepsilon}))$
(2.3)
$\displaystyle=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\varphi_{\omega,\varepsilon})\right).$

It then follows from (2.3), together with (2.2) and assumption ((P4)), that

(2.4)
$\displaystyle\left\lvert\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}\right\rvert\leq
C_{2}(\omega)\eta_{\omega,\varepsilon}.$

In particular, given assumption ((P5)), (2.4) implies

(2.5) $\displaystyle\lim_{\varepsilon\to
0}\lambda_{\omega,\varepsilon}=\lambda_{\omega,0}$

for each $\omega\in\Omega$.

######  Remark 2.1.

Note that ((P6)) and (2.4) imply that

$\displaystyle\limsup_{\varepsilon\to
0}\frac{\left\lvert\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}\right\rvert}{\Delta_{\omega,\varepsilon}}\leq
C_{2}(\omega)C_{3}(\omega)<\infty.$

For $n\geq 1$ we define the normalized operator
$\tilde{}\mathcal{L}_{\omega,\varepsilon}^{n}:\mathcal{B}_{\omega}\to\mathcal{B}_{\sigma^{n}\omega}$
by

$\displaystyle\tilde{}\mathcal{L}_{\omega,\varepsilon}^{n}:=(\lambda_{\omega,\varepsilon}^{n})^{-1}\mathcal{L}_{\omega,\varepsilon}^{n}$

where

$\displaystyle\lambda_{\omega,\varepsilon}^{n}:=\lambda_{\omega,\varepsilon}\cdot\dots\cdot\lambda_{\sigma^{n-1}\omega,\varepsilon}.$

In view of assumption ((P3)), induction gives

$\displaystyle\tilde{}\mathcal{L}_{\omega,\varepsilon}^{n}(f)=\nu_{\omega,\varepsilon}(f)\cdot\varphi_{\sigma^{n}\omega,\varepsilon}+Q_{\omega,\varepsilon}^{n}(f)$

for each $n\geq 1$ and all $f\in\mathcal{B}_{\omega}$. Similarly with (2.3),
we have that

$\displaystyle\lambda_{\omega,0}^{n}-\lambda_{\omega,\varepsilon}^{n}=\nu_{\sigma^{n}\omega,0}\left(\left(\mathcal{L}_{\omega,0}^{n}-\mathcal{L}_{\omega,\varepsilon}^{n}\right)(\varphi_{\omega,\varepsilon})\right).$

We now arrive at the main result of this section. We prove a differentiability
result for the perturbed quantities $\lambda_{\omega,\varepsilon}$ as
$\varepsilon\to 0$ in the spirit of Keller and Liverani [37].

######  Theorem 2.2.

Suppose that assumptions ((P1))-((P8)) hold. If there is some
$\varepsilon_{0}>0$ such that $\Delta_{\omega,\varepsilon}=0$ for
$\varepsilon\leq\varepsilon_{0}$ then

$\displaystyle\lambda_{\omega,0}=\lambda_{\omega,\varepsilon},$

or if ((P9)) holds then

$\displaystyle\lim_{\varepsilon\to
0}\frac{\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}}{\Delta_{\omega,\varepsilon}}=1-\sum_{k=0}^{\infty}(\lambda_{\sigma^{-(k+1)}\omega,0}^{k+1})^{-1}q_{\omega,0}^{(k)}.$

###### Proof.

Fix $\omega\in\Omega$. First, we note that assumption ((P6)) implies that if
there is some $\varepsilon_{0}$ such that $\Delta_{\omega,\varepsilon}=0$ for
all $\varepsilon\leq\varepsilon_{0}$ (which implies, by assumption, that
$\eta_{\omega,\varepsilon}=0$), then (2.4) immediately implies that

$\displaystyle\lambda_{\omega,0}=\lambda_{\omega,\varepsilon}.$

Now we suppose that $\Delta_{\omega,\varepsilon}>0$ for all $\varepsilon>0$.
Using ((P2)), ((P3)), and (2.3), for each $n\geq 1$ and all $\omega\in\Omega$
we have

$\displaystyle\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})(\lambda_{\omega,0}-\lambda_{\omega,\varepsilon})=\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\varphi_{\omega,\varepsilon})\right)$
$\displaystyle\quad=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})\cdot\varphi_{\omega,\varepsilon})\right)$
$\displaystyle\quad=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})\cdot\varphi_{\omega,\varepsilon}+Q_{\sigma^{-n}\omega,\varepsilon}^{n}(\varphi_{\sigma^{-n}\omega,0})-Q_{\sigma^{-n}\omega,\varepsilon}^{n}(\varphi_{\sigma^{-n}\omega,0}))\right)$
$\displaystyle\quad=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^{n}-Q_{\sigma^{-n}\omega,\varepsilon}^{n})(\varphi_{\sigma^{-n}\omega,0})\right)$
$\displaystyle\quad=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-n}\omega,0}^{n}-\tilde{}\mathcal{L}_{\sigma^{-n}\omega,0}^{n}+\tilde{}\mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^{n}-Q_{\sigma^{-n}\omega,\varepsilon}^{n})(\varphi_{\sigma^{-n}\omega,0})\right)$
$\displaystyle\quad=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\varphi_{\omega,0})\right)$
$\displaystyle\qquad-\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-n}\omega,0}^{n}-\tilde{}\mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^{n})(\varphi_{\sigma^{-n}\omega,0})\right)$
$\displaystyle\qquad\qquad-\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(Q_{\sigma^{-n}\omega,\varepsilon}^{n}(\varphi_{\sigma^{-n}\omega,0}))\right)$
(2.6)
$\displaystyle\quad=\Delta_{\omega,\varepsilon}-\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-n}\omega,0}^{n}-\tilde{}\mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^{n})(\varphi_{\sigma^{-n}\omega,0})\right)$
(2.7)
$\displaystyle\qquad-\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(Q_{\sigma^{-n}\omega,\varepsilon}^{n}(\varphi_{\sigma^{-n}\omega,0}))\right).$

Since

$\displaystyle(\tilde{}\mathcal{L}_{\sigma^{-n}\omega,0}^{n}-\tilde{}\mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^{n})(\varphi_{\sigma^{-n}\omega,0})=\varphi_{\omega,0}-\tilde{}\mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^{n}(\varphi_{\sigma^{-n}\omega,0}),$

using a telescoping argument, the second term of (2.6)

$D_{2}:=-\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-n}\omega,0}^{n}-\tilde{}\mathcal{L}_{\sigma^{-n}\omega,\varepsilon}^{n})(\varphi_{\sigma^{-n}\omega,0})\right)$

can be rewritten as

$\displaystyle D_{2}$
$\displaystyle=-\sum_{k=0}^{n-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\tilde{}\mathcal{L}_{\sigma^{-(k+1)}\omega,0}-\tilde{}\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)$
$\displaystyle=-\sum_{k=0}^{n-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\tilde{}\mathcal{L}_{\sigma^{-(k+1)}\omega,0}\right.$
$\displaystyle\qquad\left.-\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon}+\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon}-\tilde{}\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)$
(2.8)
$\displaystyle=-\sum_{k=0}^{n-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\tilde{}\mathcal{L}_{\sigma^{-(k+1)}\omega,0}\right.$
$\displaystyle\qquad\left.-\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon}+\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)$
(2.9)
$\displaystyle\qquad\qquad+\sum_{k=0}^{n-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\tilde{}\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right).$

Reindexing, the second summand of the above calculation, and multiplying by
$1$, (2.9), can be rewritten as

$\displaystyle\sum_{k=0}^{n-1}$
$\displaystyle\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon}^{k+1})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)$
$\displaystyle\quad=\sum_{k=1}^{n}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0})\right)$
(2.10)
$\displaystyle\quad=\sum_{k=1}^{n}\lambda_{\sigma^{-k}\omega,0}^{-1}\lambda_{\sigma^{-k}\omega,0}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0})\right).$

And (2.8) from above can again be broken into two parts and then rewritten as

$\displaystyle-\sum_{k=0}^{n-1}\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\mathcal{L}_{\sigma^{-(k+1)}\omega,0}-\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)$
(2.11)
$\displaystyle\quad-\sum_{k=1}^{n}\lambda_{\sigma^{-k}\omega,0}^{-1}\lambda_{\sigma^{-k}\omega,\varepsilon}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0})\right),$

where in the second sum we have used the fact that
$\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}=\lambda_{\sigma^{-k}\omega,\varepsilon}\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}$.
Altogether using (2.10) and (2.11), $D_{2}$ we can be written as

$\displaystyle D_{2}$
$\displaystyle=-\sum_{k=0}^{n-1}\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\mathcal{L}_{\sigma^{-(k+1)}\omega,0}-\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)$
$\displaystyle\qquad-\sum_{k=1}^{n}\lambda_{\sigma^{-k}\omega,0}^{-1}\lambda_{\sigma^{-k}\omega,\varepsilon}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0})\right)$
$\displaystyle\qquad\quad+\sum_{k=1}^{n}\lambda_{\sigma^{-k}\omega,0}^{-1}\lambda_{\sigma^{-k}\omega,0}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0})\right)$
$\displaystyle=-\sum_{k=0}^{n-1}\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\mathcal{L}_{\sigma^{-(k+1)}\omega,0}-\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)$
(2.12)
$\displaystyle\quad=+\sum_{k=1}^{n}\lambda_{\sigma^{-k}\omega,0}^{-1}\left(\lambda_{\sigma^{-k}\omega,0}-\lambda_{\sigma^{-k}\omega,\varepsilon}\right)\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0})\right).$

Now for each $k\geq 0$ we let

(2.13) $\displaystyle
q_{\omega,\varepsilon}^{(k)}:=\frac{\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\mathcal{L}_{\sigma^{-(k+1)}\omega,0}-\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)}{\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\varphi_{\omega,0})\right)}.$

Using (2.12) we can continue our rephrasing of
$\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})(\lambda_{\omega,0}-\lambda_{\omega,\varepsilon})$
from (2.6) to get

$\displaystyle\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})(\lambda_{\omega,0}-\lambda_{\omega,\varepsilon})$
$\displaystyle\quad=\Delta_{\omega,\varepsilon}-\sum_{k=0}^{n-1}\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\mathcal{L}_{\sigma^{-(k+1)}\omega,0}-\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)$
$\displaystyle\qquad+\sum_{k=1}^{n}\lambda_{\sigma^{-k}\omega,0}^{-1}\left(\lambda_{\sigma^{-k}\omega,0}-\lambda_{\sigma^{-k}\omega,\varepsilon}\right)\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0})\right)$
$\displaystyle\quad\qquad-\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(Q_{\sigma^{-n}\omega,\varepsilon}^{n}(\varphi_{\sigma^{-n}\omega,0}))\right)$
$\displaystyle\quad=\Delta_{\omega,\varepsilon}\left(1-\sum_{k=0}^{n-1}\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}(\lambda_{\sigma^{-k}\omega,\varepsilon}^{k})^{-1}q_{\omega,\varepsilon}^{(k)}\right)$
$\displaystyle\qquad+\sum_{k=1}^{n}\lambda_{\sigma^{-k}\omega,0}^{-1}\left(\lambda_{\sigma^{-k}\omega,0}-\lambda_{\sigma^{-k}\omega,\varepsilon}\right)\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0})\right)$
(2.14)
$\displaystyle\quad\qquad-\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(Q_{\sigma^{-n}\omega,\varepsilon}^{n}(\varphi_{\sigma^{-n}\omega,0}))\right).$

Dividing the calculation culminating in (2.14) by
$\Delta_{\omega,\varepsilon}$ on both sides gives

(2.15)
$\displaystyle\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})\frac{\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}}{\Delta_{\omega,\varepsilon}}$
$\displaystyle\quad=1-\sum_{k=0}^{n-1}\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}(\lambda_{\sigma^{-k}\omega,\varepsilon}^{k})^{-1}q_{\omega,\varepsilon}^{(k)}$
(2.16)
$\displaystyle\quad+\Delta_{\omega,\varepsilon}^{-1}\sum_{k=1}^{n}\lambda_{\sigma^{-k}\omega,0}^{-1}\left(\lambda_{\sigma^{-k}\omega,0}-\lambda_{\sigma^{-k}\omega,\varepsilon}\right)\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0})\right)$
(2.17)
$\displaystyle\qquad-\Delta_{\omega,\varepsilon}^{-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(Q_{\sigma^{-n}\omega,\varepsilon}^{n}(\varphi_{\sigma^{-n}\omega,0}))\right).$

Assumption ((P8)) ensures that (2.17) goes to zero as $\varepsilon\to 0$ and
$n\to\infty$. Now, using (2.2), (2.4), ((P1)), and ((P4)) we bound (2.16) by

$\displaystyle\Delta_{\omega,\varepsilon}^{-1}\sum_{k=1}^{n}\left\lvert\lambda_{\sigma^{-k}\omega,0}\right\rvert^{-1}\left\lvert\lambda_{\sigma^{-k}\omega,0}-\lambda_{\sigma^{-k}\omega,\varepsilon}\right\rvert\eta_{\omega,\varepsilon}\|\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k}(\varphi_{\sigma^{-k}\omega,0})\|_{\mathcal{B}_{\omega}}$
$\displaystyle\qquad\qquad\leq\frac{\eta_{\omega,\varepsilon}}{\Delta_{\omega,\varepsilon}}\sum_{k=1}^{n}C_{2}(\sigma^{-k}\omega)\left\lvert\lambda_{\sigma^{-k}\omega,0}\right\rvert^{-1}\eta_{\sigma^{-k}\omega,\varepsilon}\left\lvert\lambda_{\sigma^{-k}\omega,\varepsilon}^{k}\right\rvert^{-1}\|\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k}(\varphi_{\sigma^{-k}\omega,0})\|_{\mathcal{B}_{\omega}}$
(2.18)
$\displaystyle\qquad\qquad\leq\frac{\eta_{\omega,\varepsilon}}{\Delta_{\omega,\varepsilon}}\sum_{k=1}^{n}(C_{2}(\sigma^{-k}\omega))^{2}C_{1}^{k}(\sigma^{-k}\omega)\left\lvert\lambda_{\sigma^{-k}\omega,0}\right\rvert^{-1}\eta_{\sigma^{-k}\omega,\varepsilon}\left\lvert\lambda_{\sigma^{-k}\omega,\varepsilon}^{k}\right\rvert^{-1}.$

In view of ((P5)), ((P6)), and (2.5), for fixed $n$, we may continue from
(2.18) and let $\varepsilon\to 0$ to see that

(2.19) $\displaystyle\lim_{\varepsilon\to
0}\frac{\eta_{\omega,\varepsilon}}{\Delta_{\omega,\varepsilon}}\sum_{k=1}^{n}(C_{2}(\sigma^{-k}\omega))^{2}C_{1}^{k}(\sigma^{-k}\omega)\left\lvert\lambda_{\sigma^{-k}\omega,0}\right\rvert^{-1}\eta_{\sigma^{-k}\omega,\varepsilon}\left\lvert\lambda_{\sigma^{-k}\omega,\varepsilon}^{k}\right\rvert^{-1}=0.$

In light of ((P6)), ((P7)), (2.19), and (2.15)–(2.17) together with ((P8)) and
((P9)), we see that first letting $\varepsilon\to 0$ and then $n\to\infty$
gives

$\displaystyle\lim_{\varepsilon\to
0}\frac{\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}}{\Delta_{\omega,\varepsilon}}=1-\sum_{k=0}^{\infty}(\lambda_{\sigma^{-(k+1)}\omega,0}^{k+1})^{-1}q_{\omega,0}^{(k)}$

as desired. ∎

In the sequel we will refer to the quantity on the right hand side of the last
equation in the proof of the previous theorem by $\theta_{\omega,0}$, i.e. we
set

(2.20)
$\displaystyle\theta_{\omega,0}:=1-\sum_{k=0}^{\infty}(\lambda_{\sigma^{-(k+1)}\omega,0}^{k+1})^{-1}q_{\omega,0}^{(k)}.$

##  3\. Random Open Systems

In this section we introduce the general setup of random open systems to which
we can apply our perturbation theory from Section 2.

