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\leftheadtext{Quenched and annealed states
for random expanding maps}
\rightheadtext{Quenched and annealed states
for random expanding maps}
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\topmatter
%
\title \nofrills
Correlation spectrum of\\
quenched and annealed equilibrium states\\
for random expanding maps
\endtitle
\author Viviane Baladi
\endauthor
\address
Section de Math\'ematiques, Universit\'e de Gen\`eve,
CH-1211 Geneva 24, Switzerland \newline
\phantom{vb}
(on leave from CNRS, UMR 128, ENS Lyon, France)
\endaddress
\email
baladi\@sc2a.unige.ch \endemail
\date{February 1996}
\enddate
\abstract
We show that the integrated transfer operators
for positively weighted independent identically distributed
smooth expanding systems give rise
to annealed equilibrium states for a new variational principle.
The unique annealed equilibrium state
coincides with the unique annealed Gibbs state.
Using work of Ruelle [1990] and Fried [1995]
on generalised Fredholm determinants for transfer operators, we prove
that the discrete spectrum of the transfer operators coincides
with the correlation spectrum of
these invariant measures (yielding exponential decay of correlations),
and with the poles of an annealed zeta function,
defined also for complex weights.
A modified integrated transfer operator is introduced, which describes
the (relativised) quenched states studied e.g. by Kifer [1992],
and conditions (including SRB) ensuring coincidence of quenched
and annealed states are given.
For small random perturbations we obtain stability results on the
quenched and annealed measures and spectra
by applying perturbative results of Young and the
author [1993].
\endabstract
\subjclass
58F11; 58F15 82B44 58F30 58F20
\endsubjclass
\endtopmatter
\document
\medskip
\head 1. Introduction
\endhead
\smallskip
The study of equilibrium states for a single
map $f:M \to M$ and a positive weight function $g$
on $M$, i.e., the analysis of $f$-invariant Borel probability
measures $\mu$ on $M$ which maximise the expression
$$
h_f(\nu) + \int \log g (x)\, \nu(dx)\tag{1.1}
$$
(with $h_f(\nu)$ the entropy of $(f,\nu)$)
is now a well-developed subject in a variety
of settings (see e.g. Ruelle [1989] and references therein).
One of the main tools for this is a transfer operator
acting on a suitable Banach space of test functions
$\varphi : M \to \complex$ by
$$
\LL \varphi(x) = \sum_{fy=x} \varphi(y) g(y) \, .\tag{1.2}
$$
In many cases one constructs
the equilibrium state $\mu$
by combining maximal eigenfunctions of $\LL$
and its dual, and one obtains exponential decay
of the corresponding correlation functions
$$
C_{\varphi_1 \varphi_2} (n)=
\int (\varphi_1 \circ f^n (x)) \varphi_2(x) \, \mu(dx) -
\int \varphi_1 (x)\mu (dx)\, \int \varphi_2 (x)\, \mu(dx) \tag{1.3}
$$
for suitable $\varphi_1$, $\varphi_2$
by proving that there is a gap in the spectrum of $\LL$.
The discrete spectrum of $\LL$ can be shown to
correspond to the poles of the Fourier transform
of $C_{\varphi_1, \varphi_2}$ in some strip,
these poles are the {\it resonances} of Ruelle [1987].
A natural generalisation of this problem
(see e.g. Ruelle [1995] for an overview)
consists
in starting from a family of maps $f_\xi$
(or their inverse branches) and positive weights $g_\xi$
for $\xi \in E$, and defining the mixed or
generalised transfer operator
$$
\LL \varphi (x)= \sum_\xi \sum_{f_\xi(y)=x} \varphi(y) g_\xi(y)\tag{1.4}
$$
(the sum over $\xi$ being replaced by an integral
when the index set $E$ is uncountable).
This framework appears naturally when considering (weighted)
independent identically
distributed (i.i.d.) random compositions of maps $f_\xi$
associated with a probability
measure $\theta(d\xi)$ on the index set $E$,
a convenient description of the system being given by
the weighted (two-sided) skew product on $M \times E^\integer$
$$
\tau(x,\omega) = (f_{\omega_0}(x) , \sigma \omega)\, ,
\quad g(x,\omega)=g_{\omega_0} (x) \, , \tag{1.5}
$$
with $\sigma$ the shift on $E^\integer$,
or its corresponding ``one-sided'' version $\tau^+$
(see \thetag{2.3}).
For weighted random (not necessarily
i.i.d.) compositions, equilibrium states for a
relativised variational principle (Ledrappier-Walters [1977],
see \thetag{2.6} below)
have been studied,
in particular by Kifer [1992].
In the case where the maps $f_\xi$ are expanding, and
the weights are given by the
Jacobians $g_\xi(x) = 1/|\text{det} D_x f_\xi|$,
the integrated transfer operator \thetag{1.4} (see \thetag{2.16} for
a precise formula)
gives rise to this {\it relativised equilibrium state}
$\mu^{(q)}$,
which is just the SRB measure of the random system,
and has been studied in particular by Baladi-Young [1993].
The discrete spectrum of the operator is then also
related to integrated correlation functions \thetag{2.20} for the
random SRB measure.
In more general cases (consider for example a family
of one-dimensional repellors $f_\xi$, each with
an invariant Cantor set of Hausdorff dimension $\alpha_\xi$
and escape rate $P_\xi$,
and the weight $g_\xi = 1/|f'_\xi|$),
we find that the annealed integrated operator $\widehat \LL$
\thetag{1.4}-\thetag{2.16} acting on a suitably
``large'' space does {\it not} always give rise to the
relativised equilibrium state,
but to another $\tau$-invariant measure (see
\thetag{2.10} below for the corresponding variational
principle) which
we call the {\it annealed} equilibrium state $\mu^{(a)}$
(in particular, we
solve a conjecture of Ruelle [1995, Section 7], see Section 2.4).
Extending the analogy with
spin-glasses (see e.g. M\'ezard et al. [1987],
and our random Ising model example in Section 2.4)
we rename the relativised equilibrium states
{\it quenched} equilibrium states.
Using previous work of Kifer [1992] and Khanin-Kifer [1996] we
describe the modified (and less directly
accessible, for example in computer
simulations) quenched integrated transfer operator $\widehat \MM$
(see \thetag{4.12})
which gives rise to the quenched states.
We are then able to extend the very powerful transfer operator techniques
(including perturbative results
from Baladi-Young [1993], as well as the analysis
of the discrete spectrum in terms of
zeta functions or generalised Fredholm determinants
of Ruelle [1990]) to both integrated operators
$\widehat \LL$, $\widehat \MM$,
obtaining a good understanding of the ergodic properties
of both the annealed and quenched invariant measures
$\mu^{(a)}$, $\mu^{(q)}$,
including the resonances of their integrated correlation
functions.
Some of our results also apply to negative,
or even complex weights (negative
weights appear naturally e.g. in the study of renormalisation,
see e.g. Christiansen et al. [1990],
Jiang et al. [1992] and references therein)
where perturbative results are also desirable.
See Baladi et al. [1995] for a treatment of random
correlation functions (as opposed to the integrated correlation
functions \thetag{2.20}), without any i.i.d. assumption.
The Birkhoff cones used
there do not seem to be directly applicable to other Ruelle
resonances than the first one.
We refer to Ruedin [1994], Lanford-Ruedin [1995] for a
study of pressure and Gibbs state via
similar integrated transfer operators.
\smallskip
In this paper we use mainly three ingredients:
We adapt the results of Ruelle [1990] and Fried [1995]
to our (one-sided) skew product situation;
we transport the two-sided techniques of Kifer [1992] and Khanin-Kifer [1996]
to our one-sided skew product;
we apply the perturbative results
in Baladi-Young [1993] to get stochastic stability.
In some sense we are considering a
``toy model'': our uniform expansion and smoothness
assumptions are the strongest possible.
We expect and hope that the techniques developed
here may be extended to more realistic settings (expanding
in average as in Khanin-Kifer [1996], or
more generally non uniformly
hyperbolic, and/or piecewise smooth).
The theory presented here is particularly simple in the
i.i.d. setting, but most of it can be extended
to more general situations as pointed out to us
by David Ruelle (see Appendix~B).
\medskip
The outline of the paper is as follows: In Section 2 we
define precisely our model for random compositions of
expanding maps, as well as the annealed and quenched
equilibrium and Gibbs states.
We also state the main results: Theorem 1 (existence
and uniqueness of annealed states), Theorem 3 (stochastic
stability for annealed and quenched states), Theorem 4
(giving the relationship between the spectrum of the
integrated operators and the correlation functions
for annealed and quenched states),
Theorem 5 (stability of these correlation spectra)
and finally Theorem 6 on annealed zeta functions
and annealed Fredholm determinants and their stability.
Section 3 contains a proof of Theorem 1 based on
an analysis of an integrated transfer operator ($\widehat \LL$,
see Proposition 3.1)
which also yields Theorem 4, and
proofs of the stability results concerning the annealed
states in Theorems 3 and 5. Theorem 6 is also proved in Section 3.
Finally, Section 4 is devoted to the proofs of the claims
in Theorems 3, 4 and 5 on quenched states,
using the transfer operator $\widehat \MM$
(Proposition 4.2).
%\bigskip
\newpage
I am indebted to Thomas Bogensch\"utz, Konstantin Khanin,
and especially David Ruelle
for extremely useful conversations.
I would also like to thank
Yuri Kifer, Fran\c{c}ois Ledrappier,
and Laurent Ruedin for interesting comments.
It is a pleasure to acknowledge the hospitality
of ETH Z\"urich, IHES, SFB 170 in G\"ottingen, and IMPA,
where part of this work was carried through,
as well as financial support from
the Soci\'et\'e Acad\'emique de Gen\`eve.
\medskip
\head 2. Definitions and statement of results
\endhead
\subhead 2.1. Weighted random composition of expanding maps
\endsubhead
\smallskip
For fixed $r \ge 1$
let $M$ be a compact, connected, $C^r$ Riemann manifold
endowed with a Riemann metric
$d_M$. For $\gamma > 1$ let $C^r_\gamma(M,M)$ denote the space of all
$\gamma$-expanding
$C^r$ maps $f : M \to M$
(i.e., maps such that for all $x \in M$,
and all $v \in T_xM$, we have
$\| D_x f(v)\| > \gamma \| v\|$), endowed
with the $C^r$ metric.
Finally, let $C^r(M,\complex)$,
respectively $C^r(M,\real^+_*)$
be the space of all complex-valued or positive $C^r$ functions,
endowed with the $C^r$ metric $d_r$ or norm $\| \cdot \|_r$.
(Many of our results have versions
for $M$ a compact
metric space and Lipschitz or H\"older smoothness, or replacing
the inverse branches
of expanding maps by suitable families of contractions.)