We begin with a probability space $(\Omega,\mathscr{F},m)$ and an ergodic,
invertible map $\sigma:\Omega\to\Omega$ which preserves the measure $m$, i.e.

$\displaystyle m\circ\sigma^{-1}=m.$

For each $\omega\in\Omega$ let $\mathcal{J}_{\omega,0}$ be a closed subset of
a complete metrisable space $X$ such that the map

$\displaystyle\Omega\ni\omega\longmapsto\mathcal{J}_{\omega,0}$

is a closed random set, i.e. $\mathcal{J}_{\omega,0}\subseteq X$ is closed for
each $\omega\in\Omega$ and the map $\omega\mapsto\mathcal{J}_{\omega,0}$ is
measurable (see [16]), and we consider the maps

$\displaystyle
T_{\omega}:\mathcal{J}_{\omega,0}\to\mathcal{J}_{\sigma\omega,0}.$

By $T_{\omega}^{N}:\mathcal{J}_{\omega,0}\to\mathcal{J}_{\sigma^{N}\omega,0}$
we mean the $N$-fold composition

$\displaystyle T_{\sigma^{N}\omega}\circ\dots\circ
T_{\omega}:\mathcal{J}_{\omega,0}\to\mathcal{J}_{\sigma^{N}\omega,0}.$

Given a set $A\subseteq\mathcal{J}_{\sigma^{N}\omega,0}$ we let

$\displaystyle
T_{\omega}^{-N}(A):=\left\\{x\in\mathcal{J}_{\omega,0}:T_{\omega}^{N}(x)\in
A\right\\}$

denote the inverse image of $A$ under the map $T_{\omega}^{N}$ for each
$\omega\in\Omega$ and $N\geq 1$. Now let

$\displaystyle\mathcal{J}_{0}:=\bigcup_{\omega\in\Omega}\left\\{\omega\right\\}\times\mathcal{J}_{\omega,0}\subseteq\Omega\times
X,$

and define the induced skew-product map $T:\mathcal{J}_{0}\to\mathcal{J}_{0}$
by

$\displaystyle T(\omega,x)=(\sigma\omega,T_{\omega}(x)).$

Let $\mathscr{B}$ denote the Borel $\sigma$-algebra of $X$ and let
$\mathscr{F}\otimes\mathscr{B}$ be the product $\sigma$-algebra on
$\Omega\times X$. We suppose the following:

  1. (M1)

the map $T:\mathcal{J}_{0}\to\mathcal{J}_{0}$ is measurable with respect to
$\mathscr{F}\otimes\mathscr{B}$.

######  Definition 3.1.

A measure $\mu$ on $\Omega\times X$ with respect to the product
$\sigma$-algebra $\mathscr{F}\otimes\mathscr{B}$ is said to be random measure
relative to $m$ if it has marginal $m$, i.e. if

$\mu\circ\pi^{-1}_{1}=m.$

The disintegrations $\left\\{\mu_{\omega}\right\\}_{\omega\in\Omega}$ of $\mu$
with respect to the partition $\left(\\{\omega\\}\times
X\right)_{\omega\in\Omega}$ satisfy the following properties:

  1. (1)

For every $B\in\mathscr{B}$, the map
$\Omega\ni\omega\longmapsto\mu_{\omega}(B)\in X$ is measurable,

  2. (2)

For $m$-a.e. $\omega\in\Omega$, the map $\mathscr{B}\ni
B\longmapsto\mu_{\omega}(B)\in X$ is a Borel measure.

We say that the random measure
$\mu=\left\\{\mu_{\omega}\right\\}_{\omega\in\Omega}$ is a random probability
measure if for $m$-a.e. $\omega\in\Omega$ the fiber measure $\mu_{\omega}$ is
a probability measure. Given a set
$Y=\cup_{\omega\in\Omega}\left\\{\omega\right\\}\times
Y_{\omega}\subseteq\Omega\times X$, we say that the random measure
$\mu=\left\\{\mu_{\omega}\right\\}_{\omega\in\Omega}$ is supported in $Y$ if
$\operatorname{supp}(\mu)\subseteq Y$ and consequently
$\operatorname{supp}(\mu_{\omega})\subseteq Y_{\omega}$ for $m$-a.e.
$\omega\in\Omega$. We let $\mathcal{P}_{\Omega}(Y)$ denote the set of all
random probability measures supported in $Y$. We will frequently denote a
random measure $\mu$ by $\left\\{\mu_{\omega}\right\\}_{\omega\in\Omega}$.

The following proposition from Crauel [16], shows a random probability measure
$\left\\{\mu_{\omega}\right\\}_{\omega\in\Omega}$ on $\mathcal{J}_{0}$
uniquely identifies a probability measure on $\mathcal{J}_{0}$.

######  Proposition 3.2 ([16], Propositions 3.3).

If
$\left\\{\mu_{\omega}\right\\}_{\omega\in\Omega}\in\mathcal{P}_{\Omega}(\mathcal{J}_{0})$
is a random probability measure on $\mathcal{J}_{0}$, then for every bounded
measurable function $f:\mathcal{J}_{0}\to\mathbb{R}$, the function

$\Omega\ni\omega\longmapsto\int_{\mathcal{J}_{\omega,0}}f(\omega,x)\,d\mu_{\omega}(x)$

is measurable and

$\mathscr{F}\otimes\mathscr{B}\ni
A\longmapsto\int_{\Omega}\int_{\mathcal{J}_{\omega,0}}\mathbbold{1}_{A}(\omega,x)\,d\mu_{\omega}(x)\,dm(\omega)$

defines a probability measure on $\mathcal{J}_{0}$.

We assume there exists a family of Banach spaces
$\left\\{\mathcal{B}_{\omega},\|\makebox[4.30554pt]{{$\cdot$}}\|_{\mathcal{B}_{\omega}}\right\\}_{\omega\in\Omega}$
of real-valued functions on each $\mathcal{J}_{\omega,0}$ and a family of
potentials $\left\\{g_{\omega,0}\right\\}_{\omega\in\Omega}$ with
$g_{\omega,0}\in\mathcal{B}_{\omega}$ such that the (fiberwise) transfer
operator
$\mathcal{L}_{\omega,0}:\mathcal{B}_{\omega}\to\mathcal{B}_{\sigma\omega}$
given by

$\displaystyle\mathcal{L}_{\omega,0}(f)(x):=\sum_{y\in
T_{\omega}^{-1}(x)}f(y)g_{\omega,0}(y),\quad
f\in\mathcal{B}_{\omega},\,x\in\mathcal{J}_{\sigma\omega,0}$

is well defined. Using induction we see that iterates
$\mathcal{L}_{\omega,0}^{N}:\mathcal{B}_{\omega}\to\mathcal{B}_{\sigma^{N}\omega}$
of the transfer operator are given by

$\displaystyle\mathcal{L}_{\omega,0}^{N}(f)(x):=\sum_{y\in
T_{\omega}^{-N}(x)}f(y)g_{\omega,0}^{(N)}(y),\quad
f\in\mathcal{B}_{\omega},\,x\in\mathcal{J}_{\sigma^{N}\omega,0},$

where

$g_{\omega,0}^{(N)}:=\prod_{j=0}^{N-1}g_{\sigma^{j}\omega,0}\circ
T_{\omega}^{j}.$

We let $\mathcal{B}$ denote the space of functions
$f:\mathcal{J}_{0}\to\mathbb{R}$ such that $f_{\omega}\in\mathcal{B}_{\omega}$
for each $\omega\in\Omega$ and we define the global transfer operator
$\mathcal{L}_{0}:\mathcal{B}\to\mathcal{B}$ by
$(\mathcal{L}_{0}f)_{\omega}(x):=\mathcal{L}_{\sigma^{-1}\omega,0}f_{\sigma^{-1}\omega}(x)$
for $f\in\mathcal{B}$ and $x\in\mathcal{J}_{\omega,0}$. We assume the
following measurability assumption:

  1. (M2)

For every measurable function $h\in\mathcal{B}$, the map
$(\omega,x)\mapsto(\mathcal{L}_{0}h)_{\omega}(x)$ is measurable.

We suppose the following condition on the existence of a closed conformal
measure.

  1. (CCM)

There exists a random probability measure
$\nu_{0}=\left\\{\nu_{\omega,0}\right\\}_{\omega\in\Omega}\in\mathcal{P}_{\Omega}(\mathcal{J}_{0})$
and measurable functions
$\lambda_{0}:\Omega\to\mathbb{R}\backslash\left\\{0\right\\}$ and
$\varphi_{0}:\mathcal{J}_{0}\to(0,\infty)$ such that

$\displaystyle\mathcal{L}_{\omega,0}(\varphi_{\omega,0})=\lambda_{\omega,0}\varphi_{\sigma\omega,0}\quad\text{
and
}\quad\nu_{\sigma\omega,0}(\mathcal{L}_{\omega,0}(f))=\lambda_{\omega,0}\nu_{\omega,0}(f)$

for all $f\in\mathcal{B}_{\omega}$ where
$\varphi_{\omega,0}(\makebox[4.30554pt]{{$\cdot$}}):=\varphi_{0}(\omega,\makebox[4.30554pt]{{$\cdot$}})$.
Furthermore, we suppose that the fiber measures $\nu_{\omega,0}$ are non-
atomic and that
$\lambda_{\omega,0}:=\nu_{\sigma\omega,0}(\mathcal{L}_{\omega,0}\mathbbold{1})$
with $\log\lambda_{\omega,0}\in L^{1}(m)$. We then define the random
probability measure $\mu_{0}$ on $\mathcal{J}_{0}$ by

(3.1)
$\displaystyle\mu_{\omega,0}(f):=\int_{\mathcal{J}_{\omega,0}}f\varphi_{\omega,0}\
d\nu_{\omega,0},\qquad f\in L^{1}(\mathcal{J}_{\omega,0},\nu_{\omega,0}).$

From the definition, one can easily show that $\mu_{0}$ is $T$-invariant, that
is,

(3.2) $\displaystyle\int_{\mathcal{J}_{\omega,0}}f\circ T_{\omega}\
d\mu_{\omega,0}=\int_{\mathcal{J}_{\sigma\omega,0}}f\
d\mu_{\sigma\omega,0},\qquad f\in
L^{1}(\mathcal{J}_{\sigma\omega,0},\mu_{\sigma\omega,0}).$

######  Remark 3.3.

Our Assumption ((CCM)) has been shown to hold in several random settings:
random interval maps [4, 3], random subshifts [9, 42], random distance
expanding maps [43], random polynomial systems [12], and random transcendental
maps [44].

We will also assume that the fiberwise Banach spaces
$\mathcal{B}_{\omega}\subseteq
L^{\infty}(\mathcal{J}_{\omega,0},\nu_{\omega,0})$ and that there exists a
measurable $m$-a.e. finite function $K:\Omega\to[1,\infty)$ such that

  1. (B)

$\|f\|_{\infty,\omega}\leq K_{\omega}\|f\|_{\mathcal{B}_{\omega}}$

for all $f\in\mathcal{B}_{\omega}$ and each $\omega\in\Omega$, where
$\|\cdot\|_{\infty,\omega}$ denotes the supremum norm with respect to
$\nu_{\omega,0}$. We are now ready to introduce holes into the closed systems.

###  3.1. Open Systems

For each $\varepsilon>0$ we let $H_{\varepsilon}\subseteq\mathcal{J}_{0}$ be
measurable with respect to the product $\sigma$-algebra
$\mathscr{F}\otimes\mathscr{B}$ on $\mathcal{J}_{0}$ such that

  1. (A)

$H_{\varepsilon}^{\prime}\subseteq H_{\varepsilon}$ for each
$0<\varepsilon^{\prime}\leq\varepsilon$.

For each $\omega\in\Omega$ the sets
$H_{\omega,\varepsilon}\subseteq\mathcal{J}_{\omega,0}$ are uniquely
determined by the condition that

$\displaystyle\left\\{\omega\right\\}\times
H_{\omega,\varepsilon}=H_{\varepsilon}\cap\left(\left\\{\omega\right\\}\times\mathcal{J}_{\omega,0}\right).$

Equivalently we have

$\displaystyle
H_{\omega,\varepsilon}=\pi_{2}(H_{\varepsilon}\cap(\left\\{\omega\right\\}\times\mathcal{J}_{\omega,0})),$

where $\pi_{2}:\mathcal{J}_{0}\to\mathcal{J}_{\omega,0}$ is the projection
onto the second coordinate. By definition we have that the sets
$H_{\omega,\varepsilon}$ are $\nu_{\omega,0}$-measurable, and ((A)) implies
that

  1. (A’)

$H_{\omega,\varepsilon^{\prime}}\subseteq H_{\omega,\varepsilon}$ for each
$\varepsilon^{\prime}\leq\varepsilon$ and each $\omega\in\Omega$.

For each $\varepsilon>0$ we set

$\Omega_{+,\varepsilon}:=\left\\{\omega\in\Omega:\mu_{\omega,0}(H_{\omega,\varepsilon})>0\right\\}$

and then define

(3.3)
$\displaystyle\Omega_{+}:=\bigcap_{\varepsilon>0}\Omega_{+,\varepsilon}.$

######  Remark 3.4.

Note that the set $\Omega_{+}$ is measurable as it is the intersection of a
decreasing family of measurable sets and it is not necessarily
$\sigma$-invariant.

Now define

$\displaystyle\mathcal{J}_{\omega,\varepsilon}:=\mathcal{J}_{\omega,0}\backslash
H_{\omega,\varepsilon},$

and let

$\displaystyle\mathcal{J}_{\varepsilon}:=\bigcup_{\omega\in\Omega}\left\\{\omega\right\\}\times\mathcal{J}_{\omega,\varepsilon}.$

For each $\omega\in\Omega$, $N\geq 0$, and $\varepsilon>0$ we define

$\displaystyle X_{\omega,N,\varepsilon}:$
$\displaystyle=\left\\{x\in\mathcal{J}_{\omega,0}:T_{\omega}^{j}(x)\notin
H_{\sigma^{j}\omega,\varepsilon}\text{ for all }0\leq j\leq
N\right\\}=\bigcap_{j=0}^{N}T_{\omega}^{-j}\left(\mathcal{J}_{\sigma^{j}\omega,\varepsilon}\right)$

to be the set of points in $\mathcal{J}_{\omega,\varepsilon}$ which survive,
i.e. those points whose trajectories do not land in the holes, for $N$
iterates. We then naturally define

$\displaystyle
X_{\omega,\infty,\varepsilon}:=\bigcap_{N=0}^{\infty}X_{\omega,N,\varepsilon}=\bigcap_{N=0}^{\infty}T_{\omega}^{-N}(\mathcal{J}_{\sigma^{N}\omega,\varepsilon})$

to be the set of points which will never land in a hole under iteration of the
maps $T_{\omega}^{N}$ for any $N\geq 0$. We call
$X_{\omega,\infty,\varepsilon}$ the $(\omega,\varepsilon)$-surviving set. Note
that the sets $X_{\omega,N,\varepsilon}$ and $X_{\omega,\infty,\varepsilon}$
are forward invariant satisfying the properties

(3.4) $\displaystyle T_{\omega}(X_{\omega,N,\varepsilon})\subseteq
X_{\sigma\omega,N-1,\varepsilon}\qquad\text{ and }\qquad
T_{\omega}(X_{\omega,\infty,\varepsilon})\subseteq
X_{\sigma\omega,\infty,\varepsilon}.$

Now for any $0\leq\alpha\leq\infty$ we set

$\displaystyle\hat{X}_{\omega,\alpha,\varepsilon}:=\mathbbold{1}_{X_{\omega,\alpha,\varepsilon}}=\prod_{j=0}^{\alpha}\mathbbold{1}_{\mathcal{J}_{\sigma^{j}\omega,\varepsilon}}\circ
T_{\omega}^{j}.$

The global surviving set is defined as

$\displaystyle\mathcal{X}_{\alpha,\varepsilon}:=\bigcup_{\omega\in\Omega}\left\\{\omega\right\\}\times
X_{\omega,\alpha,\varepsilon}$

for each $0\leq\alpha\leq\infty$. Then
$\mathcal{X}_{\infty,\varepsilon}\subseteq\mathcal{J}_{\varepsilon}$ is
precisely the set of points that survive under forward iteration of the skew-
product map $T$. We will assume that the fiberwise survivor sets are nonempty:

  1. (X)

For each $\varepsilon>0$ sufficiently small we have
$X_{\omega,\infty,\varepsilon}\neq\emptyset$ for $m$-a.e. $\omega\in\Omega$.

As an immediate consequence of ((X)) we have that
$\mathcal{X}_{\infty,\varepsilon}\neq\emptyset$.

######  Remark 3.5.

Note that since ((A)) implies that
$X_{\omega,\infty,\varepsilon^{\prime}}\supseteq
X_{\omega,\infty,\varepsilon}$ for all $\varepsilon^{\prime}<\varepsilon$,
((X)) holds if there exists $\varepsilon>0$ such that
$X_{\omega,\infty,\varepsilon}\neq\emptyset$ for $m$-a.e. $\omega\in\Omega$.
Furthermore, since $T_{\omega}(X_{\omega,\infty,\varepsilon})\subseteq
X_{\sigma\omega,\infty,\varepsilon}$, if
$X_{\omega,\infty,\varepsilon}\neq\emptyset$ then
$X_{\sigma^{N}\omega,\infty,\varepsilon^{\prime}}\neq\emptyset$ for each
$N\geq 1$ and $\varepsilon^{\prime}\leq\varepsilon$. As
$\mathcal{X}_{\infty,\varepsilon}$ is forward invariant we have that
$X_{\omega,\infty,\varepsilon}\neq\emptyset$ not only implies that
$\mathcal{X}_{\infty,\varepsilon}\neq\emptyset$, but also that
$\mathcal{X}_{\infty,\varepsilon}$ is infinite.

The following proposition presents a setting in which the survivor set is
nonempty for random piecewise continuous open dynamics.

######  Proposition 3.6.

Suppose that for $m$-a.e. $\omega\in\Omega$ there exist
$V_{\omega},U_{\omega,1},\dots,U_{\omega,k_{\omega}}\subseteq\mathcal{J}_{\omega,\varepsilon}$
nonempty compact subsets such that for $m$-a.e. $\omega\in\Omega$

  1. (1)

$T_{\omega}\rvert_{U_{\omega},j}$ is continuous for each $1\leq j\leq
k_{\omega}$,

  2. (2)

$T_{\omega}(U_{\omega})\supseteq V_{\sigma\omega}$, where
$U_{\omega}:=\cup_{j=1}^{k_{\omega}}U_{\omega,j}\subseteq V_{\omega}$ for each
$\omega$.

Then $X_{\omega,\infty,\varepsilon}\neq\emptyset$ for $m$-a.e.
$\omega\in\Omega$ and consequently
$\mathcal{X}_{\infty,\varepsilon}\neq\emptyset$. Furthermore, if

$m(\left\\{\omega\in\Omega:k_{\omega}>1\right\\})>0$

then for $m$-a.e. $\omega\in\Omega$ the survivor set
$X_{\omega,\infty,\varepsilon}$ is uncountable.

###### Proof.

For each $1\leq j\leq k_{\omega}$ let
$T_{\omega,j}:U_{\omega,j}\to\mathcal{J}_{\sigma\omega,0}$ denote the
continuous map $T_{\omega}\rvert_{U_{\omega,j}}$, and let
$T_{\omega,U}:U_{\omega}\to\mathcal{J}_{\sigma\omega,0}$ denote the map
$T_{\omega}\rvert_{U_{\omega}}$. Since $V_{\sigma\omega}$ is compact, for each
$1\leq j\leq k_{\omega}$ we have that $T_{\omega,j}^{-1}(V_{\sigma\omega})$ is
a nonempty compact subset of $U_{\omega,j}$. Given $N\geq 1$ let
$\gamma=\gamma_{0}\gamma_{1}\dots\gamma_{N-1}$ be an $N$-length word with
$1\leq\gamma_{j}\leq k_{\sigma^{j}\omega}$ for each $0\leq j\leq N-1$. Let
$\Gamma_{\omega,N}$ denote the finite collection of all such words of length
$N$. Then for each $N\geq 1$ and each $\gamma\in\Gamma_{\omega,N}$

$T_{\omega,\gamma}^{-N}(V_{\sigma^{N}\omega}):=T_{\omega,\gamma_{0}}^{-1}\circ\dots\circ
T_{\sigma^{N-1}\omega,\gamma_{N-1}}^{-1}(V_{\sigma^{N}\omega})\subseteq
U_{\omega,\gamma_{0}}$

is compact. Furthermore, $T_{\omega,\gamma}^{-N}(V_{\sigma^{N}\omega})$ forms
a decreasing sequence in $U_{\omega,\gamma_{0}}$. Hence, we see that

$T_{\omega,U}^{-N}(V_{\sigma^{N}\omega})=\bigcup_{\gamma\in\Gamma_{\omega,N}}T_{\omega,\gamma}^{-N}(V_{\sigma^{N}\omega})$

forms a decreasing sequence of compact subsets of $U_{\omega}$. Thus,

$X_{\omega,\infty,\varepsilon}=\bigcap_{N=0}^{\infty}T_{\omega}^{-N}(V_{\sigma^{N}\omega})\supseteq\bigcap_{N=0}^{\infty}T_{\omega,U}^{-N}(V_{\sigma^{N}\omega})\neq\emptyset$

as desired. The final claim follows from the ergodicity of $\sigma$, which
ensures that for a.e. $\omega$ there are infinitely many $j\in\mathbb{N}$ such
that $k_{\sigma^{j}\omega}>1$, and the usual bijection between a point in
$X_{\omega,\infty,\varepsilon}$ and an infinite word $\gamma$ in the fiberwise
sequence space
$\Sigma_{\omega}:=\left\\{\gamma=\gamma_{1}\gamma_{2}\dots:1\leq\gamma_{j}\leq
k_{\sigma^{j}\omega}\right\\}$. ∎

Now we define the perturbed operator
$\mathcal{L}_{\omega,\varepsilon}:\mathcal{B}_{\omega}\to\mathcal{B}_{\sigma\omega}$
by

$\displaystyle\mathcal{L}_{\omega,\varepsilon}(f):=\mathcal{L}_{\omega,0}\left(f\cdot\mathbbold{1}_{\mathcal{J}_{\omega,\varepsilon}}\right)=\sum_{y\in
T_{\omega}^{-1}(x)}f(y)\mathbbold{1}_{\mathcal{J}_{\omega,\varepsilon}}g_{\omega,0}(y)=\sum_{y\in
T_{\omega}^{-1}(x)}f(y)g_{\omega,\varepsilon}(y),\quad
f\in\mathcal{B}_{\omega},$

where for each $\omega\in\Omega$ and $\varepsilon\geq 0$ we define
$g_{\omega,\varepsilon}:=g_{\omega,0}\mathbbold{1}_{\mathcal{J}_{\omega,\varepsilon}}$,
and, similarly, for each $N\in\mathbb{N}$,

$g_{\omega,\varepsilon}^{(N)}:=\prod_{j=0}^{N-1}g_{\sigma^{j}\omega,\varepsilon}\circ
T_{\omega}^{j}.$

Note that measurability of $H_{\varepsilon}\subseteq\mathcal{J}_{0}$ and
condition ((M2)) imply that for every $\varepsilon>0$ and $f\in\mathcal{B}$
the map $(\omega,x)\mapsto(\mathcal{L}_{\varepsilon}f)_{\omega}(x)$ is also
measurable. Iterates of the perturbed operator
$\mathcal{L}_{\omega,\varepsilon}^{N}:\mathcal{B}_{\omega}\to\mathcal{B}_{\sigma^{N}\omega}$
are given by

$\displaystyle\mathcal{L}_{\omega,\varepsilon}^{N}:=\mathcal{L}_{\sigma^{N-1}\omega,\varepsilon}\circ\dots\circ\mathcal{L}_{\omega,\varepsilon},$

which, using induction, we may write as

(3.5)
$\displaystyle\mathcal{L}_{\omega,\varepsilon}^{N}(f)=\mathcal{L}_{\omega,0}^{N}\left(f\cdot\hat{X}_{\omega,N-1,\varepsilon}\right),\qquad
f\in\mathcal{B}_{\omega}.$

For every $\varepsilon\geq 0$ we let

$\tilde{}\mathcal{L}_{\omega,\varepsilon}:=\lambda_{\omega,\varepsilon}^{-1}\mathcal{L}_{\omega,\varepsilon}.$

######  Definition 3.7.

We will call a closed random dynamical system that satisfies the measurability
conditions ((M1)), ((M2)), ((CCM)), and ((B)) a random open system if
assumptions ((A)) and ((X)) are also satisfied. We let $(\Omega,\allowbreak
m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$ denote the random open system generated by the random maps
$T_{\omega}:\mathcal{J}_{\omega,0}\to\mathcal{J}_{\sigma\omega,0}$ and random
holes $H_{\omega,\varepsilon}\subseteq\mathcal{J}_{\omega,0}$.

###  3.2. Some of the Terms from Sequential Perturbation Theorem

In this short section we develop a more thorough understanding of some of the
vital terms in the general sequential perturbation theorem of Section 2 in the
setting of random open systems. We begin by calculating the quantities
$\eta_{\omega,\varepsilon}$ and $\Delta_{\omega,\varepsilon}$ from Section 2.
In particular, we have that

$\displaystyle\Delta_{\omega,\varepsilon}$
$\displaystyle:=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\varphi_{\omega,0})\right)$
$\displaystyle=\nu_{\sigma\omega,0}\left(\mathcal{L}_{\omega,0}(\varphi_{\omega,0}\cdot\mathbbold{1}_{H_{\omega,\varepsilon}})\right)$
$\displaystyle=\lambda_{\omega,0}\cdot\nu_{\omega,0}(\varphi_{\omega,0}\cdot\mathbbold{1}_{H_{\omega,\varepsilon}})$
(3.6)
$\displaystyle=\lambda_{\omega,0}\cdot\mu_{\omega,0}(H_{\omega,\varepsilon})$

and

$\displaystyle\eta_{\omega,\varepsilon}:$
$\displaystyle=\|\nu_{\sigma\omega,0}\left(\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon}\right)\|_{\mathcal{B}_{\omega}}$
$\displaystyle=\sup_{\|\psi\|_{\mathcal{B}_{\omega}}\leq
1}\nu_{\sigma\omega,0}\left(\mathcal{L}_{\omega,0}(\psi\cdot\mathbbold{1}_{H_{\omega,\varepsilon}})\right)$
$\displaystyle=\lambda_{\omega,0}\cdot\sup_{\|\psi\|_{\mathcal{B}_{\omega}}\leq
1}\nu_{\omega,0}\left(\psi\cdot\mathbbold{1}_{H_{\omega,\varepsilon}}\right)$
(3.7) $\displaystyle\leq
K_{\omega}\lambda_{\omega,0}\cdot\nu_{\omega,0}\left(H_{\omega,\varepsilon}\right),$

where the last line follows from ((B)).

For $\varepsilon>0$ and $\mu_{\omega,0}(H_{\omega,\varepsilon})>0$,
calculating $q_{\omega,\varepsilon}^{(k)}$ in this setting gives

$\displaystyle
q_{\omega,\varepsilon}^{(k)}:=\frac{\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\mathcal{L}_{\sigma^{-(k+1)}\omega,0}-\mathcal{L}_{\sigma^{-(k+1)}\omega,\varepsilon})(\varphi_{\sigma^{-(k+1)}\omega,0})\right)}{\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\varphi_{\omega,0})\right)}$
$\displaystyle=\frac{\lambda_{\sigma^{-(k+1)}\omega,0}^{k+1}\cdot\mu_{\sigma^{-(k+1)}\omega,0}\left(T_{\sigma^{-(k+1)}\omega}^{-(k+1)}(H_{\omega,\varepsilon})\cap\left(\bigcap_{j=1}^{k}T_{\sigma^{-(k+1)}\omega}^{-(k+1)+j}(H_{\sigma^{-j}\omega,\varepsilon}^{c})\right)\cap
H_{\sigma^{-(k+1)}\omega,\varepsilon}\right)}{\mu_{\omega,0}(H_{\omega,\varepsilon})}$
$\displaystyle=\frac{\lambda_{\sigma^{-(k+1)}\omega,0}^{k+1}\cdot\mu_{\sigma^{-(k+1)}\omega,0}\left(T_{\sigma^{-(k+1)}\omega}^{-(k+1)}(H_{\omega,\varepsilon})\cap\left(\bigcap_{j=1}^{k}T_{\sigma^{-(k+1)}\omega}^{-(k+1)+j}(H_{\sigma^{-j}\omega,\varepsilon}^{c})\right)\cap
H_{\sigma^{-(k+1)}\omega,\varepsilon}\right)}{\mu_{\sigma^{-(k+1)}\omega,0}\left(T_{\sigma^{-(k+1)}\omega}^{-(k+1)}(H_{\omega,\varepsilon})\right)}.$

For notational convenience we define the quantity
$\hat{q}_{\omega,\varepsilon}^{(k)}$ by

(3.8)
$\displaystyle\hat{q}_{\omega,\varepsilon}^{(k)}:=\frac{\mu_{\sigma^{-(k+1)}\omega,0}\left(T_{\sigma^{-(k+1)}\omega}^{-(k+1)}(H_{\omega,\varepsilon})\cap\left(\bigcap_{j=1}^{k}T_{\sigma^{-(k+1)}\omega}^{-(k+1)+j}(H_{\sigma^{-j}\omega,\varepsilon}^{c})\right)\cap
H_{\sigma^{-(k+1)}\omega,\varepsilon}\right)}{\mu_{\sigma^{-(k+1)}\omega,0}\left(T_{\sigma^{-(k+1)}\omega}^{-(k+1)}(H_{\omega,\varepsilon})\right)},$

and thus we have that

$\displaystyle\hat{q}_{\omega,\varepsilon}^{(k)}=\left(\lambda_{\sigma^{-(k+1)}\omega,0}^{k+1}\right)^{-1}q_{\omega,\varepsilon}^{(k)}.$

In light of (3.8), one can think of $\hat{q}_{\omega,\varepsilon}^{(k)}$ as
the conditional probability (on the fiber $\sigma^{-(k+1)}\omega$) of a point
starting in the hole $H_{\sigma^{-(k+1)}\omega,\varepsilon}$, leaving and
avoiding holes for $k$ steps, and finally landing in the hole
$H_{\omega,\varepsilon}$ after exactly $k+1$ steps conditioned on the
trajectory of the point landing in $H_{\omega,\varepsilon}$. Similarly, for
$\omega\in\Omega_{+}$, we set

$\displaystyle\hat{q}_{\omega,0}^{(k)}:=\left(\lambda_{\sigma^{-(k+1)}\omega,0}^{k+1}\right)^{-1}q_{\omega,0}^{(k)}.$

##  4\. Quenched perturbation theorem and escape rate asymptotics for random
open systems

In this section we introduce versions of the assumptions ((P1))–((P9))
tailored to random open systems. Under these assumptions we then prove a
derivative result akin to Theorem 2.2 for random open systems as well as a
similar derivative result for the escape rate.