Let $E$ be a compact subspace of $C^r_\gamma(M,M)\times C^r(M,\complex)$,
for the $C^{r}$ metric $d_E$.
Let $\Omega^+$ be the compact space of one-sided
sequences $E^\natural$ endowed
with the distance
$d_\alpha(\omega,\tilde \omega) =
\sum_{k =0}^\infty\alpha^k d_E(\omega_k, \tilde \omega_k)$
for some $0 < \alpha < 1$, and
let $\Omega = E^\integer$ be the corresponding
two-sided space, with an analogous metric also denoted $d_\alpha$.
Let
$\sigma^+$ be the one-sided shift to the left on $\Omega^+$,
and $\sigma$ the two-sided shift to the left on $\Omega$.
Fix $\theta$ a Borel probability on $E$.
The product measure $\Theta^+ =
\theta^\natural$ on $\Omega^+$ is $\sigma^+$-invariant and $\sigma^+$ is
ergodic for $\Theta^+$.
For a Borel measure $\upsilon$ on $M \times \Omega^+$, we shall
write
$\pi_\upsilon$ for the marginal of $\upsilon$ on $\Omega^+$.
We let $\PP_\Theta$ denote the
space of $\tau^+$-invariant probability measures $\mu$
on $M\times \Omega^+$
with $\pi_\mu =\Theta^+$.
If $\omega \in \Omega^+$, write $f_\omega$
for the first coordinate of $\omega_0 \in E$, and $g_\omega$
for the second coordinate of $\omega_0$.
We consider the independent identically distributed compositions
$$
f^{(n)}_\omega = f_{(\sigma^+)^n \omega}
\circ \cdots \circ f_{\sigma^+ \omega} \circ f_\omega \, ,
\tag{2.1}
$$
weighted by
$$
g^{(n)}_\omega =
g _{(\sigma^+)^n\omega}\circ f^{(n-1)} _\omega \cdots
g _{\sigma^+\omega}\circ f_\omega \cdot g_\omega
\tag{2.2}
$$
where $n \ge 1$
and $(f_\omega, g_\omega)=(f_{\omega_0}, g_{\omega_0})$
is chosen in $E$ following the
distribution $\theta(d \omega_0)$.
In other words, we are iterating the (weighted)
one-sided skew-product
$\tau^+ : M \times \Omega^+ \to M \times \Omega^+$
$$
\tau^+(x,\omega) =(f_{\omega_0}(x), \sigma^+ (\omega)) \, ,
\quad g(x, \omega)= g_{\omega_0} (x) \, . \tag{2.3}
$$
The map $\tau^+$ is in general not positively expansive, but
for each $\xi \in E$ and each local inverse branch
$(f_\xi)_i^{-1}$ of $f_\xi$
the inverse branch
$$
(\tau^+)^{-1}_{\xi, i} (x,\omega)=
((f_\xi)_i^{-1} x, \xi \wedge \omega)\tag{2.4}
$$
(where $\xi \wedge \omega$, or simply $\xi\omega$,
denotes the concatenation of $\xi\in E$ and
$\omega\in \Omega^+$)
is a $\max(\alpha, 1/\gamma)$ contraction for the metric
$d_M \times d_\alpha$. In particular, we shall see that we are in
the framework of Ruelle [1990] or Fried [1995].
We call such a system
$(\tau^+, g, \theta)$ (note that the pair $(E,\theta)$ contains
all the information)
a {\it $C^r$ weighted independent identically
distributed (i.i.d.) expanding map.}
If all the $g_\omega$ are real and positive-valued (respectively
nonnegative-valued) the system is called
{\it positively weighted} (respectively nonnegatively weighted).
\medskip
A special case of a (family of) random i.i.d. expanding maps is
obtained by considering
{\it small random perturbations} of $(f_0, g_0)$,
for $f_0 \in C^r_\gamma(M,M)$
and $g_0 \in C^r(M,\complex)$:
For each small $\epsilon \ge 0$ we have a probability
measure $\theta_\epsilon$ on some fixed $E$
as above, with
$$
\text{supp}\, \theta_\epsilon
\subset B_\epsilon(f_0,g_0) \, , \tag{2.5}
$$
where $B_\epsilon$ is the $\epsilon$-ball in the $d_E$ metric.
(In particular, $\theta_0$ is the Dirac mass at $(f_0, g_0)$.)
Many of our results concern this special case.
\medskip
\remark{Remark 2.1}
Another model in which our arguments work without
modification is given
by the following data: let
$M$, $\gamma$, $r$, $C^r_\gamma(M,M)$, $C^r(M,\complex)$
be as above,
let $(E, d_E)$ be a compact metric space endowed
with a probability measure $\theta$, set
$\Omega^+=E^\natural$ endowed
with metrics $d_\alpha$ for $0 < \alpha < 1$, and consider
$$
f : \Omega^+ \to C^r_\gamma(M,M) \, , \quad
g : \Omega^+ \to C^r(M,\complex) \, ,
$$
two Lipschitz functions with $f(\omega)= f(\omega_0)$ and
$g(\omega)=g(\omega_0)$ , which we view as random
variables on $(\Omega^+, \theta^\natural)$ (or equivalently
$(E, \theta)$).
The rest of the setup is as above.
This other description is more convenient
to describe one-dimensional random Ising models in Section 2.4.
Also, it allows
generalisations to Lipschitz $g$ on $\Omega^+$
which depend on the full sequence $\omega_0, \omega_1, \ldots$
(but assuming still that $f(\omega)=f(\omega_0)$).
In this case, most of our results hold (see the
discussion on the operator
$\widehat \MM$ in Section 4). The only differences are that the
operator $\LL$ does not exist any more (we must work with
$\widehat \LL$, most notably in Theorems 4 and 6, and the maximal eigenfunction
$\hat \rho(x,\omega)$ of Proposition 3.1 \therosteritem{2}
can depend on $\omega$) and that the definition of the annealed
zeta function and determinant
\thetag{2.24-2.25}
must be slightly changed (completing periodically
the sequences $\vec \xi$ appearing in $g_{\vec \xi}$).
\endremark
\subhead
2.2. Relativised equilibrium and Gibbs states (quenched and annealed)
\endsubhead
\noindent We assume in this subsection that
all weights $g_\omega$ are {\it real and nonnegative}.
\smallskip
{\bf Quenched and annealed equilibrium states}
Recall that an {\it equilibrium state for the relativised
variational principle}
(Ledrappier-Walters [1977],
Ruelle [1978 (Sections 6.21-22)], Kifer [1992]),
for $\tau^+$,
$g(x,\omega) = g_{\omega_0} (x)$, and $\theta$
is a Borel probability measure $\mu\in \PP_\Theta$
which realises the supremum
$$
Q^{(q)} (\log g) =\sup \{ h_{\tau^+} (\nu| \Theta^+) +
\int \log g(x,\omega) \, \nu(dx, d\omega)
\mid \nu \in \PP_\Theta \} \, ,\tag{2.6}
$$
where $h_{\tau^+} (\nu| \Theta^+)$ denotes the relative entropy
of $\nu$ with respect to its marginal $\pi_\nu=\Theta^+$.
We shall apply the formula from Bogensch\"utz--Crauel [1992]
$$
h_{\tau^+} (\nu| \pi_\nu)=
\sup_{\QQ \text{ finite partition of } M}
\lim_{n \to \infty} {1\over n}
\int_{\Omega^+} H_{\nu^\omega}
(\bigvee_{i=0}^{n-1} (f^{(n)}_\omega)^{-1} \QQ ) \, \pi_\nu(d\omega) \, ,
\tag{2.7}
$$
where we use the essentially unique decomposition
$$
\nu(dx,d\omega) =\nu^\omega(dx) \pi_\nu(d\omega) \, , \tag{2.8}
$$
and where the entropy $H_\upsilon(\QQ)$ of a measure
$\upsilon$ for finite partition $\QQ$
is defined as usual by $-\sum_{Q \in \QQ} \upsilon (Q) \log \upsilon(Q)$.
We call the relativised equilibrium states defined by \thetag{2.6}
{\it quenched (relativised) equilibrium states}
for $\tau^+$, $g$, and $\theta$,
and the supremum $Q^{(q)} (\log g)$ the {\it quenched (relativised)
topological pressure} of
$\tau^+$, $g$, and $\theta$.
\smallskip
We now introduce a new type of invariant equilibrium measure.
Recall that the {\it specific entropy per site}
$h^{\theta}( \upsilon)$
of a $\sigma^+$ invariant measure $\upsilon$, relative to the {\it a priori}
measure $\theta$ on $E$ is
$$
h^{\theta}( \upsilon) =- \int \log \beta(\xi\omega) \, \upsilon(d(\xi\omega))
=- \int \log \beta(\xi\omega) \,\, \beta(\xi\omega) \,
\theta(d\xi) \, \upsilon(d\omega) \tag{2.9}
$$
if $\upsilon(d(\xi\omega))$ is absolutely continuous with respect to
$\theta(d\xi) \upsilon(d\omega)$, with Radon-Nikodym
derivative $\beta(\xi\omega)$,
and otherwise, $h^{\theta} (\upsilon)= -\infty$.
(See Georgii [1988, pp. 317--318], Pinsker [1964, Section 15.2]:
the two-sided framework there may be adapted to our one-sided shift
$\sigma^+$.)
Define now an {\it annealed (relativised) equilibrium state} for
$\tau^+$, $g$ and
the {\it a priori} measure $\theta$
to be a $\tau^+$-invariant Borel probability measure $\mu$
on $M \times \Omega^+$ realising the following
supremum:
$$
Q^{(a)} (\log g)= \sup \{ h_{\tau^+} (\nu|\pi_\nu)
+ h^{\theta}( \pi_\nu) +
\int \log g (x,\omega)\, \nu(dx, d\omega) \} \, ,\tag{2.10}
$$
the supremum being over all
$\tau^+$-invariant Borel probability measures
$\nu$ on $M\times \Omega^+$.
We call $Q^{(a)} (\log g)$ the {\it annealed topological pressure}
of $\tau^+$, $g$, and $\theta$.
Since $h^{\theta}(\pi_\mu )=h^{\theta} (\Theta^+)=0$
if and only if $\mu \in \PP_\Theta$ (see e.g. Georgii [1988]),
we have $Q^{(a)}(\log g)
\ge Q^{(q)}(\log g)$. In some cases equality holds, but not always (see
in particular Proposition 2 and Remark 3.4 below).
\medskip
{\bf Quenched and annealed Gibbs states}
We introduce first the {\it random transfer operators}
$\LL_\xi : C^r(M, \complex) \to C^r(M, \complex)$,
defined for $\xi \in E$ by
$$
\LL_\xi \varphi (x)=
\sum_{f_\xi (y)=x} \varphi(y) \, g_\xi (y) \, .\tag{2.11}
$$
We also write for $n \ge 1$ and $\omega \in \Omega^+$
$$
\LL^n_\omega = \LL_{\omega_{n-1}} \circ \cdots
\circ \LL_{\omega_1} \circ \LL_{\omega_0}\, . \tag{2.12}
$$
Define now a {\it quenched (relativised) Gibbs state} for $\tau^+$, $g$, and
$\theta$
(see Khanin-Kifer [1996], and also Bogensch\"utz-Gundlach [1995]
who used a slightly different but equivalent definition) to be a
measure $\mu\in \PP_\Theta$ such that the
probability measures $\mu^\omega$ on $M$ arising in the
essentially unique decomposition \thetag{2.8} satisfy
$$
\mu^\omega \text{ is absolutely continuous
with respect to } \nu^\omega\, ,\tag{2.13}
$$
where ($\Theta^+$-almost) each measure $\nu^\omega$ is
a {\it quenched (relativised) Gibbs measure} for
$\tau^+$, $g$, and $\theta$, i.e., satisfies
$$
\int \varphi (x) \, \nu^\omega(dx) =
\int
{\LL^n_\omega \varphi(x) \over
\LL^n_\omega {\bold 1}(x) } \,
(f^{(n)}_\omega)^* \nu^{\omega} (d x) \, ,\tag{2.14}
$$
for all continuous $\varphi : M \to \complex$, and all $n \ge 1$,
where $\bold{1}$ denotes the constant function $=1$ on $M$.
Clearly, the definition of a Gibbs measure is equivalent
to requiring that the conditional probabilities $\nu^\omega_n(dy|x)$
of the measure $\nu^\omega$ conditioned
by $f^{(n)}_\omega (y)=x$ be the discrete measures defined
on the finite set $(f_\omega^{(n)})^{-1}(x)$ by
$$
\int \varphi(y) \nu^\omega_n(dy|x)=
{ \LL_\omega^n \varphi (x,\omega) \over
\LL_\omega^n {\bold 1} (x, \omega) } \, .
\tag{2.15}
$$
Defining the {\it integrated transfer operator} $\widehat \LL$
acting on measurable functions
$\varphi : M \times \Omega^+ \to \complex$
(we write $\varphi(x,\omega)=\varphi_\omega(x)$) by:
$$
\widehat \LL \varphi (x,\omega) =
\int_E ( \LL_\xi \varphi_{\xi \wedge \omega} ) (x)\, \theta(d\xi)
\, ,\tag{2.16}
$$
we define an {\it annealed (relativised) Gibbs measure} for
$\tau^+$, $g$ and
the a priori measure $\theta$
to be a Borel probability measure $\nu$ on $M\times \Omega^+$
such that for
all measurable $\varphi : M \times \Omega^+$ and all $n \ge 0$
$$
\int \varphi (x, \omega) \, \nu(dx, d\omega)=
\int
{\widehat \LL^n \varphi(x,\omega)
\over \widehat \LL^n {\bold 1} (x,\omega)}
\, ((\tau^+)^n)^* \nu (dx,d \omega)\, .\tag{2.17}
$$
Again, there is an interpretation in terms of
conditional probabilities: a Borel probability
measure $\nu$ on
$M \times \Omega^+$ is an annealed Gibbs measure if
for any integer $n \ge 1$ the conditional
probability $\nu_n((dy,d\eta) |(x,\omega))$
under the condition
$(\tau^+)^{n}(y,\eta)=(x,\omega)$
(in particular $\eta_{n+j} =\omega_j$ for $j \ge 0$)
is equal to the Radon measure
$$
\int \varphi(y,\eta) \nu_n(dy,d\eta |(x,\omega))
=
{\widehat \LL^n \varphi (x,\omega) \over
\widehat \LL^n {\bold 1} (x, \omega) } \, .\tag{2.18}
$$
Finally we define an {\it annealed (relativised) Gibbs state} for
$\tau^+$, $g$ and the a priori measure $\theta$ to be a Borel probability
measure $\mu$ on $M \times \Omega^+$ which is
$\tau^+$-invariant and absolutely continuous
with respect to an annealed Gibbs measure on $M\times \Omega^+$ for
$\tau^+$, $g$, and $\theta$.
\medskip
\subhead 2.3 Results
\endsubhead
\smallskip
\noindent Let us first recall results due to Kifer, Khanin-Kifer,
and Bogensch\"utz-Gundlach:
\proclaim{Theorem (Unique quenched Gibbs and equilibrium states $\mu^{(q)}$)}
A $C^r$ positively weighted i.i.d. expanding map
$(\tau^+, g,\theta)$
admits a unique quenched (relativised)
Gibbs state and a unique quenched (relativised) equilibrium state.
These two measures coincide.
\endproclaim
For a proof, see Kifer [1992, Theorem A], Khanin-Kifer [1996, Theorem C] and
Bogensch\"utz-Gundlach [1995] (their results are for the two-sided
skew product $\tau$ in \thetag{1.5}, but give readily our claim by integration).
Our first main result is:
\proclaim{Theorem 1 (Unique annealed Gibbs and equilibrium states $\mu^{(a)}$)}
A $C^r$ positively weighted i.i.d. expanding map
$(\tau^+, g,\theta)$
admits a unique annealed
Gibbs state and a unique annealed equilibrium state.
These two measures coincide.
\endproclaim
We prove Theorem 1 in Section 3.2. In fact, we also construct there
(non necessarily unique) annealed states for nonnegative weights.
In the case of SRB measure
the quenched and annealed states are the same:
\proclaim{Proposition 2 (SRB)}
For a $C^r$ weighted i.i.d. expanding map
$(\tau^+, g,\theta)$ with $g_\omega(x)= 1/|\text{Jac} D_x f_\omega|$,
the unique annealed equilibrium state and the unique quenched equilibrium state
coincide.
\endproclaim
Essentially in the same setting, Kifer [1992] proved
that for a $C^r$ weighted i.i.d. expanding map
$(\tau^+, g, \theta)$ with $g_\omega(x)= 1/|\text{Jac} D_x f_\omega|$,
the (relativised) equilibrium state is a direct product
$\rho \times \Theta^+$, with $\rho$ a measure
equivalent with Riemannian volume on $M$ and invariant
for the {\it Markov chain} corresponding
to $(\tau^+,\theta)$, which is defined by the transition probabilities
$$
\PP (x,A) = \int \chi_A(f_{\omega_0} (x)) \, \theta(d\omega_0)\tag{2.19}
$$
for $x \in M$ and $A \subset M$ Borel.
(The marginal on $M$ of an arbitrary $\tau^+$ invariant measure is
not invariant for the Markov chain in general.)
Proposition 2 is a consequence of Remark 3.4 in Section 3.
One-dimensional i.i.d. random Ising
models give simple examples where the quenched and
annealed states differ (see Section 2.4).
\noindent We obtain the following stability result in Section 3.4
(the second claim in Theorem 3 was obtained
previously by Kifer [1992, Section 4; 1990], see also Bogensch\"utz [1994]):
\proclaim{Theorem 3 (Stochastic stability for annealed and quenched states)}
Let $\mu_0$ be the equilibrium state for
$f_0\in C^r_\gamma(M,M)$ and
$\log g_0$ with $g_0\in C^r(M,\real^+_*)$. Consider
a positively weighted small random perturbation of $f_0, g_0$ given by a family
$\theta_\epsilon$ ($\epsilon \ge 0$).
\roster
\item
The annealed equilibrium states $\mu^{(a)}_\epsilon$ weakly converge
to $\mu_0 \times \delta_{f_0,g_0}^\natural$,
where $\delta_{f_0,g_0}$ is the Dirac measure at $(f_0,g_0)$
as $\epsilon \to 0$.
The annealed relativised pressure $Q^{(a)}_\epsilon(\log g)$
converges to the topological pressure $P(\log g_0)$
of $f_0$.
\item
The quenched equilibrium states $\mu^{(q)}_\epsilon$ weakly converge
to $\mu_0 \times \delta_{f_0, g_0}^\natural$ as $\epsilon \to 0$.
The quenched relativised pressure $Q^{(q)}_\epsilon(\log g)$
converges to the topological pressure $P(\log g_0)$
of $f_0$.
\endroster
\endproclaim
\medskip
{\bf Integrated correlation functions}
If $\mu$ is a $\tau^+$-invariant probability measure, we define
its {\it integrated} random
{\it correlation function} for $\varphi_1, \varphi_2 \in L^2(\mu)$,
and any integer $n \ge 0$ by
$$
\eqalign
{
C_{\varphi_1 \varphi_2} (n)
&= \int
(\varphi_1 \circ (\tau^+)^{n} ) (x,\omega) \,
\varphi_2(x,\omega) \, \mu(dx,d\omega)\cr
&\qquad\qquad-
\int \varphi_1 (x,\omega) \, \mu(dx,d\omega)
\int \varphi_2(x,\omega) \, \mu(dx,d\omega) \, .}
\tag{2.20}
$$
For $\varphi_1, \varphi_2$ in some function class $\FF$,
we may ask if $|C_{\varphi_1 \varphi_2}(n)|$
goes to zero exponentially fast, i.e., if there
exists $\tau < 1$ so that for any $\varphi_1, \varphi_2 \in \FF$
there is $K(\varphi_1, \varphi_2)$
with $|C_{\varphi_1 \varphi_2}(n)|
\le K(\varphi_1, \varphi_2) \cdot \tau^n$ for all
$n$ (the smallest such
$\tau$ is called the (exponential) {\it rate of decay of
correlations} for $\mu$ and $\FF$).
More generally, we can ask if the formal Fourier
transform (see Pollicott [1985], Ruelle [1987] for corrresponding
objects in a nonrandom setting)
$$
\widehat C_{\varphi_1 \varphi_2}(\eta)=
\sum_{n=0}^\infty C_{\varphi_1 \varphi_2} (n)\, e^{i\eta n}
+\sum_{n=1}^\infty C_{\varphi_2 \varphi_1} (n)\, e^{-i\eta n}
\tag{2.21}
$$
admits an analytic extension to
a strip and a meromorphic extension to a larger domain of the complex plane.
Using
the notation $\LL$ to represent the restriction to $C^r(M,\complex)$
of the operator $\widehat \LL$ defined in \thetag{2.16},
and referring to \thetag{4.12} Section 4 for the definition
of $\widehat \MM$ (and to Section 3.1 for the definition
of the Banach space $\BB(\alpha)$) we have
(see Sections 3.4, 3.6 and 4 for proofs):
\proclaim{Theorem 4 (Annealed and quenched
integrated correlation spectrum)}
Set $\FF =C^r(M,\complex)$.