Suppose $(\Omega,\allowbreak
m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$ is a random open system. We assume the following conditions
hold:

  1. (C1)

There exists a measurable $m$-a.e. finite function
$C_{1}:\Omega\to\mathbb{R}_{+}$ such that for $f\in\mathcal{B}_{\omega}$ we
have

$\displaystyle\sup_{\varepsilon\geq
0}\|\mathcal{L}_{\omega,\varepsilon}(f)\|_{\mathcal{B}_{\sigma\omega}}\leq
C_{1}(\omega)\|f\|_{\mathcal{B}_{\omega}}.$

  2. (C2)

For each $\varepsilon\geq 0$ there is a random measure
$\left\\{\nu_{\omega,\varepsilon}\right\\}_{\omega\in\Omega}$ supported in
$\mathcal{J}_{0}$ and measurable functions
$\lambda_{\varepsilon}:\Omega\to(0,\infty)$ with
$\log\lambda_{\omega,\varepsilon}\in L^{1}(m)$ and
$\varphi_{\varepsilon}:\mathcal{J}_{0}\to\mathbb{R}$ such that

$\displaystyle\mathcal{L}_{\omega,\varepsilon}(\varphi_{\omega,\varepsilon})=\lambda_{\omega,\varepsilon}\varphi_{\sigma\omega,\varepsilon}\quad\text{
and
}\quad\nu_{\sigma\omega,\varepsilon}(\mathcal{L}_{\omega,\varepsilon}(f))=\lambda_{\omega,\varepsilon}\nu_{\omega,\varepsilon}(f)$

for all $f\in\mathcal{B}_{\omega}$. Furthermore we assume that for $m$-a.e.
$\omega\in\Omega$

$\nu_{\omega,0}(\varphi_{\omega,\varepsilon})=1\quad\text{ and
}\quad\nu_{\omega,0}(\mathbbold{1})=1.$

  3. (C3)

There is an operator
$Q_{\omega,\varepsilon}:\mathcal{B}_{\omega}\to\mathcal{B}_{\sigma\omega}$
such that for $m$-a.e. $\omega\in\Omega$ and each $f\in\mathcal{B}_{\omega}$
we have

$\displaystyle\lambda_{\omega,\varepsilon}^{-1}\mathcal{L}_{\omega,\varepsilon}(f)=\nu_{\omega,\varepsilon}(f)\cdot\varphi_{\sigma\omega,\varepsilon}+Q_{\omega,\varepsilon}(f).$

Furthermore, for $m$-a.e. $\omega\in\Omega$ we have

$\displaystyle
Q_{\omega,\varepsilon}(\varphi_{\omega,\varepsilon})=0\quad\text{ and
}\quad\nu_{\sigma\omega,\varepsilon}(Q_{\omega,\varepsilon}(f))=0.$

  4. (C4)

For each $f\in\mathcal{B}$ there exist measurable functions
$C_{f}:\Omega\to(0,\infty)$ and $\alpha:\Omega\times\mathbb{N}\to(0,\infty)$
with $\alpha_{\omega}(N)\to 0$ as $N\to\infty$ such that for $m$-a.e.
$\omega\in\Omega$ and all $N\in\mathbb{N}$

$\displaystyle\sup_{\varepsilon\geq
0}\|Q_{\omega,\varepsilon}^{N}f_{\omega}\|_{\infty,\sigma^{N}\omega}$
$\displaystyle\leq
C_{f}(\omega)\alpha_{\omega}(N)\|f_{\omega}\|_{\mathcal{B}_{\omega}},$
$\displaystyle\sup_{\varepsilon\geq
0}\|Q_{\sigma^{-N}\omega,\varepsilon}^{N}f_{\sigma^{-N}\omega}\|_{\infty,\omega}$
$\displaystyle\leq
C_{f}(\omega)\alpha_{\omega}(N)\|f_{\sigma^{-N}\omega}\|_{\mathcal{B}_{\sigma^{-N}\omega}},$

and
$\|\varphi_{\sigma^{N}\omega,0}\|_{\infty,\sigma^{N}\omega}\alpha_{\omega}(N)\to
0$,
$\|\varphi_{\sigma^{-N}\omega,0}\|_{\infty,\sigma^{-N}\omega}\alpha_{\omega}(N)\to
0$ as $N\to\infty$.

  5. (C5)

There exists a measurable $m$-a.e. finite function $C_{2}:\Omega\to[1,\infty)$
such that

$\displaystyle\sup_{\varepsilon\geq
0}\|\varphi_{\omega,\varepsilon}\|_{\infty,\omega}\leq
C_{2}(\omega)\quad\text{ and
}\quad\|\varphi_{\omega,0}\|_{\mathcal{B}_{\omega}}\leq C_{2}(\omega).$

  6. (C6)

For $m$-a.e. $\omega\in\Omega$ we have

$\displaystyle\lim_{\varepsilon\to
0}\nu_{\omega,0}(H_{\omega,\varepsilon})=0.$

  7. (C7)

There exists a measurable $m$-a.e. finite function $C_{3}:\Omega\to[1,\infty)$
such that for all $\varepsilon>0$ sufficiently small we have

$\displaystyle\inf\varphi_{\omega,0}\geq C_{3}^{-1}(\omega)>0\qquad\text{ and
}\qquad\operatorname*{ess\inf}_{\omega}\inf\varphi_{\omega,\varepsilon}\geq
0.$

  8. (C8)

For $m$-a.e. $\omega\in\Omega_{+}$ we have that the limit
$\hat{q}_{\omega,0}^{(k)}:=\lim_{\varepsilon\to
0}\hat{q}_{\omega,\varepsilon}^{(k)}$ exists for each $k\geq 0$, where
$\hat{q}_{\omega,\varepsilon}^{(k)}$ is as in (3.8).

######  Remark 4.1.

To obtain the scaling required in ((C2)) and ((C3)), in particular to obtain
the assumption that $\nu_{\omega,0}(\varphi_{\omega,\varepsilon})=1$, suppose
that $(\Omega,\allowbreak
m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$ is a random open system satisfying the following properties:

  1. (O1)

For each $\varepsilon\geq 0$ there is a random measure
$\left\\{\nu_{\omega,\varepsilon}^{\prime}\right\\}_{\omega\in\Omega}$ with
$\nu^{\prime}_{\omega,\varepsilon}\in\mathcal{B}_{\omega}^{*}$, the dual space
of $\mathcal{B}_{\omega}$,
$\lambda^{\prime}_{\omega,\varepsilon}\in\mathbb{C}\backslash\left\\{0\right\\}$,
and $\varphi^{\prime}_{\omega,\varepsilon}\in\mathcal{B}_{\omega}$ such that

$\displaystyle\mathcal{L}_{\omega,\varepsilon}(\varphi^{\prime}_{\omega,\varepsilon})=\lambda^{\prime}_{\omega,\varepsilon}\varphi^{\prime}_{\sigma\omega,\varepsilon}\quad\text{
and
}\quad\nu_{\sigma\omega,\varepsilon}^{\prime}(\mathcal{L}_{\omega,\varepsilon}(f))=\lambda^{\prime}_{\omega,\varepsilon}\nu^{\prime}_{\omega,\varepsilon}(f)$

for all $f\in\mathcal{B}_{\omega}$.

  2. (O2)

There is an operator
$Q_{\omega,\varepsilon}^{\prime}:\mathcal{B}_{\omega}\to\mathcal{B}_{\sigma\omega}$
such that for $m$-a.e. $\omega\in\Omega$ and each $f\in\mathcal{B}_{\omega}$
we have

$\displaystyle(\lambda^{\prime}_{\omega,\varepsilon})^{-1}\mathcal{L}_{\omega,\varepsilon}(f)=\nu^{\prime}_{\omega,\varepsilon}(f)\cdot\varphi^{\prime}_{\sigma\omega,\varepsilon}+Q^{\prime}_{\omega,\varepsilon}(f).$

Furthermore, for $m$-a.e. $\omega\in\Omega$ we have

$\displaystyle
Q^{\prime}_{\omega,\varepsilon}(\varphi^{\prime}_{\omega,\varepsilon})=0\quad\text{
and
}\quad\nu_{\sigma\omega,\varepsilon}^{\prime}(Q^{\prime}_{\omega,\varepsilon}(f))=0.$

Then, for every $\varepsilon>0$ and each $f\in\mathcal{B}_{\omega}$, by
setting

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

we see that ((C2)) and ((C3)) hold, and in particular,
$\nu_{\omega,0}(\varphi_{\omega,\varepsilon})=1$. Furthermore, ((O1)) and
((O2)) together imply that
$\nu^{\prime}_{\omega,\varepsilon}(\varphi^{\prime}_{\omega,\varepsilon})=\nu_{\omega,\varepsilon}(\varphi_{\omega,\varepsilon})=1$
for $m$-a.e. $\omega\in\Omega$. Note that the measures
$\nu_{\omega,\varepsilon}$ (for $\varepsilon>0$) described in ((C2)) are not
necessarily probability measures and should not be confused with the conformal
measures for the open systems.

Now given $N\in\mathbb{N}$ and $\omega\in\Omega$, for
$\psi_{\omega}\in\mathcal{B}_{\omega}$ such that
$\nu_{\omega,0}(\psi_{\omega})=1$ we have

$\displaystyle\int_{X_{\omega,N-1,\varepsilon}}\psi_{\omega}\,d\nu_{\omega,0}=\int_{\mathcal{J}_{\omega,0}}\psi_{\omega}\cdot\prod_{j=0}^{N-1}\mathbbold{1}_{\mathcal{J}_{\sigma^{j-N}\omega,\varepsilon}}\circ
T_{\omega}^{j}\,d\nu_{\omega,0}$
$\displaystyle\qquad=\left(\lambda_{\omega,0}^{N}\right)^{-1}\int_{\mathcal{J}_{\sigma^{N}\omega,0}}\mathcal{L}_{\omega,0}^{N}\left(\psi_{\omega}\cdot\prod_{j=0}^{N-1}\mathbbold{1}_{\mathcal{J}_{\sigma^{j-N}\omega,\varepsilon}}\circ
T_{\omega}^{j}\right)\,d\nu_{\sigma^{N}\omega,0}$
$\displaystyle\qquad=\frac{\lambda_{\omega,\varepsilon}^{N}}{\lambda_{\omega,0}^{N}}\int_{\mathcal{J}_{\sigma^{N}\omega,0}}\tilde{}\mathcal{L}_{\omega,\varepsilon}^{N}\left(\psi_{\omega}\right)\,d\nu_{\sigma^{N}\omega,0}$
$\displaystyle\qquad=\frac{\lambda_{\omega,\varepsilon}^{N}}{\lambda_{\omega,0}^{N}}\int_{\mathcal{J}_{\sigma^{N}\omega,0}}\nu_{\omega,\varepsilon}(\psi_{\omega})\cdot\varphi_{\sigma^{N}\omega,\varepsilon}\,d\nu_{\sigma^{N}\omega,0}+\frac{\lambda_{\omega,\varepsilon}^{N}}{\lambda_{\omega,0}^{N}}\int_{\mathcal{J}_{\sigma^{N}\omega,0}}Q_{\omega,\varepsilon}^{N}\left(\psi_{\omega}\right)\,d\nu_{\sigma^{N}\omega,0}.$

Thus if $\psi_{\omega}=\mathbbold{1}$ we have

(4.1)
$\nu_{\omega,0}(X_{\omega,N-1,\varepsilon})=\frac{\lambda_{\omega,\varepsilon}^{N}}{\lambda_{\omega,0}^{N}}\nu_{\omega,\varepsilon}(\mathbbold{1})+\frac{\lambda_{\omega,\varepsilon}^{N}}{\lambda_{\omega,0}^{N}}\int_{\mathcal{J}_{\sigma^{N}\omega,0}}Q_{\omega,\varepsilon}^{N}\left(\mathbbold{1}\right)\,d\nu_{\sigma^{N}\omega,0},$

and if $\psi_{\omega}=\varphi_{\omega,0}$, then we have

(4.2)
$\displaystyle\mu_{\omega,0}(X_{\omega,N-1,\varepsilon})=\frac{\lambda_{\omega,\varepsilon}^{N}}{\lambda_{\omega,0}^{N}}\nu_{\omega,\varepsilon}(\varphi_{\omega,0})+\frac{\lambda_{\omega,\varepsilon}^{N}}{\lambda_{\omega,0}^{N}}\int_{\mathcal{J}_{\sigma^{N}\omega,0}}Q_{\omega,\varepsilon}^{N}\left(\varphi_{\omega,0}\right)\,d\nu_{\sigma^{N}\omega,0}.$

######  Remark 4.2.

Using (2.3) and ((C7)) we see that

$\displaystyle\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}$
$\displaystyle=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\varphi_{\omega,\varepsilon})\right)=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}(\mathbbold{1}_{H_{\omega,\varepsilon}}\varphi_{\omega,\varepsilon})\right)$
$\displaystyle=\lambda_{\omega,0}\nu_{\omega,0}(\mathbbold{1}_{H_{\omega,\varepsilon}}\varphi_{\omega,\varepsilon})\geq\lambda_{\omega,0}\nu_{\omega,0}(H_{\omega,\varepsilon})\inf\varphi_{\omega,\varepsilon}\geq
0.$

Thus, we have that $\lambda_{\omega,0}\geq\lambda_{\omega,\varepsilon}$. Note
that if
$\nu_{\omega,0}(\mathbbold{1}_{H_{\omega,\varepsilon}}\varphi_{\omega,\varepsilon})>0$
then we have that $\lambda_{\omega,0}>\lambda_{\omega,\varepsilon}$.
Consequently, if $\nu_{\omega,0}(H_{\omega,\varepsilon})=0$ then we have that
$\lambda_{\omega,0}=\lambda_{\omega,\varepsilon}$.

######  Definition 4.3.

Given a random probability measure $\zeta$ on $\mathcal{J}_{0}$, for each
$\varepsilon>0$, we define the lower and upper escape rates on the fiber
$\omega\in\Omega$ respectively by the following:

$\displaystyle\underline{R}_{\varepsilon}(\zeta_{\omega}):=-\limsup_{N\to\infty}\frac{1}{N}\log\zeta_{\omega}(X_{\omega,N,\varepsilon})\quad\text{
and
}\quad\overline{R}_{\varepsilon}(\zeta_{\omega}):=-\liminf_{N\to\infty}\frac{1}{N}\log\zeta_{\omega}(X_{\omega,N,\varepsilon}).$

If
$\underline{R}_{\varepsilon}(\zeta_{\omega})=\overline{R}_{\varepsilon}(\zeta_{\omega})$
we say the escape rate exists and denote the common value by
$R_{\varepsilon}(\zeta_{\omega})$.

As an immediate consequence of (4.1), (4.2), and assumptions ((C4)), ((C5)),
and ((C7)) for each (fixed) $\varepsilon>0$ we have that

(4.3)
$\displaystyle\lim_{N\to\infty}\frac{1}{N}\log\nu_{\omega,0}(X_{\omega,N,\varepsilon})=\lim_{N\to\infty}\frac{1}{N}\log\mu_{\omega,0}(X_{\omega,N,\varepsilon})=\lim_{N\to\infty}\frac{1}{N}\log\lambda_{\omega,\varepsilon}^{N}-\lim_{N\to\infty}\frac{1}{N}\log\lambda_{\omega,0}^{N}.$

Since $\log\lambda_{\omega,\varepsilon}\in L^{1}(m)$ for all $\varepsilon\geq
0$ by ((C2)), the following proposition follows directly from Birkhoff’s
Ergodic Theorem.

######  Proposition 4.4.

Given a random open system $(\Omega,\allowbreak
m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$ satisfying conditions ((C1))-((C7)), for $m$-a.e.
$\omega\in\Omega$ we have that

(4.4) $\displaystyle
R_{\varepsilon}(\nu_{\omega,0})=R_{\varepsilon}(\mu_{\omega,0})=\int_{\Omega}\log\lambda_{\omega,0}\,dm(\omega)-\int_{\Omega}\log\lambda_{\omega,\varepsilon}\,dm(\omega).$

We now begin to work toward an application of Theorem 2.2 to random open
systems. The following implications are immediate: ((C1))$\implies$((P1)),
((C2))$\implies$((P2)), ((C3))$\implies$((P3)), ((C5))$\implies$((P4)), and in
light of (3.6) and (3.7) we also have that ((C6))$\implies$((P5)) and
((C7))$\implies$((P6)). Thus, in order to ensure that Theorem 2.2 applies for
the random open dynamical setting we need only to check assumptions ((P7)) and
((P8)). We now prove the following lemma showing that ((P7)) and ((P8)) follow
from assumptions ((C1))-((C7)).