\roster
\item
The Fourier transform $\widehat C_{\varphi_1 \varphi_2}(\eta)$
of the integrated correlation
function of the annealed equilibrium state $\mu^{(a)}$
of a $C^r$ positively weighted i.i.d. expanding map
$(\tau^+, g,\theta)$
for test functions in $\FF$ is analytic in a strip
$|\Im(\eta)| \le \delta^{(a)}$ for some $\delta^{(a)} > 0$
and admits a meromorphic extension to the strip
$|\Im(\eta)| \le \log \gamma^r$ where
its poles appear at points $\eta$, where
$\lambda=\exp(-i\eta +Q^{(a)}(\log g))$
is an eigenvalue of $\LL$ acting on $\FF$ with
$\exp(Q^{(a)}(\log g))/\gamma^r< |\lambda| < \exp(Q^{(a)}(\log g))$.
\item
Let $\alpha > 1/\gamma$.
The Fourier transform of the integrated correlation function
of the quenched equilibrium state $\mu^{(q)}$
of a $C^r$ positively weighted i.i.d. expanding map
$(\tau^+, g,\theta)$
for test functions in $\FF$ or $\BB(\alpha)$
is analytic in a strip
$|\Im(\eta)| \le \delta^{(q)}$ for some $\delta^{(q)} > 0$
and admits a meromorphic extension to the strip $|\Im(\eta)| \le \log 1/\alpha$
where
its poles appear at points $\eta$, where
$\lambda=\exp(-i\eta)$
is an eigenvalue of $\widehat \MM$ acting on $\BB(\alpha)$ satisfying
$\alpha< |\lambda| <1$.
\endroster
\endproclaim
In particular, the rate of decay of $\mu^{(a)}$ for $\FF=C^r(M,\complex)$
coincides with the ratio of
the moduli of the ``first two eigenvalues''
of $\LL$ acting on $\FF$ and similarly for
$\mu^{(q)}$ and $\widehat \MM$.
Motivated by Theorem 4, and since we will show
in Proposition 3.1 (respectively Proposition 4.2)
that the essential spectral radius
of $\LL$ on $\FF$ (respectively $\widehat \MM$ on
$\BB(\alpha)$ with $\alpha > 1/\gamma$) is not bigger than
$\exp(Q^{(a)}(\log g))/\gamma^r$ (respectively $\alpha$)
we call the (discrete) spectrum of $\LL$
in the annulus
$\exp(Q^{(a)}(\log g))/\gamma^r< |\lambda| < \exp(Q^{(a)}(\log g))$
(respectively of $\widehat \MM$ in
$\alpha< |\lambda| <1$)
the {\it annealed
(respectively quenched) integrated correlation spectrum} of the measure
$\mu^{(a)}$
and the function class $\FF$ (respectively
$\mu^{(q)}$ and $\BB(\alpha)$).
\smallskip
\noindent Regarding small random perturbations, we shall show
in Sections 3.4, 3.6, and 4:
\proclaim{Theorem 5 (Stability of the annealed/quenched
integrated correlation spectrum)}
Let $\mu_0$ be the equilibrium state for
$f_0\in C^r_\gamma(M,M)$ and
$\log g_0$, with $g_0\in C^r(M,\real^+_*)$,
let $P(\log g_0)$ be the
corresponding pressure, and let
$\tau_0 < 1$ be the rate of decay of correlations for
$\mu_0$ and
$\FF=C^r(M,\complex)$. Consider
a positively weighted small random perturbation of $f_0, g_0$ given by a
family
$\theta_\epsilon$ ($\epsilon \ge 0$).
\roster
\item
The rate of decay $\tau^{(a)}_\epsilon$
of correlations for
the annealed equilibrium state $\mu^{(a)}_\epsilon$
of $(\tau^+,g,\theta_\epsilon)$,
and test functions in $\FF$, satisfies
$
\limsup_{\epsilon \to 0} \tau^{(a)}_\epsilon \le\tau_0 \
$.
In fact, outside of
any disc of radius $e^{P(\log g_0)}/\gamma^r+\delta$, ($\delta>0$)
the integrated correlation spectrum of $\mu^{(a)}_\epsilon$
for $\FF$ converges to the correlation spectrum of $\mu_0$
for $\FF$.
\item
The rate of decay $\tau^{(q)}_\epsilon$
of correlations for
the quenched equilibrium state $\mu^{(q)}_\epsilon$ of
$(\tau^+,g,\theta_\epsilon)$, and test functions in $\BB(\alpha)$
($\alpha> 1/\gamma$), satisfies
$
\limsup_{\epsilon \to 0} \tau^{(q)}_\epsilon \le\tau_0
$.
In fact, outside of any disc of radius $e^{P(\log g_0)}/\gamma+\delta$
($\delta > 0$)
the integrated
correlation spectrum of $\mu^{(q)}_\epsilon$
for $\BB(\alpha)$ converges to the correlation spectrum of $\mu_0$
for $\FF$.
\endroster
\endproclaim
\smallskip
{\bf Annealed zeta functions and annealed generalised Fredholm determinants}
First consider the deterministic system $f$, $g$
($g$ not necessarily real or positive),
and define the formal {\it zeta function}
(see e.g. Ruelle [1989] and references therein)
by
$$
\zeta(z)=\exp \sum_{m\ge 1}{z^m\over m} \zeta_m
\quad \text{ where }
\zeta_m = \sum_{f^m (x)=x} \prod_{j=0}^{m-1} g(f^j(x)) \, .
\tag{2.22}
$$
A second formal series, the {\it generalised
Fredholm determinant} (Ruelle [1990]), may be associated with the
deterministic system by setting
$$
d(z)=\exp - \sum_{m\ge 1}{z^m\over m} d_m
\quad \text{ where }
d_m = \sum_{f^m (x)=x} {\prod_{j=0}^{m-1} g(f^j(x))
\over \det{ (1-D_x f^{-m})}}\, ,\tag{2.23}
$$
where $D_x f^{-m}$ denotes the derivative of the local
inverse branch of $f^m$ associated to the $m$-periodidic orbit of $x$.
For a weighted i.i.d. map ($\tau^+,g, \theta$), we define
the formal {\it annealed zeta function} by
$$
\eqalign
{
\zeta^{(a)}(z)&=\exp \sum_{m\ge 1}{z^m\over m} \zeta^{(a)}_m \cr
&\quad \text{ where }
\zeta^{(a)}_m =
\int_{E^m} \sum_{f^{(m)}_{\vec \xi} (x)=x}
\prod_{j=0}^{m-1}
g_{\xi_j}(f^{(j)}_{\vec \xi}(x)) \, \theta(d\xi_0) \ldots
\theta(d\xi_{m-1})\, .
}
\tag{2.24}
$$
Similarly, we get an {\it annealed Fredholm determinant}
by setting
$$
\eqalign
{
d^{(a)}(z)&=\exp -\sum_{m\ge 1}{z^m\over m} d^{(a)}_m \cr
&\quad \text{ where }
d^{(a)}_m =
\int_{E^m} \sum_{f^{(m)}_{\vec \xi} (x)=x}
{\prod_{j=0}^{m-1} g_{\xi_j}(f^{(j)}_{\vec \xi}(x)) \over
\det{ (1-D_x f_{\vec \xi}^{(-m)})}}
\, \theta(d\xi_0) \ldots
\theta(d\xi_{m-1}) \, .
}
\tag{2.25}
$$
The following stability result will be a consequence of the proofs in
Ruelle [1990]
and the spectral stability
obtained in Proposition 3.5 below (see Section 3.5):
\proclaim{Theorem 6 (Annealed zeta functions and Fredholm determinants)}
Consider a $C^r$ weighted i.i.d. expanding map
$(\tau^+, g,\theta)$, write
$R^{(a)}=\exp Q^{(a)}(\log |g|)$,
and let $\FF=\CC^r(M,\complex)$.
\roster
\item
The annealed zeta function $\zeta^{(a)}(z)$ is analytic
in the disc of radius $1/R^{(a)}$
and admits a meromorphic, zero-free,
extension to the open disc of radius $\gamma / R^{(a)}$,
where its poles are exactly the inverses of the eigenvalues of $\LL$ acting
on $\FF$ of modulus $>R^{(a)} / \gamma$
(including multiplicities).
\item
The annealed Fredholm determinant $d^{(a)}(z)$
admits an analytic
extension to the disc of radius $\gamma^r / R^{(a)}$
where its zeroes are exactly the
inverses of the eigenvalues of $\LL$ acting
on $\FF$ of modulus $>R^{(a)}/ \gamma^r$
(including multiplicities).
\item
In the case of a weighted small random perturbation
$(\tau^+, g, \theta_\epsilon)$ of $(f_0,g_0)$,
writing $R_0 = \exp P(\log |g_0|)$,
the functions $(\zeta^{(a)}_\epsilon (z))^{-1}$,
respectively $d^{(a)}_\epsilon (z)$ converge
to $(\zeta(z))^{-1}$, respectively $d(z)$,
as $\epsilon \to 0$
in any compact subset of
the disc of radius $\gamma/R_0$, respectively
$\gamma^r/R_0$,
in the sense of analytic functions.
\endroster
\endproclaim
\medskip
\subhead 2.4 Two examples
\endsubhead
\smallskip
{\bf A conjecture of Ruelle}
Our first example is taken from Ruelle [1995, Section 7.4].
Assume that $E$ is countable or finite
(with $\theta(\xi) > 0$ for all $\xi\in E$)
and that $g$ is nonnegative. Assume also that the spectral
radius $R > 0$ of the operator $\LL$ acting on $C^r(M,\complex)$ is the
only eigenvalue of modulus $R$ and is simple (this is true
for example if $g$ is positive, see Section 3.1).
We start by giving a different characterisation
of the annealed equilibrium state $\mu$ for
$(\tau^+, g)$ and $\theta$, assuming
further that $\mu$ has the property that
$h_{\sigma^+}(\pi_\mu) < \infty$.
In this case we have on the one hand
$$
h_{\tau^+} (\mu|\pi_\mu)=
h_{\tau^+} (\mu) - h_{\sigma^+}(\pi_\mu) \, ,\tag{2.26}
$$
and on the other
$$
h_{\sigma^+}(\pi_\mu)
= -\int_{\Omega^+} \log {\pi_\mu(\omega) \over \pi_\mu(\sigma^+ \omega)} \,
\pi_\mu(d \omega) \, ,\tag{2.27}
$$
where
$\pi_\mu(\omega) / \pi_\mu(\sigma^+ \omega)$
denotes the Radon-Nikodym derivative
(recall that $E$ is countable, $\pi_\mu$ is
$\sigma^{+}$-invariant, and use Theorem 4.14 in Walters [1982]).
It thus follows from the definition of the specific entropy
per site that
$$
\eqalign
{
h_{\tau^+} (\mu|\pi_\mu) + & h^\theta(\pi_\mu)+
\int_{M \times \Omega^+}
\log g_{\omega_0} (x) \, \mu(d x,d \omega) \cr
&=
h_{\tau^+} (\mu) +
\int_{M \times \Omega^+}
\log (\theta(\omega_0)g_{\omega_0} (x)) \, \mu(d x,d \omega) \, .
}\tag{2.28}
$$
In other words, the annealed equilibrium state $\mu$ is
the (almost ordinary\footnote{Restricting to
$\tau^+$ invariant measures with finite entropy $h_{\tau^+}(\nu)$
in the variational principle.}) equilibrium state for $\tau^+$ and the
weight $G(x, \omega) = \theta(\omega_0)g_{\omega_0} (x)$
on $M \times \Omega^+$, whenever $h_{\sigma^+}(\pi_\mu) < \infty$.
(Note that this finiteness property does {\it not} always hold for
$E$ countable infinite:
just consider $g_\xi = 1/|\text{Det}\, Df_\xi|$ so that the annealed
equilibrium
state satisfies $\pi_\mu =\theta^\natural$,
and take $\theta(n)$ of the order of $1/(n(\log n)^2)$;
an example where it does hold would be given by an a priori
measure of the order of $1/n^2$.)
To obtain a $\tau^+$-invariant
measure, Ruelle starts from $\rho$,
the nonnegative eigenfunction
of $\LL$ associated to its spectral radius $R \ge 0$
(Ruelle [1989, 1990]),
and constructs a measure $\bar \nu$ on $M \times \Omega^+$ by
iterating the corresponding eigenfunctional $\nu$ of $\LL^*$
(noting that $\nu$ is a positive measure):
$$
\eqalign
{
\bar \nu(&dx, d\omega_0, d\omega_1, \ldots)\cr
&= \lim_{m \to \infty}
\theta(d\omega_0) \ldots \theta(d\omega_m)
\sum_{i_0}(f_{\omega_0})^{-1}_{i_0}(R^{-1} g_{\omega_0} \cdot( \cdots
\sum_{i_m} (f_{\omega_m})^{-1}_{i_m} (R^{-1} g_{\omega_m} \nu(dx)) \cdots )\, ,
}
\tag{2.29}
$$
where we use the notation $(f_\xi)^{-1}_i$ for the finitely many
inverse
branches of $f_\xi$. He then considers the normalisation
$\upsilon(dx,d\omega)$ of the
$\tau^+$-invariant
measure $\rho(x) \bar \nu(dx,d\omega)$
and formulates the
{\bf Conjecture (Ruelle [1995, Section 7.4]):}
{\it The spectral radius $R$ is the exponential
of the topological pressure of $\log G$ for $\tau^+$.
The $\tau^+$-invariant probability measure
$\upsilon=\rho\bar\nu/\bar\nu(\rho)$
is an equilibrium state for the dynamical system $\tau^+$ on
$(M \times \Omega^+)$ and the function} $\log G : M \times \Omega^+
\to \real \cup {-\infty}$.
It is not difficult to check that $\widehat \LL^* \bar \nu = R \bar \nu$,
so that $\bar \nu = \hat \nu$, the maximal
eigenfunctional of $\widehat \LL^*$ (Proposition 3.1).
The invariant measure $\upsilon$ is thus the annealed
equilibrium state $\mu$ for $\tau^+$, $g$ and $\theta$
by Proposition 3.2 below.
We have therefore proved the above conjecture,
under the {\it assumptions}, nontrivial when
$E$ is infinite, that
$h_{\sigma^+}(\pi_\upsilon) < \infty$
(equivalently $h_{\tau^+}(\upsilon) < \infty$,
note that the conjecture needs to be reformulated
otherwise)
and that
$E=\{ (f_\xi, g_\xi) \, , \xi \in \natural\}
\subset C^r_\gamma(M,M) \times C^r(M ,\complex)$ is compact for
the induced metric.
Ruelle actually works with a countable
family of inverse branches instead
of expanding maps $f_\omega$
defined on the entire space $M$. However, the
assumptions he makes on the support of the corresponding
weights ensure that our arguments in Sections 2 and 3 carry through.
\medskip
{\bf One dimensional exponentially decaying random Ising model}
Our second example is a one-di\-mensional Ising model
with i.i.d. random external field and coupling constants
(see e.g. Ledrappier [1977] for a description in terms of
relative variational principle and references).
More precisely, we work on the half-lattice $\natural$
and consider the full shift $f$ on the metric space $M =\{\pm 1\}^\natural$
with a metric
$d_M(x,y) = \sum_{k\ge 0} |x_k-y_k|/ \gamma^{k}$ for
$\gamma> 1$
(a compact set of continuous spins could also be considered).
It is more convenient to work in the
setup described in Remark 2.1, considering
of course Lipschitz (instead of $C^r$) functions on $M$.
For the weight, we can fix for example some $\beta \ge 0$,
consider a
probability law $\theta=\theta_1\times \theta_2$
on a compact square $E=[-A, A]^2 \subset \real^2$
(see Ledrappier [1977] on
how to remove the compactness assumption in the nearest
neighbour interaction case),
and set
$$
g_{\omega_0}(x) =
\exp (-\beta \cdot (h_{\omega_0} x_0 + J_{\omega_0}
x_0 \cdot x_1 )) \, , \, x \in M \, ,\tag{2.30}
$$
with $\omega_0= (h_{\omega_0},J_{\omega_0})$ picked
in $E$ with law $\theta$.
(At the end of this subsection we explain how to generalise
to long-range {\it exponentially decaying} interactions.)
The physical interpretation is that
$\log g_\omega(f^k x)$ is the random contribution to the Hamiltonian associated
with the $k^{\text {th}}$ site of the
configuration $x$ (i.e., the sum of the interaction
between the $k^{\text{th}}$ site and the $(k+j)^{\text{th}}$ sites
for $j \ge 0$, as well as the term from the external random
field acting on $x_k$).
Note that since the skew product is in fact a direct product
here, the marginal
on $M$ of an annealed or quenched state will be a shift invariant
measure on $M$.
In other words, if we define by the usual formula the
partition function $Z_n(\omega,x)=
Z_n(\omega_0, \ldots, \omega_{n-1},x)$ ($\omega \in \Omega^+$,
$x \in M$)
of a finite one-sided box $[0, n-1]$ corresponding to the random
Hamiltonian
with fixed boundary condition $y_{n+j}=x_j$, $j \ge 0$,
we find that $Z_n(\omega,x)= \LL^{(n)}_\omega {\bold 1} (x)$.
The results of Ledrappier [1977] for finite range interaction and
more generally (a slight modification)
of Kifer [1992, Theorem 3.2 iii] imply that
for $\theta^\natural$-almost all $\omega$
and all $x \in M$
$$
\lim_{n \to \infty} {1\over n} \log Z_n(\omega,x)
= Q^{(q)} (\log g) \, .\tag{2.31}
$$
Therefore it follows from Proposition 3.2 below that
for $\theta^\natural$-almost all $\omega$
and all $x \in M$
$$
\eqalign
{
\lim_{n \to \infty} {1\over n} \log Z_n(\omega,x)
&= Q^{(q)} (\log g) \cr
&\le
Q^{(a)}(\log g)
=
\lim_{n \to \infty} {1\over n} \log \int_{E^n} Z_n(\omega,x)
\theta(d\omega_0)\cdots \theta(d\omega_{n-1}) \, ,
}\tag{2.32}
$$
with a {\it strict inequality} in general
(for example, in the simplest case \thetag{2.30}, with $h \equiv 0$
we see that the necessary condition stated in Remark 3.4 below is
only satisfied when $J$ takes $\theta$-almost everywhere at
most two values).
Our definitions of one-sided quenched and annealed Gibbs states
are consistent with the standard terminology and we recover
in particular by Theorem 4
the folklore theorem of exponential decay of correlations
for both states (note that the integrated correlations
\thetag{2.20} are simply
the space-correlation functions of observables in phase space
$M$ for the shift-invariant $M$-marginal).
Even when the quenched and annealed states are different, it
is not obvious that they have different marginals on $M$.
Since the physically observable measure is this $M$-marginal,
it would be of interest if possible to find conditions ensuring that
the quenched and annealed marginal are the same.
We point out also that considering partition
functions $Z^{(\text {per})}_n(\omega)$ with periodic boundary conditions
$x_{n+j}=x_j$, $j \ge 0$ yields
$$
Q^{(a)}(\log g)
=
\lim_{n \to \infty} {1\over n} \log \int_{E^n} Z^{(\text {per})}_n(\omega)
\theta(d\omega_0)\cdots \theta(d\omega_{n-1})
\tag{2.33}
$$
(because $\zeta^{(a)}_m =\int_{E^n} Z^{(\text {per})}_n(\omega)
\theta(d\omega_0)\cdots \theta(d\omega_{n-1})$
and by Theorem 6 \therosteritem{1} on the annealed zeta
function).
We may also consider exponentially decaying
long-range interactions such as
$$
g_{\omega_0}(x) =
\exp \biggl [
-\beta \cdot \bigl ( h_{\omega_0^0} x_0 +
\sum_{j=1}^\infty J_{\omega_0^j}
{x_0\cdot x_j \over \gamma^j} \bigr )\biggr ] \, , \, x \in M \, ,\tag{2.34}
$$
with $\omega_0=(h_{\omega_0^0}, J_{\omega_0^j}, j \ge 1)$
chosen in $E=[-A,A]\times [-A,A]^\natural$ with law $\theta_1
\times \theta_2^\natural$ where $\theta_1$ and $\theta_2$ are
two probabilities on $[-A,A]$.
Since we are in a purely Lipschitz context, there is
no need to modify the results of Ruelle [1990]
(see beginning of the proof of Proposition 3.1 (2))
and it suffices to check that $g$ is a Lipschitz
function on $M \times \Omega^+$
if we endow $E$ with a metric $d_\alpha$ for $\alpha> 1/\gamma$
in order to apply our results.
\medskip
\head 3. The annealed transfer operators $\widehat \LL$
\endhead
\subhead 3.1 The integrated annealed transfer operators $\widehat \LL$
\endsubhead
\smallskip
Let $\BB=\BB(\alpha)$ denote the Banach space of Lipschitz
functions
$\varphi : \Omega^+ \to C^r(M,\complex)$
(for the metric $d_\alpha$ on $\Omega$ and $d_r$ on $C^r(M,\complex)$)
endowed
with the norm $\|\varphi\|_\alpha=
\sup_{\omega} \| \varphi\|_r
+ \text{Lip}_\omega \varphi$, where $\text{Lip}_\omega \varphi$ denotes the
smallest Lipschitz constant. We may view an element of
$\BB$ as a function on $M \times \Omega^+$ by setting
$\varphi(x,\omega) = \varphi(\omega)(x)$ and
it is easy to see that
the operator $\widehat \LL$ defined
by \thetag{2.16} preserves the Banach space $\BB$.