Recall that the set $\Omega_{+}$, defined in (3.3), is the set of all fibers
$\omega$ such that $\mu_{\omega,0}(H_{\omega,\varepsilon})>0$ for all
$\varepsilon>0$.

######  Lemma 4.5.

Given a random open system $(\Omega,\allowbreak
m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$ satisfying conditions ((C1))-((C7)), for $m$-a.e.
$\omega\in\Omega_{+}$ we have that

(4.5) $\displaystyle\lim_{\varepsilon\to
0}\nu_{\omega,\varepsilon}(\varphi_{\omega,0})=1$

and

(4.6) $\displaystyle\lim_{N\to\infty}\limsup_{\varepsilon\to
0}\Delta_{\omega,\varepsilon}^{-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(Q_{\sigma^{-N}\omega,\varepsilon}^{N}\varphi_{\sigma^{-N}\omega,0})\right)=0.$

###### Proof.

First, using (3.6), we note that if $\mu_{\omega,0}(H_{\omega,\varepsilon})>0$
then so is $\Delta_{\omega,\varepsilon}$. To prove (4.5), we note that for
fixed $N\in\mathbb{N}$ we have

(4.7) $\displaystyle\lim_{\varepsilon\to
0}\frac{\lambda_{\omega,0}^{N}}{\lambda_{\omega,\varepsilon}^{N}}\mu_{\omega,0}(X_{\omega,N-1,\varepsilon})=1$

since $\lambda_{\omega,0}^{N}/\lambda_{\omega,\varepsilon}^{N}\to 1$ (by
(2.4), (3.7), ((C5)), and ((C6))) and non-atomicity of $\nu_{\omega,0}$
((CCM)) together with ((C3)) imply that
$\mu_{\omega,0}(X_{\omega,N-1,\varepsilon})\to 1$ as $\varepsilon\to 0$. Using
(4.2) we can write

(4.8)
$\displaystyle\nu_{\omega,\varepsilon}(\varphi_{\omega,0})=\frac{\lambda_{\omega,0}^{N}}{\lambda_{\omega,\varepsilon}^{N}}\mu_{\omega,0}(X_{\omega,N-1,\varepsilon})-\nu_{\sigma^{N}\omega,0}(Q_{\omega,\varepsilon}^{N}(\varphi_{\omega,0})),$

and thus using (4.7) and ((C4)), for each $\omega\in\Omega$ and each
$N\in\mathbb{N}$ we can write

$\displaystyle\lim_{\varepsilon\to 0}\left\lvert
1-\nu_{\omega,\varepsilon}(\varphi_{\omega,0})\right\rvert$
$\displaystyle\leq\lim_{\varepsilon\to 0}\left\lvert
1-\frac{\lambda_{\omega,0}^{N}}{\lambda_{\omega,\varepsilon}^{N}}\mu_{\omega,0}(X_{\omega,N-1,\varepsilon})\right\rvert+\|Q_{\omega,\varepsilon}^{N}(\varphi_{\omega,0})\|_{\infty,\sigma^{N}\omega}$
$\displaystyle\leq
C_{\varphi_{0}}(\omega)\alpha_{\omega}(N)\|\varphi_{\omega,0}\|_{\mathcal{B}_{\omega}}.$

As this holds for each $N\in\mathbb{N}$ and as the right-hand side of the
previous equation goes to zero as $N\to\infty$, we must in fact have that

$\displaystyle\lim_{\varepsilon\to 0}\left\lvert
1-\nu_{\omega,\varepsilon}(\varphi_{\omega,0})\right\rvert=0,$

which yields the first claim.

Now, for the second claim, using (3.6), we note that ((C7)) implies

$\displaystyle\Delta_{\omega,\varepsilon}^{-1}\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(Q_{\sigma^{-N}\omega,\varepsilon}^{N}\varphi_{\sigma^{-N}\omega})\right)$
$\displaystyle=\frac{\nu_{\omega,0}\left(\mathbbold{1}_{H_{\omega,\varepsilon}}\cdot
Q_{\sigma^{-N}\omega,\varepsilon}^{N}(\varphi_{\sigma^{-N}\omega,0})\right)}{\mu_{\omega,0}(H_{\omega,\varepsilon})}$
$\displaystyle\leq\frac{\nu_{\omega,0}(H_{\omega,\varepsilon})}{\mu_{\omega,0}(H_{\omega,\varepsilon})}\|Q_{\sigma^{-N}\omega,\varepsilon}^{N}(\varphi_{\sigma^{-N}\omega,0})\|_{\infty,\omega}$
$\displaystyle\leq
C_{3}(\omega)\|Q_{\sigma^{-N}\omega,\varepsilon}^{N}(\varphi_{\sigma^{-N}\omega,0})\|_{\infty,\omega}.$

Thus, letting $\varepsilon\to 0$ first and then $N\to\infty$, the second claim
follows from ((C4)). ∎

Now recall from (2.20) that if $\mu_{\omega,0}(H_{\omega,\varepsilon})>0$ for
each $\varepsilon>0$, then $\theta_{\omega,0}$ is given by

$\displaystyle\theta_{\omega,0}:=1-\sum_{k=0}^{\infty}(\lambda_{\sigma^{-(k+1)}\omega,0}^{k+1})^{-1}q_{\omega,0}^{(k)}=1-\sum_{k=0}^{\infty}\hat{q}_{\omega,0}^{(k)}.$

In light of Lemma 4.5, we see that ((C1))-((C8)) imply ((P1))-((P9)), and thus
we have the following Theorem and first main result of this section.

######  Theorem 4.6.

Suppose that ((C1))-((C7)) hold for a random open system $(\Omega,\allowbreak
m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$. For $m$-a.e. $\omega\in\Omega$ if there is some
$\varepsilon_{0}>0$ such that $\mu_{\omega,0}(H_{\omega,\varepsilon})=0$ for
$\varepsilon\leq\varepsilon_{0}$ then

$\displaystyle\lambda_{\omega,0}=\lambda_{\omega,\varepsilon},$

or if ((C8)) holds, then

$\displaystyle\lim_{\varepsilon\to
0}\frac{\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}}{\lambda_{\omega,0}\mu_{\omega,0}(H_{\omega,\varepsilon})}=\theta_{\omega,0}.$

Furthermore, the map $\Omega_{+}\ni\omega\mapsto\theta_{\omega,0}$ is
measurable.

###### Proof.

All statements follow directly from Theorem 2.2, except measurability of
$\theta_{\omega,0}$, which follows from its construction as a limit of
measurable objects. ∎

######  Remark 4.7.

As Theorem 2.2 does not require any measurability, one could restate Theorem
4.6 to hold for sequential open systems satisfying the sequential versions of
hypotheses ((C1))-((C8)).

The following lemma will be useful for bounding $\theta_{\omega,0}$.

######  Lemma 4.8.

Suppose $m(\Omega\backslash\Omega_{+})=0$. Then for each $\varepsilon>0$ and
$m$-a.e. $\omega\in\Omega$ we have

$\displaystyle\sum_{k=0}^{\infty}(\lambda_{\sigma^{-(k+1)}\omega,0}^{k+1})^{-1}q_{\omega,\varepsilon}^{(k)}=\sum_{k=0}^{\infty}\hat{q}^{(k)}_{\omega,\varepsilon}=1.$

###### Proof.

Recall from Section 3.1 that the hole
$H_{\varepsilon}\subseteq\mathcal{J}_{0}$ is given by

$H_{\varepsilon}:=\bigcup_{\omega\in\Omega}\left\\{\omega\right\\}\times
H_{\omega,\varepsilon},$

and the skew map $T:\mathcal{J}_{0}\to\mathcal{J}_{0}$ is given by

$T(\omega,x)=(\sigma\omega,T_{\omega}(x)).$

Let $\tau_{H_{\varepsilon}}(\omega,x)$ denote the first return time of the
point $(\omega,x)\in H_{\varepsilon}$ into $H_{\varepsilon}.$ For each $k\geq
0$ let $B_{k}:=\left\\{(\omega,x)\in
H_{\varepsilon}:\tau_{H_{\varepsilon}}(\omega,x)=k+1\right\\}$ be the set of
points $(\omega,x)$ that remain outside of the hole $H_{\varepsilon}$ for
exactly $k$ iterates, and for each $\omega\in\Omega$ we set
$B_{k,\omega}:=\pi_{2}(B_{k}\cap\left\\{\omega\right\\}\times\mathcal{J}_{\omega,0})$.
Then $B_{k,\omega}$ is precisely the set

$B_{k,\omega}=\left\\{x\in H_{\omega,\varepsilon}:T_{\omega}^{k+1}(x)\in
H_{\sigma^{k+1}\omega,\varepsilon}\text{ and }T_{\omega}^{j}(x)\not\in
H_{\sigma^{j}\omega,\varepsilon}\text{ for all }1\leq j\leq k\right\\}.$

For each $k\geq 0$ we can disintegrate $\mu_{0}$ as

$\displaystyle\mu_{0}\left(B_{k}\right)$
$\displaystyle=\int_{\Omega}\mu_{\omega,0}(B_{k,\omega})\ dm(\omega),$

where the last equality follows from the $\sigma$-invariance $m$. In
particular, for fixed $\omega\in\Omega_{+}$ and each $k\geq 0$ we have

(4.9)
$\displaystyle\mu_{\sigma^{-(k+1)}\omega,0}(B_{k,\sigma^{-(k+1)}\omega})=\mu_{\sigma^{-(k+1)}\omega,0}\left(T_{\sigma^{-(k+1)}\omega}^{-(k+1)}(H_{\omega,\varepsilon})\cap\left(\bigcap_{j=1}^{k}T_{\sigma^{-(k+1)}\omega}^{-(k+1)+j}(H_{\sigma^{-j}\omega,\varepsilon}^{c})\right)\cap
H_{\sigma^{-(k+1)}\omega,\varepsilon}\right).$

Recall from (3.8) that $\hat{q}_{\omega,\varepsilon}^{(k)}$ is given by

$\displaystyle\hat{q}_{\omega,\varepsilon}^{(k)}:=\frac{\mu_{\sigma^{-(k+1)}\omega,0}\left(T_{\sigma^{-(k+1)}\omega}^{-(k+1)}(H_{\omega,\varepsilon})\cap\left(\bigcap_{j=1}^{k}T_{\sigma^{-(k+1)}\omega}^{-(k+1)+j}(H_{\sigma^{-j}\omega,\varepsilon}^{c})\right)\cap
H_{\sigma^{-(k+1)}\omega,\varepsilon}\right)}{\mu_{\sigma^{-(k+1)}\omega,0}\left(T_{\sigma^{-(k+1)}\omega}^{-(k+1)}(H_{\omega,\varepsilon})\right)},$

which is well defined since $\omega\in\Omega_{+}$ by assumption. Note that
since
$\sum_{k=0}^{\infty}\mu_{\sigma^{-(k+1)}\omega,0}(B_{k,\sigma^{-(k+1)}\omega})\leq\mu_{\omega,0}(H_{\omega,\varepsilon})$
we must have that $\sum_{k=0}^{\infty}\hat{q}^{(k)}_{\omega,\varepsilon}\leq
1$. Now, as the measure $\mu_{0}$ is $T$-invariant, the right-hand side of
(4.9) is equal to
$\hat{q}^{(k)}_{\omega,\varepsilon}\mu_{\omega,0}(H_{\omega,\varepsilon})$,
and therefore

(4.10) $\displaystyle\int_{\Omega}\hat{q}^{(k)}_{\omega,\varepsilon}\
\mu_{\omega,0}(H_{\omega,\varepsilon})\
dm(\omega)=\int_{\Omega}\mu_{\omega,0}(B_{k,\omega})\
dm(\omega)=\mu_{0}(B_{k}).$

By the Poincaré recurrence theorem we have

$\sum_{k=0}^{\infty}\mu_{0}(B_{k})=\mu_{0}(H_{\varepsilon})=\int_{\Omega}\mu_{\omega,0}(H_{\omega,\varepsilon})\,dm(\omega).$

By interchanging the sum with the integral above (possible by Tonelli’s
Theorem) and using (4.10), we have

$\int_{\Omega}\left(\sum_{k=0}^{\infty}\hat{q}^{(k)}_{\omega,\varepsilon}\
\mu_{\omega,0}(H_{\omega,\varepsilon})\right)\
dm(\omega)=\int_{\Omega}\mu_{\omega,0}(H_{\omega,\varepsilon})\ dm(\omega),$

which implies

$\int_{\Omega}\left(\mu_{\omega,0}(H_{\omega,\varepsilon})\left(\sum_{k=0}^{\infty}\hat{q}^{(k)}_{\omega,\varepsilon}-1\right)\right)dm(\omega)=0.$

Since we already have that
$\sum_{k=0}^{\infty}\hat{q}^{(k)}_{\omega,\varepsilon}\leq 1$ for $m$-a.e.
$\omega\in\Omega$, we must in fact have
$\sum_{k=0}^{\infty}\hat{q}^{(k)}_{\omega,\varepsilon}=1$, which completes the
proof. ∎

By definition, we have that $\hat{q}_{\omega,\varepsilon}^{(k)}\in[0,1]$ and
thus ((C8)) implies that $\hat{q}_{\omega,0}^{(k)}\in[0,1]$ for each $k\geq 0$
as well. In light of the previous lemma, dominated convergence implies that

(4.11) $\displaystyle\theta_{\omega,0}\in[0,1].$

###  4.1. Escape Rate Asymptotics

If we have some additional $\omega$-control on the size of the holes we obtain
the following corollary of Theorem 4.6, which provides a formula for the
escape rate asymptotics for small random holes. The scaling of the holes
(4.13) takes a similar form to the scaling we will shortly use in the next
section for our quenched extreme value theory.