We consider the operator
$\LL=\LL_g$ defined by restricting
$\widehat \LL$ (see \thetag{2.16}) to measurable
functions on $M$:
$$
\LL \varphi(x) = \int (\LL_\xi \varphi) (x) \theta(d\xi) \, .
\tag{3.1}
$$
The transfer operator $|\LL|$ obtained by replacing
$g_\xi$ by $|g_\xi|$ in \thetag{3.1}
is bounded when acting on the Banach space of bounded functions
on $M$ endowed with the supremum norm $\|\varphi\|_\infty$.
Denote by $R=R(|g|)$ its spectral radius which satisfies by definition
$$
R = \lim_{m \to \infty} (\| |\LL|^m 1 \|_\infty)^{1/m}\, .\tag{3.2}
$$
The basic properties of $\LL$ and $\widehat \LL$
that we shall use are:
\proclaim {Proposition 3.1 (Quasicompacity)} Set $\FF=C^r(M,\complex)$.
\roster
\item
The spectral radius of $\LL$ acting
on $\FF$ is bounded above by $R$,
its essential spectral radius is bounded above by $R/\gamma^r$.
If $g$ is nonnegative and $R > 0$, then $R$ is an eigenvalue of $\LL$
with a nonnegative eigenfunction $\rho\in \FF$.
If $g$ is positive then $\rho$ is positive
and $R$ is a simple eigenvalue;
moreover it is the only
eigenvalue of modulus $R$, and the
corresponding eigenfunctional for $\LL^*$ is a positive measure $\nu$
such that
$$
\lim_{n \to \infty}
\sup_{x \in M}
\biggl |{\LL^n \varphi(x) \over R^n} -\rho(x) \cdot \nu( \varphi)
\biggr |=0\tag{3.3}
$$
for all $\varphi \in L^1(\nu)$.
\item
The essential spectral radius of the operator
$\widehat \LL$ acting on $\BB(\alpha)$
is not larger than
$R \cdot \max(\alpha, \gamma^{-r})$.
The spectra of $\LL$ acting on $\FF$
and
$\widehat \LL$ acting on $\BB(\alpha)$ coincide, including multiplicities
in the domain
$\{ |z| > R \cdot \max(\alpha, \gamma^{-r})\}$.
If $g$ is positive, then $R$
is a simple eigenvalue of $\widehat \LL$
with eigenfunction $\hat \rho$ equal to the eigenfunction $\rho$
of $\LL$, the
corresponding positive eigenfunctional $\hat \nu$
is a positive measure with marginal $\nu$ on $M$.
Also, when $g$ is positive
$$
\lim_{n \to \infty}
\sup_{(x,\omega) \in M\times \Omega^+}
\biggl |{\widehat \LL^n \varphi(x,\omega)\over R^n }
- \rho(x) \cdot \hat \nu( \varphi) \biggr | =0\tag{3.4}
$$
for all $\varphi \in L^1(\hat \nu)$.
\endroster
\endproclaim
\demo{Proof of Proposition 3.1}
\roster
\item
The bounds on the spectral and essential spectral radius are proved
in Ruelle [1990, Theorem 1.1, Theorem 1.3]
(condition (ii) of Ruelle is satisfied up to
using a partition of unity). If $g$ is nonnegative, the spectral
radius of $\LL$ acting on $\FF$ is equal to $R$ by \thetag{3.2}.
To prove that there is a corresponding nonnegative eigenfunction,
just use the algebra in Ruelle [1989, \thetag{4.10--4.12}]
(the stronger assumptions of that paper were not used in this particular
argument, see also
Baladi--Kitaev--Ruelle--Semmes [1995, Proof of Theorem 2.5] for
more details). If $g$ is positive then since each
$f_\omega$ is transitive
we may use easy modifications of standard arguments
(see e.g. Parry-Pollicott [1990, pp. 23-24]) to show that
each nonnegative eigenfunction is positive.
To show that there is a nonnegative eigenfunctional
which is a positive measure, one may consider as usual
the weight defined by
$\bar g(y,\omega)= g(y,\omega) \rho(y)/(R \rho(f_\omega(y))$.
The constant function $1$ is then
fixed by the integrated operator $\LL_{\bar g}$,
so that the dual of this operator preserves the compact
convex space of Borel
probability measures. By Schauder-Tychonoff,
$\LL_{\bar g}^*$ therefore has a fixed point $\bar \nu$ and then the
normalisation of the measure defined by $\nu = \bar \nu/\rho$
is the desired maximal eigenmeasure for $\LL$.
Since the iterates
of $\LL_{\bar g}$ satisfy
a classical Yorke-type inequality
(see e.g. Lemma 4.2 in Baladi et al. [1995]) and
each $f_\omega$ is topologically mixing, the standard convexity argument
(see Parry-Pollicott [1990, pp. 25-26]) may be applied to $\LL_{\bar g}$,
yielding $\LL^n_{\bar g} \varphi \to \nu( \varphi )$
for continuous $\varphi$,
the result for $\nu$-integrable $\varphi$ follows from Lusin's theorem.
Standard arguments
(see Parry-Pollicott [1990, pp. 25-26]) then
show that $\LL_{\bar g}$
restricted to $\{ \varphi\in C^r(M) \mid \nu( \varphi )=0\}$
has spectral radius $< 1$, so that the spectrum of $\LL_{\bar g}$
is formed of the simple eigenvalue $1$ and a subset of
a disc with radius strictly less than $1$.
\item
Theorem 1.1 in Ruelle [1990] yields the upper bound
$R \cdot \max(\alpha, \gamma^{-1})$ for the essential
spectral radius of $\widehat \LL$ acting on the Banach space of Lipschitz
functions on $M \times \Omega^+$ with the metric $d+d_\alpha$.
To get the better bound claimed for the
space $\BB(\alpha)$, we could adapt Ruelle's original computation,
but have chosen to follow Fried's [1995] subsequent presentation.
Our setting is much simpler than the one
considered by Fried, in particular, we are only considering
a graph $(V,A)$ with vertex set $V$ reduced to a point
$M\times \Omega^+$ so that all arrows $A$ have initial and final vertex equal
to $M\times \Omega^+$.
The arrows are simply an index set for the contractions
which are the inverse branches
$\psi_{\xi,i}(x,\omega)=
((f_\xi)_i^{-1}x, \xi\omega)$ (for $\xi \in E$ and $i$ in a finite set
depending on $\xi$) as in \thetag{2.4},
that we view as being defined
on a closed subset $\hat M_{\xi,i}\times \Omega^+$ of $M\times \Omega^+$.
We say that an $n$-tuple of local inverse branches of $\tau_+$
is admissible if the corresponding composition
$\psi_{\vec \xi, \vec \imath}^n
=\psi_{\xi_n, i_n} \circ \cdots\circ \psi_{\xi_1, i_1}$
has a non empty domain of definition $D_{\vec \xi,
\vec \imath} \times \Omega^+$ in $M \times \Omega^+$.
We need to refine Lemma 1 from Fried [1995], adapting
it to our skew-product situation: We claim that
there is a constant $C> 0$ so that for all $n \ge 0$,
each admissible composition $\psi^n=\psi^n_{\xi,\vec \imath}$
of $n$ local inverse branches of $\tau^+$, and any
$(\tilde x, \tilde \omega)$ in the image of
$\psi^n (D_{\vec \xi,
\vec \imath} \times \Omega^+)$, then
if $T=T_{\psi^n}(\tilde x, \tilde \omega)$
denotes the finite rank
operator on $\BB$ given by $T\varphi=$ the Taylor
expansion of order $r$ about $\tilde x$ of
$\varphi_{\tilde \omega}$, we have:
$$
\| (\psi^n)^* (I-T) \varphi \|_\BB\le
C \max (\alpha, \gamma^{-r})^n \| \varphi\|_\BB
\tag{3.5}
$$
(see Appendix A for a proof, where we explain the slight
differences with Fried's assumptions and bounds).
Using \thetag{3.5} in place of Lemma 1 in Fried [1995], the proof of
Proposition 1 from Fried [1995]
combined with the
Leibniz-telescoping argument
in the proof of Ruelle [1990, Proposition 2.5]
(useful to replace the growth rate appearing in Fried [1995]
by the better bound $R$)
then yields our claim.
(We may ensure Fried's [1995, p. 1064] gap condition by
using a suitable partition of unity.)
Applying Theorem 1.1 in Ruelle [1990], we see that the eigenvalues of
both $\LL$ and $\widehat \LL$ acting
on Lipschitz functions
(on $M$, respectively $M \times \Omega^+$)
in the domain $|z| > R \max(\alpha, \gamma^{-1})$
are exactly (including multiplicities)
the inverses of the poles of the zeta function $\zeta^{(a)} (z)$
\thetag{2.24}.
Since any eigenfunction for $\LL$
is clearly an eigenfunction for $\widehat \LL$,
the statements on $\hat \rho$ and $\hat \nu$ in the case
of a positive weight $g$
follow from the simplicity of the eigenvalue $R$ for $\LL$.
Using \thetag{3.5} to generalise the main theorem
in Fried [1995, Section 3, p. 1067],
we obtain a bijection between the spectra
of $\LL$, $\widehat \LL$ and the zeroes of the determinant
$d^{(a)}(z)$ \thetag{2.25}
in the bigger domain $|z| > R \max(\alpha, \gamma^{-r})$.
(To find the formula for the trace of
each finite rank operator $\KK=\LL_{\psi^n} T$ associated to an
admissible composition $\psi^n$ and the corresponding operator $T
=T_{\psi^n}(\tilde x, \tilde \omega)$,
where we choose $(\tilde x, \tilde \omega)$ to be a fixed
point of $\psi^n$ if possible, and otherwise a point
in $\psi^n (D_{\vec \xi, \vec \imath} \times \Omega^+)\setminus
D_{\vec \xi, \vec \imath} \times \Omega^+$,
with $\KK$ acting on $\BB$,
we may compute instead the trace of $\KK$ as
an operator on functions $\BB_{\tilde \omega}$
depending only on the $x\in M$ variable,
setting the random argument to be equal to $\tilde \omega$ in
the notation above. This is possible because the corresponding
projection $\Pi=\Pi_{\tilde \omega}:\BB\to\BB_{\tilde \omega}$
satisfies $\KK = \Pi \KK \Pi$ so that $\text{Tr } \KK = \text{Tr }
\Pi \KK \Pi = \text{Tr } \KK|_{\BB(\tilde \omega)}$
and this last trace is computed as in page 1067 of Fried [1995].)
\qed
\endroster
\enddemo
\medskip
\subhead 3.2 Annealed equilibrium and Gibbs states
\endsubhead
\smallskip
\noindent Theorem 1 will be an immediate consequence of Proposition 3.2
and Proposition 3.3:
\proclaim{Proposition 3.2}
Assume that $g$ is positive, and let
$\rho$ be the maximal eigenfunction
and $\hat \nu$ the maximal eigenmeasure of $\widehat \LL$
from Proposition 3.1\therosteritem{2}. Then the probability measure
$\mu = \rho \hat \nu/\hat\nu(\rho)$ is the unique annealed equilibrium state
for $(\tau^+, g, \theta)$.