######  Corollary 4.9.

Suppose that ((C1))–((C8)) hold for a random open system $(\Omega,\allowbreak
m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$ and there exists $A\in L^{1}(m)$ such that for $m$-a.e.
$\omega\in\Omega$ and all $\varepsilon>0$ sufficiently small we have

(4.12)
$\displaystyle\frac{\log\lambda_{\omega,0}-\log\lambda_{\omega,\varepsilon}}{\mu_{\omega,0}(H_{\omega,\varepsilon})}\leq
A(\omega).$

Further suppose that there is some $\kappa(\varepsilon)$ with
$\kappa(\varepsilon)\to\infty$ as $\varepsilon\to 0$, $t\in L^{\infty}(m)$
with $t_{\omega}>0$, and $\xi_{\omega,\varepsilon}\in L^{\infty}(m)$ with
$\xi_{\omega,\varepsilon}\to 0$ as $\varepsilon\to 0$ for $m$-a.e.
$\omega\in\Omega$ such that

(4.13)
$\displaystyle\mu_{\omega,0}(H_{\omega,\varepsilon})=\frac{t_{\omega}+\xi_{\omega,\varepsilon}}{\kappa(\varepsilon)}$

and $|\xi_{\omega,\varepsilon}|<Ct_{\omega}$ for all $\varepsilon>0$
sufficiently small for some $C>0$. Then for $m$-a.e. $\omega\in\Omega$ we have

$\displaystyle\lim_{\varepsilon\to
0}\frac{R_{\varepsilon}(\mu_{\omega,0})}{\mu_{\omega,0}(H_{\omega,\varepsilon})}=\frac{\int_{\Omega}\theta_{\omega,0}t_{\omega}\,dm(\omega)}{t_{\omega}}.$

In particular, if $t_{\omega}$ is constant $m$-a.e. then

$\displaystyle\lim_{\varepsilon\to
0}\frac{R_{\varepsilon}(\mu_{\omega,0})}{\mu_{\omega,0}(H_{\omega,\varepsilon})}=\int_{\Omega}\theta_{\omega,0}\,dm(\omega).$

###### Proof.

First, we note that (4.13) implies that
$\mu_{\omega,0}(H_{\omega,\varepsilon})>0$ for $m$-a.e. $\omega\in\Omega$ and
all $\varepsilon>0$ sufficiently small, i.e.
$m(\Omega\backslash\Omega_{+})=0$. Using Theorem 4.6 we have that

(4.14) $\displaystyle\lim_{\varepsilon\to
0}\frac{\log\lambda_{\omega,0}-\log\lambda_{\omega,\varepsilon}}{\mu_{\omega,0}(H_{\omega,\varepsilon})}=\theta_{\omega,0}$

since

$\displaystyle\frac{\log\lambda_{\omega,0}-\log\lambda_{\omega,\varepsilon}}{\mu_{\omega,0}(H_{\omega,\varepsilon})}$
$\displaystyle=\frac{\log\lambda_{\omega,0}-\log\lambda_{\omega,\varepsilon}}{\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}}\cdot\frac{\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}}{\mu_{\omega,0}(H_{\omega,\varepsilon})}\longrightarrow\frac{1}{\lambda_{\omega,0}}\cdot\lambda_{\omega,0}\theta_{\omega,0}=\theta_{\omega,0}$

as $\varepsilon\to 0$. In light of (4.3) and (4.4), we use (4.13) to get that

$\displaystyle\lim_{\varepsilon\to
0}\frac{R_{\varepsilon}(\mu_{\omega,0})}{\mu_{\omega,0}(H_{\omega,\varepsilon})}$
$\displaystyle=\lim_{\varepsilon\to
0}\lim_{N\to\infty}\frac{1}{N}\frac{\log\lambda_{\omega,0}^{N}-\log\lambda_{\omega,\varepsilon}^{N}}{\mu_{\omega,0}(H_{\omega,\varepsilon})}$
$\displaystyle=\lim_{\varepsilon\to
0}\lim_{N\to\infty}\frac{1}{N}\sum_{j=0}^{N-1}\frac{\log\lambda_{\sigma^{j}\omega,0}-\log\lambda_{\sigma^{j}\omega,\varepsilon}}{\mu_{\sigma^{j}\omega,0}(H_{\sigma^{j}\omega,\varepsilon})}\cdot\frac{\mu_{\sigma^{j}\omega,0}(H_{\sigma^{j}\omega,\varepsilon})}{\mu_{\omega,0}(H_{\omega,\varepsilon})}$
$\displaystyle=\lim_{\varepsilon\to
0}\lim_{N\to\infty}\frac{1}{N}\sum_{j=0}^{N-1}\frac{\log\lambda_{\sigma^{j}\omega,0}-\log\lambda_{\sigma^{j}\omega,\varepsilon}}{\mu_{\sigma^{j}\omega,0}(H_{\sigma^{j}\omega,\varepsilon})}\cdot\frac{t_{\sigma^{j}\omega}+\xi_{\sigma^{j}\omega,\varepsilon}}{t_{\omega}+\xi_{\omega,\varepsilon}}$
$\displaystyle=\lim_{\varepsilon\to
0}\frac{1}{t_{\omega}+\xi_{\omega,\varepsilon}}\lim_{N\to\infty}\frac{1}{N}\sum_{j=0}^{N-1}\frac{\log\lambda_{\sigma^{j}\omega,0}-\log\lambda_{\sigma^{j}\omega,\varepsilon}}{\mu_{\sigma^{j}\omega,0}(H_{\sigma^{j}\omega,\varepsilon})}\cdot(t_{\sigma^{j}\omega}+\xi_{\sigma^{j}\omega,\varepsilon})$
$\displaystyle=\lim_{\varepsilon\to
0}\frac{1}{t_{\omega}+\xi_{\omega,\varepsilon}}\int_{\Omega}\frac{\log\lambda_{\omega,0}-\log\lambda_{\omega,\varepsilon}}{\mu_{\omega,0}(H_{\omega,\varepsilon})}\cdot(t_{\omega}+\xi_{\omega,\varepsilon})\,dm(\omega),$

where the last line follows from Birkhoff (which is applicable thanks to
(4.12) and the fact that $t_{\omega},\xi_{\omega,\varepsilon}\in
L^{\infty}(m)$). As $\xi_{\omega,\varepsilon}\to 0$ by assumption, (4.12)
allows us to apply Dominated Convergence which, in view of (4.14), implies

$\displaystyle\lim_{\varepsilon\to
0}\frac{R_{\varepsilon}(\mu_{\omega,0})}{\mu_{\omega,0}(H_{\omega,\varepsilon})}=\frac{1}{t_{\omega}}\lim_{\varepsilon\to
0}\int_{\Omega}\frac{\log\lambda_{\omega,0}-\log\lambda_{\omega,\varepsilon}}{\mu_{\omega,0}(H_{\omega,\varepsilon})}\cdot(t_{\omega}+\xi_{\omega,\varepsilon})\,dm(\omega)=\frac{\int_{\Omega}\theta_{\omega,0}t_{\omega}\,dm(\omega)}{t_{\omega}},$

completing the proof. ∎

In the next section we will present easily checkable assumptions which will
imply the hypotheses of Corollary 4.9.

##  5\. Quenched extreme value law

###  5.1. Gumbel’s law in the quenched regime

Suppose that $h_{\omega}:\mathcal{J}_{\omega,0}\to\mathbb{R}$ is a continuous
function for each $\omega\in\Omega$. In our quenched random extreme value
theory, $h_{\omega}$ is an observation function that is allowed to depend on
$\omega$. For each $\omega\in\Omega$ let $\overline{z}_{\omega}$ be the
essential supremum of $h_{\omega}$ with respect to $\nu_{\omega,0}$, that is

$\displaystyle\overline{z}_{\omega}:=\sup\left\\{z\in\mathbb{R}\;:\nu_{\omega,0}(\left\\{x\in\mathcal{J}_{\omega,0}:h_{\omega}(x)\geq
z\right\\})>0\right\\}.$

Similarly we define $\underline{z}_{\omega}$ to be the essential infimum of
$h_{\omega}$ with respect to $\nu_{\omega,0}$. Suppose
$\underline{z}_{\omega}<\overline{z}_{\omega}$ for $m$-a.e. $\omega\in\Omega$,
and for each $z\in[\underline{z}_{\omega},\overline{z}_{\omega}]$ we define
the set

$\displaystyle
V_{\omega,z_{\omega}}:=\left\\{x\in\mathcal{J}_{\omega,0}:h_{\omega}(x)-z_{\omega}>0\right\\},$

which represents points $x$ in our phase space where the observation
$h_{\omega}$ exceeds a random threshold $z_{\omega}$ at base configuration
$\omega$. Our theory allows for random thresholds $z_{\omega}$ so that we may
consider both “anomalous” and “absolute” exceedances. For example in a real-
world application, $h_{\omega}(x)$ may represent the surface ocean temperature
at a spatial location $x$ for an ocean system configuration $\omega$. Random
temperature exceedances above $z_{\omega}$ allow one to describe extreme value
statistics for temperature anomalies, e.g. those above a climatological
seasonal mean (on average the surface ocean is warmer in summer and colder in
winter). On the other hand, non-random absolute temperature exceedances above
$z$ are more relevant for marine life. We suppose that

$\displaystyle\nu_{\omega,0}(V_{\omega,\overline{z}_{\omega}})=0.$

As is standard in extreme value theory, to develop an exponential law we will
consider an increasing sequence of thresholds
$z_{\omega,0}<z_{\omega,1}<\cdots$. For each $N\geq 0$ we take
$z_{\omega,N}\in[\underline{z}_{\omega},\overline{z}_{\omega}]$ and the choice
will be made explicit in a moment. For each $k,N\in\mathbb{N}$ define
$G_{\omega,N}^{(k)}:\mathcal{J}_{\omega,0}\to\mathbb{R}$ by

$\displaystyle
G_{\omega,N}^{(k)}(x):=h_{\sigma^{N}\omega}(T_{\omega}^{N}(x))-z_{\sigma^{N}\omega,k}.$

If $G_{\omega,N}^{(k)}>0$ then
$h_{\sigma^{N}\omega}(T_{\omega}^{N}(x))>z_{\sigma^{N}\omega,k}$. Our extreme
value law concerns the large $N$ limit of the likelihood of continued
threshold non-exceedances:

(5.1)
$\nu_{\omega,0}\left(\left\\{x\in\mathcal{J}_{\omega,0}:\max\left(G_{\omega,0}^{(N)}(x),\dots,G_{\omega,N-1}^{(N)}(x)\right)\leq
0\right\\}\right).$

We may easily transform (5.1) into the language of random open systems: one
immediately has that $G_{\omega,N}^{(k)}>0$ is equivalent to
$T_{\omega}^{N}(x)\in V_{\sigma^{N}\omega,z_{\sigma^{N}\omega,k}}$. Therefore,
for each $N\geq 0$ we have

(5.2)
$(\ref{evtexpression})=\nu_{\omega,0}\left(\left\\{x\in\mathcal{J}_{\omega,0}:T_{\omega}^{j}(x)\notin
V_{\sigma^{j}\omega,z_{\sigma^{j}\omega,N}}\text{ for
}j=0,\dots,N-1\right\\}\right)\\\
=\nu_{\omega,0}\left(X_{\omega,N-1,\varepsilon_{N}}\right),$

where to obtain the second equality we identify the sets
$V_{\omega,z_{\omega,N}}$ with holes
$H_{\omega,\varepsilon_{N}}\subset\mathcal{J}_{\omega,0}$ for each
$N\in\mathbb{N}$ 111 In Section 5 we consider a decreasing sequence of holes,
whereas in previous sections we considered a decreasing family of holes
$H_{\omega,\varepsilon}$ parameterised by $\varepsilon>0$. For the sake of
notational continuity, in this section, and in the sequel, we denote a
decreasing sequence of holes by $H_{\omega,\varepsilon_{N}}$. Note that
$\varepsilon_{N}$ here is just a parameter and should not be thought of as a
real number, but instead as an index and that $H_{\omega,\varepsilon_{N}}$
serves as an alternative notation to $H_{\omega,N}$. Furthermore, note that
the measure of the hole depends on the fiber $\omega$ with
$\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})\to 0$ as $N\to\infty$. . Now using
(4.1) and (5.2) we may convert (5.1) into the spectral expression:

(5.3)
$(\ref{evtexpression})=\frac{\lambda_{\omega,\varepsilon_{N}}^{N}}{\lambda_{\omega,0}^{N}}\left(\nu_{\omega,\varepsilon_{N}}(\mathbbold{1})+\nu_{\sigma^{N}\omega,0}\left(Q_{\omega,\varepsilon_{N}}^{N}(\mathbbold{1})\right)\right).$

Similarly, using (4.2), we can write

$\displaystyle\mu_{\omega,0}\left(\left\\{x\in\mathcal{J}_{\omega,0}:T_{\omega}^{j}(x)\notin
V_{\sigma^{j}\omega,z_{\sigma^{j}\omega,N}}\text{ for
}j=0,\dots,N-1\right\\}\right)=\mu_{\omega,0}\left(X_{\omega,N-1,\varepsilon_{N}}\right)$
(5.4)
$\displaystyle\qquad=\frac{\lambda_{\omega,\varepsilon_{N}}^{N}}{\lambda_{\omega,0}^{N}}\left(\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})+\nu_{\sigma^{N}\omega,0}\left(Q_{\omega,\varepsilon_{N}}^{N}(\varphi_{\omega,0})\right)\right).$

Before we state our main result concerning the $N\to\infty$ limit, in addition
to ((C2)), ((C3)), and ((C8)), we make the following uniform adjustments to
some of the assumptions in Section 4 as well as an assumption on the choice of
the sequence of thresholds, condition ((S)). At the end of this section, we
will compare it with the Hüsler condition, which is the usual prescription for
non stationary processes, as we anticipated in the Introduction.

  1. (S)

For any fixed random scaling function $t\in L^{\infty}(m)$ with $t>0$, we may
find sequences of functions $z_{N},\xi_{N}\in L^{\infty}(m)$ and a constant
$W<\infty$ satisfying

$\mu_{\omega,0}(\\{h_{\omega}(x)-z_{\omega,N}>0\\})=(t_{\omega}+\xi_{\omega,N})/N,\mbox{
for a.e.\ $\omega$ and each $N\geq 1$}$

where:
(i) $\lim_{N\to\infty}\xi_{\omega,N}=0$ for a.e. $\omega$ and
(ii) $|\xi_{\omega,N}|\leq W$ for a.e. $\omega$ and all $N\geq 1$.

  1. (C1’)

There exists $C_{1}\geq 1$ such that for $m$-e.a. $\omega\in\Omega$ we have

$\displaystyle C_{1}^{-1}\leq\mathcal{L}_{\omega,0}\mathbbold{1}\leq C_{1}.$

  2. (C4’)

For each $f\in\mathcal{B}$ and each $N\in\mathbb{N}$ there exists $C_{f}>0$
and $\alpha(N)>0$ (independent of $\omega$) with
$\alpha:=\sum_{N=1}^{\infty}\alpha(N)<\infty$ such that for $m$-a.e.
$\omega\in\Omega$, all $N\in\mathbb{N}$

$\displaystyle\sup_{\varepsilon\geq
0}\|Q_{\omega,\varepsilon}^{N}f_{\omega}\|_{\infty,\sigma^{N}\omega}\leq
C_{f}\alpha(N)\|f_{\omega}\|_{\mathcal{B}_{\omega}}.$

  3. (C5’)

There exists $C_{2}\geq 1$ such that

$\displaystyle\sup_{\varepsilon\geq
0}\|\varphi_{\omega,\varepsilon}\|_{\infty,\omega}\leq C_{2}\quad\text{ and
}\quad\|\varphi_{\omega,0}\|_{\mathcal{B}_{\omega}}\leq C_{2}$

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

  4. (C7’)

There exists $C_{3}\geq 1$ such that for all $\varepsilon>0$ sufficiently
small we have

$\displaystyle\operatorname*{ess\inf}_{\omega}\inf\varphi_{\omega,0}\geq
C_{3}^{-1}>0\qquad\text{ and
}\qquad\operatorname*{ess\inf}_{\omega}\inf\varphi_{\omega,\varepsilon}\geq
0.$

######  Remark 5.1.

Note that ((S)) implies that $\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})>0$
for each $N\in\mathbb{N}$ since $t_{\omega}>0$, and in particular we have
$m(\Omega\backslash\Omega_{+})=0$.

######  Remark 5.2.

Note that since
$\lambda_{\omega,0}=\nu_{\sigma\omega,0}(\mathcal{L}_{\omega,0}\mathbbold{1})$,
((C1’)) implies that

$\displaystyle C_{1}^{-1}\leq\lambda_{\omega,0}\leq C_{1}$

for $m$-a.e. $\omega$.

######  Remark 5.3.

Note that conditions ((S)), ((C5’)), and ((C7’)) together imply ((C6)), thus
Theorem 4.6 applies. Furthermore, these same conditions along with Remark 5.2
imply that there exists $A\geq 1$ such that

$\displaystyle\frac{\log\lambda_{\omega,0}-\log\lambda_{\omega,\varepsilon}}{\mu_{\omega,0}(H_{\omega,\varepsilon})}\leq
A$

for $m$-a.e. $\omega\in\Omega$ and all $\varepsilon>0$ sufficiently small, and
thus (4.12) and (4.13) hold, meaning that Corollary 4.9 applies as well.

The following lemma shows that
$\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})$ converges to $1$ uniformly
in $\omega$ under our assumptions ((C1’)), ((C4’)), ((C5’)), ((C7’)), and
((S)).