The maximal eigenvalue $R$ of $\widehat \LL$
is equal to $e^{Q^{(a)}(\log g)}$.
\endproclaim
\proclaim{Proposition 3.3}
Assume that $g$ is positive.
The probability measure
$\mu$ in Proposition 3.2
is the unique annealed Gibbs state
for $(\tau^+, g, \theta)$.
\endproclaim
\demo{Proof of Proposition 3.2}
To check that the measure $\mu$ is $\tau^+$-invariant,
consider the following chain of equalities, which holds for any
$\varphi \in L^1(\mu)$ (assume that $\hat \nu(\rho)=1$):
$$
\eqalign{
\int \varphi (x,\omega) \mu(dx,d\omega) &=
\int \varphi_\omega(x) \rho(x) \hat\nu(dx,d\omega)\cr
&=
{1 \over R}
\int \varphi_\omega(x) \int \LL_\xi(\rho)(x) \theta(d\xi)
\hat\nu(dx,d\omega)\cr
&=
{1\over R}
\int \int
\LL_\xi ((\varphi_{\omega}\circ f_{\xi \omega}) \cdot \rho )(x)
\theta(d\xi)
\hat\nu(dx,d\omega)\cr
&=
\int
\varphi_{\sigma^+ \omega} (f_\omega x) \rho (x)
\hat\nu(dx,d\omega)
=\int \varphi\circ \tau^+ (x,\omega) \mu(dx,d\omega) \, .
}\tag{3.6}
$$
\medskip
The basic strategy now is to go to the two-sided situation in order
to apply the arguments in Kifer [1992] and Khanin-Kifer [1996].
We will use two other random transfer operators
to construct an invariant measure $\upsilon$ for the
two-sided skew product $\tau$ with same relative
entropy as $\mu$. Consider first the random operator
$\LL'_\xi$, defined by formula \thetag{2.11} for the weight
$$
g'_{\xi}(x)=
g_\xi(x) \rho(x)/(R \cdot \rho ( f_\xi x) )
$$
(note that $g' \in \BB$).
The operator $\LL'_\xi$
has by definition the property that for
all $\varphi \in L^1(\mu)$:
$$
\int \int (\LL'_\xi \varphi(\xi\omega)) (x) \theta(d\xi) \mu(dx,d\omega)
= \int \varphi(x,\omega) \mu(dx,d\omega)\, .
\tag{3.7}
$$
It follows from the definitions that the measure
$\pi_{\mu}(d\xi\omega)$ is equivalent with the product measure
$\theta(d\xi) \pi_{\mu}(d\omega)$ with a density
denoted by $\beta(\xi\omega)\in L^1(\pi_{\mu})$.
In fact, from \thetag{3.7} for functions $\varphi(x,\xi\omega)$
independent of $x$, we obtain by Fubini the explicit formula
for $\pi_\mu$ almost all $\xi\omega \in \Omega^+$
$$
\beta(\xi \omega)
= \int_M {1 \over R \rho(x) } \sum_{f_\xi (y)=x} g_\xi(y) \rho(y) \,
\mu^\omega(dx) \, ,\tag{3.8}
$$
(where we use the decomposition \thetag{2.8} for $\mu$).
It is clear from \thetag{3.8} that $\beta(\xi\omega)$
is $\Theta^+$-almost everywhere uniformly bounded and bounded away from zero
(combining the (uniform) smoothness and positivity of $\rho$
and $g_\xi$, together with the fact that the
number of inverse branches of the $f_\xi$ is
uniformly bounded).
We then define the second modified random
operator associated to
$g''_\omega=g'_\xi / \beta(\xi\omega)$ by
$$
\LL''_{\xi\omega} \varphi(x)
= {\LL'_{\xi} \varphi(x) \over \beta(\xi\omega)} \, ,\tag{3.9}
$$
whose dual has the key property that for
$\pi_\mu$-almost all $\xi\omega$:
$$
(\LL''_{\xi\omega})^* \mu^\omega = \mu^{\xi\omega}\, .\tag{3.10}
$$
To prove \thetag{3.10}, consider an arbitrary $\varphi \in L^1(\mu)$
and write (using \thetag{3.7})
$$
\eqalign
{
\int \int (\LL''_{\xi\omega}& \varphi(\xi\omega))(x) \,
\mu^\omega(dx)\pi_\mu (d\xi\omega)\cr
&=
\int \int
\beta(\xi\omega)^{-1}( \LL'_{\xi} \varphi(\xi\omega))(x)
\mu^\omega(dx)\pi_\mu (d\xi\omega)\cr
&=
\int \int \int
(\LL'_{\xi} \varphi(\xi\omega))(x) \theta(d\xi)
\mu^\omega(dx) \pi_\mu(d\omega)\cr
&=
\int \int \varphi(x,\omega) \mu^\omega(dx)\pi_\mu(d\omega)
=
\int \int \varphi(x,\xi\omega)
\mu^{\xi\omega}(dx)\pi_{\mu}(d\xi\omega)\, . \cr
}\tag{3.11}
$$
Consider now two-sided sequences $\omega\in \Omega$
viewing
$
g''_\omega=g'_\xi/\beta(\xi\omega)
$
as a function of $\omega$ depending
only on the $\omega_i$ with $i \ge 0$,
and let $\pi_\upsilon$ denote the natural extension of
$\pi_{\mu}$ to $\Omega$.
Since the family of positive weights $g''_\omega$
satisfies the equi-H\"older continuous
property of Kifer [1992, \thetag{1.7}]
we are now in a position to apply Kifer [1992, Proposition 2.5]
to the operators $\LL''_\omega$.
Recall that in this two-sided and not necessarily i.i.d.
setting Kifer constructs
for $\pi_\upsilon$ almost all $\omega \in \Omega$
uniquely defined
numbers $\lambda^\omega > 0$, probability measures
$(\mu'')^\omega$ on $M$,
and positive H\"older functions
$h''_\omega : M \to \real$ with $(\mu'')^\omega(h''_\omega)=1$ and such that
$$
\LL''_{\omega} h''_\omega = \lambda^\omega h''_{\sigma\omega}\, , \quad
(\LL''_{\omega})^* (\mu'')^{\sigma\omega}
= \lambda^\omega (\mu'')^{\omega} \, .
\tag{3.12}
$$
It follows from \thetag{3.10} and the uniqueness
statement in Kifer that $\lambda^\omega \equiv 1$
and $\mu^\omega=(\mu'')^\omega$.
By construction, the
two-sided probability measure
$$
\upsilon(dx,d\omega) =
h''_\omega (x) \mu^\omega (dx)\, \pi_{\upsilon} (d\omega)
\tag{3.13}
$$
is invariant under the two-sided skew product $\tau$,
and from Theorem 3.2 in Kifer [1992] it is the unique (quenched) equilibrium
state with marginal
$\pi_\upsilon$ for the pair $(\tau, g'')$ on $M \times \Omega$.
We claim that the relative entropy of $(\tau,\upsilon)$ over
$(\sigma,\pi_\upsilon)$ coincides with
the relative entropy of $(\tau^+,\mu)$ over $(\sigma^+,\pi_\mu)$.
This follows from formula \thetag{2.7} applied both
in the one-sided and two-sided settings,
and the fact that $\sup_x |\log h''_{\omega} (x)|$
is bounded uniformly $\pi_\upsilon$-almost everywhere by
Kifer [1992, Proposition 2.5, \thetag{2.16}].
(Indeed, this uniform bound implies that there is a positive constant $C > 0$
so that
$C \cdot H_{\upsilon^\omega}(\QQ) \le H_{\mu^\omega} (\QQ)
\le H_{\upsilon^\omega}(\QQ)/C$
for any finite partition $\QQ$ and $\pi_\upsilon$ almost all
$\omega\in \Omega$ where
we used the decomposition $\upsilon (dx,d\omega) = \upsilon^\omega(dx)
\pi_\upsilon(d\omega)$.)
We now show that
$$
0 = h_{\tau} (\upsilon |\pi_\upsilon)+ \upsilon( \log g'')\, . \tag {3.14}
$$
Equality \thetag{3.14} follows from Kifer [1992, Proposition 3.1] which tells
us in particular that if we set
$g'''_\omega= g''_\omega h''_\omega/(h''_{\sigma \omega}
\circ f_\omega)$ then we have for almost all $\omega$
$$
0 = \upsilon^\omega(I_{\upsilon^\omega}(\BB_M | f^{-1}_\omega \BB_M)
+ \log g'''_\omega) \, , \tag{3.15}
$$
where
$I_{\eta}(\BB_M | f^{-1}_\omega \BB_M)$ denotes the conditional
information of the partition $\EE$ of $M$ into points with respect
to the partition $f^{-1}_\omega \EE$ for the probability measure
$\eta$.
(Just integrate \thetag{3.15}
with respect to $\pi_\upsilon$, use the definitions
of $g'''$ and $\upsilon$
and the fact that
$h_{\tau} (\upsilon |\pi_\upsilon)=
\int \upsilon^\omega(I_{\upsilon^\omega}(\BB_M | f^{-1}_\omega \BB_M))
\pi_\upsilon(d\omega)$.)
Since we have
$$
\eqalign
{
h_{\tau} (\upsilon |\pi_\upsilon)+ \upsilon( \log g'' )
&= h_{\tau^+} (\mu |\pi_\mu)+ \mu(\log g'' )\cr
&= h_{\tau^+} (\mu |\pi_\mu)+ \mu( \log g')
+ h^{\theta} (\pi_\mu) \, ,
}\tag {3.16}
$$
Equation \thetag{3.14} implies by definition of $g'$ that
$$
h_{\tau^+} (\mu |\pi_\mu)+ \mu(\log g )
+ h^{\theta} (\pi_\mu) = \log R\, .\tag{3.17}
$$
We now check that $\log R= Q^{(a)}(\log g)$.
Let then $\hat \mu(dx, d\omega)=\hat \mu^\omega (dx)\pi_{\hat \mu}(d\omega)$
be a one-sided $\tau^+$-invariant measure
on $M\times \Omega^+$
with $h^{\theta} (\pi_{\hat \mu}) > -\infty$, and let
$\hat \beta(\omega)$ be the Radon-Nikodym derivative of
$\pi_{\hat \mu}(d\omega)$ with respect to
$\theta(d\omega_0) \pi_{\hat \mu}(d\sigma \omega)$
(note that $\hat \beta$
is $\pi_{\hat \mu}$-almost everywhere nonzero since
$1/\hat \beta(\omega)$ is in $L^1(\pi_{\hat\mu})$
because the specific entropy per site is finite).