######  Lemma 5.4.

For each $N\in\mathbb{N}$ there exists $C_{\varepsilon_{N}}\geq 1$ with
$C_{\varepsilon_{N}}\to 1$ as $N\to\infty$ such that

(5.5) $\displaystyle
C_{\varepsilon_{N}}^{-1}\leq\nu_{\omega,\varepsilon_{N}}(\mathbbold{1}),\,\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})\leq
C_{\varepsilon_{N}}$

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

###### Proof.

Note that

(5.6)
$\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon_{N}})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon_{N}}^{k}\varphi_{\sigma^{-k}\omega,0})\right)=\lambda_{\omega,0}\nu_{\omega,0}\left(\mathbbold{1}_{H_{\omega,\varepsilon_{N}}}\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon_{N}}^{k}\varphi_{\sigma^{-k}\omega,0}\right).$

Further, by ((C7’)) we have

(5.7) $\nu_{\omega,0}(H_{\omega,\varepsilon_{N}})\leq
C_{3}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})=\frac{C_{3}(t_{\omega}+\xi_{\omega,N})}{N}$

for $m$-a.e. $\omega\in\Omega$. Following the same derivation of (2.3), and
using ((C2)) gives that

(5.8) $\displaystyle\lambda_{\omega,0}-\lambda_{\omega,\varepsilon_{N}}$
$\displaystyle=\nu_{\sigma\omega,0}\left((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon_{N}})(\varphi_{\omega,\varepsilon_{N}})\right)=\lambda_{\omega,0}\nu_{\omega,0}(\mathbbold{1}_{H_{\omega,\varepsilon_{N}}}\varphi_{\omega,\varepsilon_{N}}).$

Thus it follows from Remark 4.2, (5.8), ((C5’)), and (5.7) that

(5.9)
$0\leq\lambda_{\omega,0}-\lambda_{\omega,\varepsilon_{N}}\leq\frac{C_{2}C_{3}\lambda_{\omega,0}(t_{\omega}+\xi_{\omega,N})}{N}.$

Using Remark 5.2 and (5.9), for all $N$ sufficiently large, we see that

(5.10) $\displaystyle
1\leq\frac{\lambda_{\omega,0}}{\lambda_{\omega,\varepsilon_{N}}}=\frac{\lambda_{\omega,0}}{\lambda_{\omega,0}-\lambda_{\omega,0}\nu_{\omega,0}(\mathbbold{1}_{H_{\omega,\varepsilon_{N}}}\varphi_{\omega,\varepsilon_{N}})}\leq\frac{1}{1-C_{2}\nu_{\omega,0}(H_{\omega,\varepsilon_{N}})}\leq\frac{1}{1-E_{N}},$

where

$\displaystyle E_{N}:=\frac{C_{2}C_{3}(|t|_{\infty}+W)}{N}\to 0$

as $N\to\infty$.

######  Claim 5.4.1.

For every $n$, $\varepsilon_{N}$, and $m$-a.e. $\omega\in\Omega$ we have

$\displaystyle\left\lvert
1-\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})\right\rvert\leq
n\left(\frac{1}{1-E_{N}}\right)^{n}\cdot\left(C_{1}E_{N}+\frac{C_{1}C_{2}E_{N}}{1-E_{N}}\right)+C_{2}C_{\varphi_{0}}\alpha(n).$

###### Proof.

Using ((C2)) and a telescoping argument, we can write

$\displaystyle\left\lvert
1-\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})\right\rvert$
$\displaystyle=\left\lvert\nu_{\sigma^{n}\omega,0}(\varphi_{\sigma^{n}\omega,0})-\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})\right\rvert$
$\displaystyle=\left\lvert\nu_{\sigma^{n}\omega,0}(\varphi_{\sigma^{n}\omega,0})-\nu_{\sigma^{n}\omega,0}\big{(}\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})\cdot\varphi_{\sigma^{n}\omega,\varepsilon_{N}}\big{)}\right\rvert$
$\displaystyle=\left\lvert\nu_{\sigma^{n}\omega,0}\left(\varphi_{\sigma^{n}\omega,0}-\tilde{}\mathcal{L}_{\omega,\varepsilon_{N}}^{n}(\varphi_{\omega,0})+Q_{\omega,\varepsilon_{N}}^{n}(\varphi_{\omega,0})\right)\right\rvert$
$\displaystyle\leq\sum_{k=0}^{n-1}\left\lvert\nu_{\sigma^{n}\omega,0}\left(\tilde{}\mathcal{L}_{\sigma^{n-k}\omega,\varepsilon_{N}}^{k}(\varphi_{\sigma^{n-k}\omega,0})-\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}^{k+1}(\varphi_{\sigma^{n-(k+1)}\omega,0})\right)\right\rvert$
$\displaystyle\qquad+\left\lvert\nu_{\sigma^{n}\omega,0}\left(Q_{\omega,\varepsilon_{N}}^{n}(\varphi_{\omega,0})\right)\right\rvert$
(5.11)
$\displaystyle=\sum_{k=0}^{n-1}\left\lvert\nu_{\sigma^{n}\omega,0}\left(\left(\tilde{}\mathcal{L}_{\sigma^{n-k}\omega,\varepsilon_{N}}^{k}\right)\left(\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}-\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\right)(\varphi_{\sigma^{n-(k+1)}\omega,0})\right)\right\rvert$
(5.12)
$\displaystyle\qquad+\left\lvert\nu_{\sigma^{n}\omega,0}\left(Q_{\omega,\varepsilon_{N}}^{n}(\varphi_{\omega,0})\right)\right\rvert.$

First we note that we can estimate (5.12) as

(5.13)
$\displaystyle\left\lvert\nu_{\sigma^{n}\omega,0}\left(Q_{\omega,\varepsilon_{N}}^{n}(\varphi_{\omega,0})\right)\right\rvert\leq\|Q_{\omega,\varepsilon_{N}}^{n}(\varphi_{\omega,0})\|_{\infty,\sigma^{n}\omega}\leq
C_{2}C_{\varphi_{0}}\alpha(n).$

Now recall that

$\displaystyle\hat{X}_{\sigma^{-k}\omega,k,\varepsilon_{N}}:=\mathbbold{1}_{\bigcap_{j=0}^{k}T_{\sigma^{-k}\omega}^{-j}(\mathcal{J}_{\sigma^{j-k}\omega,\varepsilon_{N}})}.$

Using ((C2)), we can write (5.11) as

$\displaystyle\sum_{k=0}^{n-1}\left\lvert\nu_{\sigma^{n}\omega,0}\left(\left(\tilde{}\mathcal{L}_{\sigma^{n-k}\omega,\varepsilon_{N}}^{k}\right)\left(\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}-\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\right)(\varphi_{\sigma^{n-(k+1)}\omega,0})\right)\right\rvert$
$\displaystyle\quad=\sum_{k=0}^{n-1}\left\lvert\nu_{\sigma^{n}\omega,0}\left(\left(\lambda_{\sigma^{n-k}\omega,\varepsilon_{N}}^{k}\right)^{-1}\mathcal{L}_{\sigma^{n-k}\omega,0}^{k}\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\cdot\left(\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}-\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\right)(\varphi_{\sigma^{n-(k+1)}\omega,0})\right)\right)\right\rvert$
(5.14)
$\displaystyle\quad=\sum_{k=0}^{n-1}\frac{\lambda_{\sigma^{n-k}\omega,0}^{k}}{\lambda_{\sigma^{n-k}\omega,\varepsilon_{N}}^{k}}\left\lvert\nu_{\sigma^{n-k}\omega,0}\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\cdot\left(\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}-\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\right)(\varphi_{\sigma^{n-(k+1)}\omega,0})\right)\right\rvert.$

Now, since
$\nicefrac{{\lambda_{\omega,0}}}{{\lambda_{\omega,\varepsilon_{N}}}}\geq 1$
(by Remark 4.2) for $m$ a.e. $\omega$ and $\varepsilon\geq 0$, and thus
$\nicefrac{{\lambda_{\sigma^{n-k}\omega,0}^{k}}}{{\lambda_{\sigma^{n-k}\omega,\varepsilon_{N}}^{k}}}\geq
1$ for each $k\geq 1$, we can write

$\displaystyle\eqref{unifeq1.0}$
$\displaystyle\leq\frac{\lambda_{\omega,0}^{n}}{\lambda_{\omega,\varepsilon_{N}}^{n}}\sum_{k=0}^{n-1}\left\lvert\nu_{\sigma^{n-k}\omega,0}\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\cdot\left(\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}-\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\right)(\varphi_{\sigma^{n-(k+1)}\omega,0})\right)\right\rvert$
(5.15)
$\displaystyle\leq\left(\frac{1}{1-E_{N}}\right)^{n}\cdot\sum_{k=0}^{n-1}\left\lvert\nu_{\sigma^{n-k}\omega,0}\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\cdot\left(\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}-\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\right)(\varphi_{\sigma^{n-(k+1)}\omega,0})\right)\right\rvert,$

Using

$\displaystyle\tilde{}\mathcal{L}_{\omega,0}-\tilde{}\mathcal{L}_{\omega,\varepsilon}$
$\displaystyle=\tilde{}\mathcal{L}_{\omega,0}-\lambda_{\omega,0}^{-1}\mathcal{L}_{\omega,\varepsilon}+\lambda_{\omega,0}^{-1}\mathcal{L}_{\omega,\varepsilon}-\tilde{}\mathcal{L}_{\omega,\varepsilon}$
$\displaystyle=\lambda_{\omega,0}^{-1}\left(\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon}\right)+(\lambda_{\omega,0}^{-1}-\lambda_{\omega,\varepsilon}^{-1})\mathcal{L}_{\omega,\varepsilon}$
$\displaystyle=\lambda_{\omega,0}^{-1}\left(\left(\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon}\right)+\lambda_{\omega,0}\lambda_{\omega,\varepsilon}(\lambda_{\omega,0}^{-1}-\lambda_{\omega,\varepsilon}^{-1})\tilde{}\mathcal{L}_{\omega,\varepsilon}\right)$
$\displaystyle=\lambda_{\omega,0}^{-1}\left(\left(\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon}\right)+(\lambda_{\omega,\varepsilon}-\lambda_{\omega,0})\tilde{}\mathcal{L}_{\omega,\varepsilon}\right),$

the fact that

$\displaystyle\mathcal{L}_{\omega,\varepsilon_{N}}\left((f_{\sigma\omega}\circ
T_{\omega})\cdot
h_{\omega}\right)=f_{\sigma\omega}\cdot\mathcal{L}_{\omega,\varepsilon_{N}}(h_{\omega}),$

for all $\varepsilon_{N}\geq 0$ and $\omega\in\Omega$, and Remark 5.2, we may
estimate the sum in (5.15) by

$\displaystyle\sum_{k=0}^{n-1}\left\lvert\nu_{\sigma^{n-k}\omega,0}\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\cdot\left(\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}-\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\right)(\varphi_{\sigma^{n-(k+1)}\omega,0})\right)\right\rvert$
$\displaystyle\quad\leq\sum_{k=0}^{n-1}\lambda_{\sigma^{n-(k+1)}\omega,0}^{-1}\left\lvert\nu_{\sigma^{n-k}\omega,0}\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\cdot\left(\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}-\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\right)(\varphi_{\sigma^{n-(k+1)}\omega,0})\right)\right\rvert$
$\displaystyle\qquad+\sum_{k=0}^{n-1}\lambda_{\sigma^{n-(k+1)}\omega,0}^{-1}\left\lvert(\lambda_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}-\lambda_{\sigma^{n-(k+1)}\omega,0})\cdot\nu_{\sigma^{n-k}\omega,0}\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\cdot\tilde{}\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}(\varphi_{\sigma^{n-(k+1)}\omega,0})\right)\right\rvert$
(5.16)
$\displaystyle\quad=\sum_{k=0}^{n-1}\lambda_{\sigma^{n-(k+1)}\omega,0}^{-1}\left\lvert\nu_{\sigma^{n-k}\omega,0}\left(\left(\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}-\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\right)\left(\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\circ
T_{\sigma^{n-(k+1)}\omega}\right)\cdot\varphi_{\sigma^{n-(k+1)}\omega,0}\right)\right)\right\rvert$
$\displaystyle\qquad+\sum_{k=0}^{n-1}\bigg{(}\lambda_{\sigma^{n-(k+1)}\omega,0}^{-1}\left\lvert
1-\frac{\lambda_{\sigma^{n-(k+1)}\omega,0}}{\lambda_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}}\right\rvert$
(5.17)
$\displaystyle\quad\qquad\cdot\left\lvert\nu_{\sigma^{n-k}\omega,0}\left(\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\left(\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\circ
T_{\sigma^{n-(k+1)}\omega}\right)\cdot\varphi_{\sigma^{n-(k+1)}\omega,0}\right)\right)\right\rvert\bigg{)}.$

Using (5.8), (5.9), and Remark 5.2 we can estimate (5.16) to get

$\displaystyle\sum_{k=0}^{n-1}\lambda_{\sigma^{n-(k+1)}\omega,0}^{-1}\left\lvert\nu_{\sigma^{n-k}\omega,0}\left(\left(\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}-\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\right)\left(\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\circ
T_{\sigma^{n-(k+1)}\omega}\right)\cdot\varphi_{\sigma^{n-(k+1)}\omega,0}\right)\right)\right\rvert$
$\displaystyle\qquad=\sum_{k=0}^{n-1}\left\lvert\nu_{\sigma^{n-(k+1)}\omega,0}\left(\mathbbold{1}_{H_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}}\cdot\left(\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\circ
T_{\sigma^{n-(k+1)}\omega}\right)\cdot\varphi_{\sigma^{n-(k+1)}\omega,0}\right)\right)\right\rvert$
(5.18)
$\displaystyle\qquad\leq\sum_{k=0}^{n-1}\nu_{\sigma^{n-(k+1)}\omega,0}\left(\mathbbold{1}_{H_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}}\right)\|\varphi_{\sigma^{n-(k+1)}\omega,0}\|_{\infty,\sigma^{n-(k+1)}\omega}\leq\frac{C_{2}C_{3}(t_{\omega}+\xi_{\omega,N})\cdot
n}{N}\leq nE_{N}.$

Since
$\mathcal{L}_{\omega,\varepsilon}(f)=\mathcal{L}_{\omega,0}(\hat{X}_{\omega,0,\varepsilon}f)$
we can rewrite the second product in the sum in (5.17) so that we have

$\displaystyle\nu_{\sigma^{n-k}\omega,0}\left(\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\left(\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\circ
T_{\sigma^{n-(k+1)}\omega}\right)\cdot\varphi_{\sigma^{n-(k+1)}\omega,0}\right)\right)$
$\displaystyle\qquad=\nu_{\sigma^{n-k}\omega,0}\left(\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}\left(\hat{X}_{\sigma^{n-(k+1)}\omega,0,\varepsilon_{N}}\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\circ
T_{\sigma^{n-(k+1)}\omega}\right)\cdot\varphi_{\sigma^{n-(k+1)}\omega,0}\right)\right)$
(5.19)
$\displaystyle\qquad\leq\|\mathcal{L}_{\sigma^{n-(k+1)}\omega,0}\mathbbold{1}\|_{\infty,\sigma^{n-k}\omega}\|\varphi_{\sigma^{n-(k+1)}\omega,0}\|_{\infty,\sigma^{n-(k+1)}\omega}\leq
C_{1}C_{2}.$

Inserting (5.19) into (5.17) and using (5.10) yields

$\displaystyle\sum_{k=0}^{n-1}\left\lvert
1-\frac{\lambda_{\sigma^{n-(k+1)}\omega,0}}{\lambda_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}}\right\rvert\cdot\left\lvert\nu_{\sigma^{n-k}\omega,0}\left(\mathcal{L}_{\sigma^{n-(k+1)}\omega,\varepsilon_{N}}\left(\left(\hat{X}_{\sigma^{n-k}\omega,k,\varepsilon_{N}}\circ
T_{\sigma^{n-(k+1)}\omega}\right)\cdot\varphi_{\sigma^{n-(k+1)}\omega,0}\right)\right)\right\rvert$
(5.20)
$\displaystyle\qquad\leq\sum_{k=0}^{n-1}\frac{C_{1}C_{2}E_{N}}{1-E_{N}}=n\cdot\frac{C_{1}C_{2}E_{N}}{1-E_{N}}.$

Thus, collecting the estimates (5.16)-(5.20) together with (5.13) and
inserting into (5.15) yields

(5.21) $\displaystyle\left\lvert
1-\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})\right\rvert\leq\left(\frac{1}{1-E_{N}}\right)^{n}\cdot\left(nE_{N}+\frac{C_{1}C_{2}E_{N}\cdot
n}{1-E_{N}}\right)+C_{2}C_{\varphi_{0}}\alpha(n),$

which finishes the proof of the claim. ∎

To finish the proof of Lemma 5.4, we note that (5.21) holds for $m$-a.e.
$\omega$, every $N$ sufficiently large, and each $n\geq 1$. Given a
$\delta>0$, choose and fix $n$ so that
$C_{2}C_{\varphi_{0}}\alpha(n)<\delta/2$. Because $\lim_{N\to\infty}E_{N}=0$,
we may choose $N$ large enough so that first summand in (5.21) is also smaller
than $\delta/2$. Thus,
$\lim_{N\to\infty}\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})=1$,
uniformly in $\omega$. This proves (5.5) for
$\nu_{\omega,\varepsilon_{N}}(\varphi_{\omega,0})$; we immediately obtain the
other inequality using ((C5’)) and ((C7’)), and thus the proof of Lemma 5.4 is
complete.

∎

We now obtain a formula for the explicit form of Gumbel law for the extreme
value distribution.

######  Theorem 5.5.

Given a random open system $(\Omega,\allowbreak
m,\allowbreak\sigma,\allowbreak\mathcal{J}_{0},\allowbreak
T,\allowbreak\mathcal{B},\allowbreak\mathcal{L}_{0},\allowbreak\nu_{0},\allowbreak\varphi_{0},\allowbreak
H_{\varepsilon})$ satisfying ((C1’)), ((C2)), ((C3)), ((C4’)), ((C5’)),
((C7’)), ((C8)), and ((S)), for almost every $\omega\in\Omega$ one has

(5.22)
$\lim_{N\to\infty}\nu_{\omega,0}\left(X_{\omega,N-1,\varepsilon_{N}}\right)=\lim_{N\to\infty}\mu_{\omega,0}\left(X_{\omega,N-1,\varepsilon_{N}}\right)=\lim_{N\to\infty}\frac{\lambda_{\omega,\varepsilon_{N}}^{N}}{\lambda_{\omega,0}^{N}}=\exp\left(-\int_{\Omega}t_{\omega}\theta_{\omega,0}\,dm(\omega)\right).$

###### Proof.

Step 1: Estimating $\lambda_{\omega,\varepsilon_{N}}/\lambda_{\omega,0}$. To
work towards constructing an estimate for
$\lambda^{N}_{\omega,\varepsilon_{N}}/\lambda^{N}_{\omega,0}$, we first
estimate $\lambda_{\omega,\varepsilon_{N}}/\lambda_{\omega,0}$. For brevity,
in this step we drop the $N$ subscripts on $\varepsilon$. Following the proof
of Theorem 2.2 up to equation (2.15), which uses assumptions ((C2)) and
((C3)), we have

$\displaystyle\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})\frac{\lambda_{\omega,0}-\lambda_{\omega,\varepsilon}}{\Delta_{\omega,\varepsilon}}$
(5.23) $\displaystyle=$
$\displaystyle\underbrace{1-\sum_{k=0}^{n-1}\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}(\lambda^{k}_{\sigma^{-k}\omega,\varepsilon})^{-1}q^{(k)}_{\omega,\varepsilon}}_{=:\theta_{\omega,\varepsilon,n}}$
(5.24) $\displaystyle+$
$\displaystyle\Delta^{-1}_{\omega,\varepsilon}\underbrace{\sum_{k=1}^{n}\lambda^{-1}_{\sigma^{-k}\omega,0}(\lambda_{\sigma^{-k}\omega,0}-\lambda_{\sigma^{-k}\omega,\varepsilon})\nu_{\sigma\omega,0}((\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(\tilde{}\mathcal{L}_{\sigma^{-k}\omega,\varepsilon}^{k})(\varphi_{\sigma^{-k}\omega,0}))}_{=:\theta^{\prime}_{\omega,\varepsilon,n}}$
(5.25) $\displaystyle+$
$\displaystyle\Delta^{-1}_{\omega,\varepsilon}\underbrace{\nu_{\sigma\omega,0}(\mathcal{L}_{\omega,0}-\mathcal{L}_{\omega,\varepsilon})(Q^{n}_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0}))}_{=:\theta^{\prime\prime}_{\omega,\varepsilon,n}}.$

By first rearranging to solve for $\lambda_{\omega,\varepsilon}$ we have

$\displaystyle\lambda_{\omega,\varepsilon}$ $\displaystyle=$
$\displaystyle\lambda_{\omega,0}-\frac{\theta_{\omega,\varepsilon,n}\Delta_{\omega,\varepsilon}+\theta^{\prime}_{\omega,\varepsilon,n}+\theta^{\prime\prime}_{\omega,\varepsilon,n}}{\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})},$

and thus

(5.26)
$\displaystyle\frac{\lambda_{\omega,\varepsilon}}{\lambda_{\omega,0}}=1-\underbrace{\frac{\theta_{\omega,\varepsilon,n}\Delta_{\omega,\varepsilon}}{\lambda_{\omega,0}\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})}}_{=:Y^{(1)}_{\omega,\varepsilon,n}}-\underbrace{\frac{\theta^{\prime}_{\omega,\varepsilon,n}}{\lambda_{\omega,0}\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})}}_{Y^{(2)}_{\omega,\varepsilon,n}}-\underbrace{\frac{\theta^{\prime\prime}_{\omega,\varepsilon,n}}{\lambda_{\omega,0}\nu_{\sigma^{-n}\omega,\varepsilon}(\varphi_{\sigma^{-n}\omega,0})}}_{=:Y^{(3)}_{\omega,\varepsilon,n}}.$

Setting
$Y_{\omega,\varepsilon,n}:=Y^{(1)}_{\omega,\varepsilon,n}+Y^{(2)}_{\omega,\varepsilon,n}+Y^{(3)}_{\omega,\varepsilon,n}$
applying Taylor to $\log(1-\cdot)$, we obtain

(5.27)
$\frac{\lambda_{\omega,\varepsilon}}{\lambda_{\omega,0}}=\exp\left(-Y_{\omega,\varepsilon,n}-\frac{Y_{\omega,\varepsilon,n}^{2}}{2(1-y)^{2}}\right),$

where $0\leq y\leq Y_{\omega,\varepsilon,n}$. Setting
$\varepsilon=\varepsilon_{N}$ in (5.27) we obtain

(5.28)
$\displaystyle\frac{\lambda^{N}_{\omega,\varepsilon_{N}}}{\lambda_{\omega,0}^{N}}$
$\displaystyle=$
$\displaystyle\exp\left(-\sum_{i=0}^{N-1}Y_{\sigma^{i}\omega,\varepsilon_{N},n}-\sum_{i=0}^{N-1}\frac{Y_{\sigma^{i}\omega,\varepsilon_{N},n}^{2}}{2(1-y)^{2}}\right)$
(5.29) $\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{N}\sum_{i=0}^{N-1}\left(g^{(1)}_{N,n}(\sigma^{i}\omega)+g^{(2)}_{N,n}(\sigma^{i}\omega)+g^{(3)}_{N,n}(\sigma^{i}\omega)\right)-\sum_{i=0}^{N-1}\frac{Y_{\sigma^{i}\omega,\varepsilon_{N},n}^{2}}{2(1-y)^{2}}\right),$

where $g^{(j)}_{N,n}(\omega)=NY^{(j)}_{\omega,\varepsilon_{N},n}$ for
$j=1,2,3$ and $0\leq y\leq Y_{\omega,\varepsilon_{N},n}$.

Step 2: A non-standard ergodic lemma. In preparation for estimating the
products along orbits contained in
$\lambda^{N}_{\omega,\varepsilon_{N}}/\lambda^{N}_{\omega,0}$, we state and
prove a non-standard ergodic lemma.

######  Lemma 5.6.

For $N\geq 0$, let $g_{N}\in L^{1}(m)$. Suppose that as $N\to\infty$,
$g_{N}\to g$ $m$-almost everywhere for some $g\in L^{1}(m)$ and that
$\lim_{N\to\infty}\int_{\Omega}|g_{N}-g|\,dm=0$. Then for $m$-a.e.
$\omega\in\Omega$,
$\lim_{N\to\infty}\frac{1}{N}\sum_{i=0}^{N-1}g_{N}(\sigma^{i}\omega)$ exists
and equals $\mathbb{E}(g)=\int_{\Omega}g\ dm$.

###### Proof.

We write

$\left|\frac{1}{N}\sum_{i=0}^{N-1}g_{N}(\sigma^{i}\omega)-\mathbb{E}(g)\right|\leq\frac{1}{N}\sum_{i=0}^{N-1}|g_{N}(\sigma^{i}\omega)-g(\sigma^{i}\omega)|+\left|\frac{1}{N}\sum_{i=0}^{N-1}g(\sigma^{i}\omega)-\mathbb{E}(g)\right|.$

First, we note that the Birkhoff Ergodic Theorem implies that

$\displaystyle\lim_{N\to\infty}\left|\frac{1}{N}\sum_{i=0}^{N-1}g(\sigma^{i}\omega)-\mathbb{E}(g)\right|=0.$

Now, to deal with the remaining term, since $g_{N}\to g$ almost everywhere,
given $\delta>0$, we let $N_{\delta}\in\mathbb{N}$ be sufficiently large such
that

$\displaystyle
m\left(\Omega_{\delta}:=\left\\{\omega\in\Omega:|g(\omega)-g_{N}(\omega)|<\delta\text{
for all }N\geq N_{\delta}\right\\}\right)>1-\delta.$

Birkhoff applied to $\mathbbold{1}_{\Omega_{\delta}}$ then gives that for each
$\delta>0$ and all $N\geq N_{\delta}$ sufficiently large we have that

(5.30) $\displaystyle\frac{1}{N}\\#\left\\{0\leq
k<N:\sigma^{k}\omega\in\Omega_{\delta}\right\\}>1-\delta.$

Since $\lim_{N\to\infty}\int_{\Omega}|g_{N}-g|\,dm(\omega)=0$, for each
$\delta>0$ there exists $N_{\delta}^{\prime}>0$ such that
$\int_{\Omega}|g_{N}-g|\,dm<\delta$ for all $N\geq N_{\delta}^{\prime}$. We
apply the Birkhoff Ergodic Theorem to the functions
$|g_{N}-g|\cdot\mathbbold{1}_{\Omega_{\delta}}$ and
$|g_{N}-g|\cdot\mathbbold{1}_{\Omega_{\delta}^{c}}$. Using (5.30) gives that
for $m$-a.e. $\omega\in\Omega$ and all
$N\geq\max(N_{\delta},N_{\delta}^{\prime})$ sufficiently large (so that the
Birkhoff error is less than $\delta$, noting that $N$ depends on $\omega$) we
have

$\displaystyle\frac{1}{N}\sum_{i=0}^{N-1}|g_{N}(\sigma^{i}\omega)-g(\sigma^{i}\omega)|$
$\displaystyle\qquad=\frac{1}{N}\sum_{i=0}^{N-1}|g_{N}(\sigma^{i}\omega)-g(\sigma^{i}\omega)|\mathbbold{1}_{\Omega_{\delta}}(\sigma^{i}\omega)+\frac{1}{N}\sum_{i=0}^{N-1}|g_{N}(\sigma^{i}\omega)-g(\sigma^{i}\omega)|\mathbbold{1}_{\Omega_{\delta}^{c}}(\sigma^{i}\omega)$
$\displaystyle\qquad<\delta+\int_{\Omega_{\delta}^{c}}|g_{N}-g|\,dm+\delta\leq
2\delta+\int_{\Omega}|g_{N}-g|\,dm<3\delta.$

As this holds for every $\delta>0$, we must in fact have that

$\displaystyle\lim_{N\to\infty}\left|\frac{1}{N}\sum_{i=0}^{N-1}g_{N}(\sigma^{i}\omega)-\mathbb{E}(g)\right|=0$

as desired. ∎

Step 3: Estimating $g^{(1)}$.

In this step we construct estimates of $g^{(1)}$ that are required to apply
Lemma 5.6.

In preparation for the first use of Lemma 5.6 we recall that

$g_{N,n}^{(1)}(\omega):=\frac{N\theta_{\omega,\varepsilon_{N},n}\Delta_{\omega,\varepsilon_{N}}}{\lambda_{\omega,0}\nu_{\sigma^{-n}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-n}\omega,0})}\\\
=\frac{N\theta_{\omega,\varepsilon_{N},n}\mu_{\omega,0}(H_{\omega,\varepsilon_{N}})}{\nu_{\sigma^{-n}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-n}\omega,0})}=\frac{\theta_{\omega,\varepsilon_{N},n}(t_{\omega}+\xi_{\omega,N})}{\nu_{\sigma^{-n}\omega,\varepsilon_{N}}(\varphi_{\sigma^{-n}\omega,0})},$

where $\lim_{N\to\infty}\xi_{\omega,N}=0$ for a.e. $\omega$ and
$|\xi_{\omega,N}|\leq W$ by ((S)). We also set
$g_{n}^{(1)}(\omega):=\theta_{\omega,0,n}t_{\omega}$, where

(5.31)
$\displaystyle\theta_{\omega,0,n}:=1-\sum_{k=0}^{n-1}\left(\lambda_{\sigma^{-(k+1)}\omega,0}^{k+1}\right)^{-1}q_{\omega,0}^{(k)}=1-\sum_{k=0}^{n-1}\hat{q}_{\omega,0}^{(k)}.$

By Lemma 4.8 and ((C8)) we see that
$0\leq\hat{q}_{\omega,0}^{(k)},\hat{q}_{\omega,\varepsilon_{N}}^{(k)}\leq 1$
for each $k$, $N$, and $m$-a.e. $\omega$. Thus, ((C8)) and the Dominated
Convergence Theorem imply that
$\lim_{N\to\infty}\|\hat{q}_{\omega,\varepsilon_{N}}^{(k)}-\hat{q}_{\omega,0}^{(k)}\|_{1}=0$.
From (5.23) we have that

(5.32)
$\displaystyle\theta_{\omega,\varepsilon_{N},n}:=1-\sum_{k=0}^{n-1}\lambda_{\sigma^{-(k+1)}\omega,0}^{-1}(\lambda^{k}_{\sigma^{-k}\omega,\varepsilon_{N}})^{-1}q_{\omega,\varepsilon_{N}}^{(k)}=1-\sum_{k=0}^{n-1}\frac{\lambda_{\sigma^{-k}\omega,0}^{k}}{\lambda_{\sigma^{-k}\omega,\varepsilon_{N}}^{k}}\hat{q}_{\omega,\varepsilon_{N}}^{(k)},$

and thus that $\theta_{\omega,0,N}\in[0,1]$ and
$\theta_{\omega,\varepsilon_{N},n}\leq 1$ for each $n$ and $m$-a.e. $\omega$.
Using (5.10) and the fact that
$\nicefrac{{\lambda_{\omega,0}}}{{\lambda_{\omega,\varepsilon_{N}}}}\geq 1$
(by Remark 4.2), we have that

$\displaystyle\theta_{\omega,\varepsilon_{N},n}\geq
1-\frac{\lambda_{\sigma^{-(n-1)}\omega,0}^{n-1}}{\lambda_{\sigma^{-(n-1)}\omega,\varepsilon_{N}}^{n-1}}\sum_{k=0}^{n-1}\hat{q}_{\omega,\varepsilon_{N}}^{(k)}\geq
1-\left(1+\frac{C_{1}E_{N}}{C_{1}-E_{N}}\right)^{n}.$

For fixed $n$, again we apply Dominated Convergence to get that

(5.33)
$\displaystyle\lim_{N\to\infty}\|\theta_{\omega,\varepsilon_{N},n}-\theta_{\omega,0,n}\|_{1}=0.$