We consider $\hat \upsilon(dx,d \omega)=
\hat \upsilon^\omega(dx) \pi_{\hat \upsilon}(d\omega)$ the
$\Omega$-natural extension of
$(\hat\mu,\tau^+)$, i.e., the unique
$\tau$ invariant measure $\hat \upsilon$
such that
$\hat \upsilon(\varphi) = \hat\mu( \varphi)$
for all $\varphi\in L^1(\hat\mu)$
which depend only on ($x$ and)
$\omega_j$ for $j \ge 0$. The $\Omega$-natural
extension
is constructed just like the standard natural extension
and has the property that it leaves
the relativised entropy invariant. (It is possible although
not necessary to show
that $\upsilon$ above is the $\Omega$-natural extension of $\mu$.)
Note that $\pi_{\hat \upsilon}$
is the (ordinary) natural extension of $\pi_{\hat \mu}$.
Next, let $\hat h_\omega >0$
be the functions associated
to the
weight $g'_{\omega_0} (x)/\hat \beta(\omega)$
by Theorem 3.1 in Khanin--Kifer [1996]
(note that the corresponding $\hat \lambda^\omega$
are equal to one $\pi_{\hat \upsilon}$
almost everywhere by a uniqueness argument
and a computation identical to \thetag{3.11}).
Set
$$
\hat g_{\omega}(x) =
{g'_{\omega_0} (x)\hat h_\omega(x)
\over
\hat\beta(\omega)
(\hat h_{\sigma \omega} \circ f_{\omega_0}(x)) } \, . \tag{3.18}
$$
By construction, $\sum_{f_\omega y=x} \hat g_\omega(y) =1$
for any $x \in M$ and $\pi_{\hat \upsilon}$ almost all $\omega\in \Omega$.
Applying finally the arguments
of Kifer [1992, (3.8)--(3.11)] to $\hat \upsilon$
we get for $\pi_{\hat\upsilon}$ almost every $\omega$
$$
\eqalign
{
\hat \upsilon^\omega\bigl
( I_{\hat \upsilon^\omega} (\BB_M |f^{-1}_\omega \BB_M)
&+ \log \hat g_{\omega} \bigr )
\le\cr
&\int \sum_{y \in f_\omega^{-1} f_\omega(x)}
\hat g_{\omega}(y) \, \hat \upsilon^\omega(dx) -1=0\, ,
}\tag{3.19}
$$
Integrating both sides of \thetag{3.19}
with respect to $\pi_{\hat \upsilon}$ and using the definition
of $\hat g$, we get
$$
\eqalign
{
h_{\tau} (\hat \upsilon |\pi_{\hat \upsilon})&+
\hat \upsilon(\log g' )
+ h^{\theta} (\pi_{\hat \mu}) \cr
&=
h_{\tau^+} (\hat \mu |\pi_{\hat \mu})+ \hat \mu (\log g')
+ h^{\theta} (\pi_{\hat \mu})
\le 0 \, .
}\tag{3.20}
$$
Since $\hat \mu( \log g' )= \hat \mu(\log g ) - \log R$,
we are done.
It remains to prove uniqueness of the annealed state.
Let $\hat \mu(dx,d\omega)=\hat \mu^\omega(dx) \pi_{\hat \mu}(d\omega)$
be a $\tau^+$ invariant probability measure
with $h^{\theta} (\pi_{\hat \mu}) > -\infty$ and such that the inequality in
\thetag{3.20} is an equality, so that the inequality
for the corresponding $\hat \upsilon$
on $M \times \Omega$
in \thetag{3.19} is $\pi_{\hat\upsilon}$
almost everywhere an equality, i.e.,
$$
\eqalign
{
\hat \upsilon^\omega
\bigl ( I_{\hat \upsilon^\omega} (\BB_M |f^{-1}_\omega \BB_M)
&+ \log \hat g_\omega \bigr )
=\cr
&\int \sum_{y \in f^{-1}_\omega f_\omega(x)}
\hat g_\omega \, \hat \upsilon^\omega(dx) -1
=0\, .
}\tag{3.21}
$$
Starting from \thetag{3.21}, we may proceed exactly
as Kifer [1992, \thetag{3.12}]
obtaining that $\LL_{\hat g_{\xi \omega}}^* \hat \upsilon^{\omega}
= \hat \upsilon^{\xi\omega}$ ($\pi_{\hat \upsilon}$
almost everywhere) for $\LL_{\hat g_{\omega}}$
the random operator associated to $\hat g_\omega$.
Integration then shows that \thetag{3.7} holds with
$\hat \mu$ instead of $\mu$:
$$
\int \sum_{f_\xi y=x}
{ g'_{\xi} (y) \over \hat \beta(\xi\omega)}
\hat h_{\xi\omega}(y) \varphi_{\xi\omega}(y) \, \hat \mu^{\omega}(dx) \,
\pi_{\hat \mu}(d\xi\omega)
=
\int \varphi_\omega(x) \hat h_\omega(x) \, \hat \mu^\omega(dx)
\pi_{\hat \mu}(d\omega)
\, .\tag{3.22}
$$
Since the simplicity of the maximal
eigenvalue statement
in Proposition 3.1 \therosteritem{2} applies to the dual of the integrated
operator $\hat \LL'$ associated to $g'_\xi$, we get the claimed equality
$\mu=\hat \mu$ from \thetag{3.22} by definition
of $\hat \beta$.\qed
\enddemo
\smallskip
\demo{Proof of Proposition 3.3}
We essentially follow the path laid out by Khanin and Kifer [1996, Section 4],
proving first the existence and uniqueness of the annealed Gibbs measure.
Observe that any limit point of the probability measures
$$
\varphi \mapsto
{\widehat \LL^n \varphi(x,\omega)_n \over
\widehat \LL^n {\bold 1}(x,\omega)_n}\, ,
\quad \varphi \in C^0(M\times \Omega^+, \complex) \, ,
\tag{3.23}
$$
as $n \to \infty$
with $(x,\omega)_n \in
M \times \Omega^+$
is an annealed Gibbs measure, and that all annealed
Gibbs measures are constructed with this procedure.
Such a limit point must exist by standard compactness
arguments. In fact it follows from the proof of Proposition 3.1 that
for all continuous $\varphi : M \times \Omega^+ \to \complex$
$$
\eqalign
{
\lim_{n \to \infty}
\sup_{(x,\omega),
(y,\tilde \omega)}
\biggl |{\widehat \LL^n \varphi(x,\omega)
\over \widehat \LL^n {\bold 1}(x,\omega)}
-{\widehat \LL^n \varphi(y,\tilde \omega)
\over \widehat \LL^n {\bold 1}(y,\tilde \omega)}\biggr |
=\lim_{n \to \infty}
\biggl |{\widehat \LL^n \varphi(x,\omega) \over
\widehat \LL^n {\bold 1}(x,\omega)}
-\hat \nu (\varphi)\biggr |
=0 \, ,
}\tag{3.24}
$$
uniformly in $(x,\omega)$,
where $\hat \nu$ is defined in Proposition 3.1.
(Indeed, the difference
$\widehat \LL^n \varphi(x,\omega)/(R^n \rho(x))- \hat\nu (\varphi)$
converges to zero uniformly in $(x,\omega)$.)
In particular, we also get uniqueness of the Gibbs measure,
which coincides with $\hat \nu$.
Clearly,
the annealed equilibrium state $\mu=\rho \hat \nu$ is therefore
also an annealed Gibbs state.
To prove that there
is no other annealed Gibbs state we note that any
such state $\mu'$
has a density $\rho'\in L^1(\hat \nu)$ with respect to $\hat \nu$
which satisfies $\widehat \LL \rho' = R \rho'$ in $L^1(\hat \nu)$
(indeed, we have by assumption $\hat \nu((\varphi \circ \tau^+) \rho' ) =
\hat\nu( \varphi \rho' )$
for all $\varphi \in L^1(\hat \nu)$, and we may use
$\widehat \LL \hat \nu = R \hat \nu$). Since
$(\widehat \LL^n \varphi (x))/R^n$ converges to $\rho(x)$ for all
continuous $\varphi$ with $\hat \nu(\varphi)=1$ and
continuous functions
are dense in $L^1(\hat \nu)$ by Lusin's Theorem, we get
(Theorem VIII.5.1 in Dunford-Schwartz [1988]) that
${1\over n}\sum_{k=0}^{n-1} (\widehat \LL^k \varphi (x)/R^k)$ converges
to $\rho(x)$ for all $\varphi \in L^1(\hat \nu)$ with $\hat \nu(\varphi)=1$,
in particular $\rho'=\rho$ as desired.
\qed
\enddemo
\medskip
\remark{Remark 3.4}
The annealed equilibrium state $\mu$
is also a quenched equilibrium state if and only
if $\pi_\mu=\Theta^+$ (if and only
if $Q^{(a)}(\log g)=Q^{(q)}(\log g)$). A {\it necessary} condition for this
is the existence of a probability measure $\nu(dx)$ on $M$
so that $\int_M \LL_\xi {\bold 1} (x)\, \nu(dx)$ is $\theta$-almost everywhere
constant (because $\mu \in \PP_\Theta$ and
$\widehat \LL^* \mu \in \PP_\Theta$, in particular when integrating
functions independent of $x \in M$).
This constancy condition is violated for example if the number
of branches of the $f_\xi$ is constant in $\xi$, and
$g_\xi(x)$ is constant in $x$ but depends (essentially) on $\xi$.
A {\it sufficient} condition ensuring $\pi_\mu=\Theta^+$
is the existence of a probability $\nu(dx)$ on $M$ and
a constant $\lambda>0$
such that $\LL_\xi^*(\nu)=\lambda (\nu)$ for $\theta$ almost all
$\xi \in E$. By definition
of the Jacobian, this property holds with $\lambda=1$ for
$g_\xi=1/|\det D(f_\xi)|$ and
$\nu$ Lebesgue measure, proving Proposition 2.
In Section 4, Remark 4.3, we mention a weaker sufficient
condition.
\endremark
\medskip
\subhead {3.4. Stability of the discrete spectrum and annealed state}
\endsubhead
\smallskip
The stability claims in Theorems 3 and 5 will be a
consequence of the following
proposition and results from Baladi-Young [1993]:
\proclaim{Proposition 3.4}
Consider
a weighted small random perturbation of
$f_0\in C^r_\gamma(M,M)$, $g_0\in C^r(M,\complex)$
given by a family
$\theta_\epsilon$ ($\epsilon \ge 0$)
and write $\widehat \LL_\epsilon$, $\LL_\epsilon$
($\epsilon \ge 0$) for the corresponding transfer operators
acting on $\BB(\alpha)$, respectively $\FF=C^r(M,\complex)$.
Write $R=\exp P(\log |g_0|)$ as usual.
\roster
\item
For any fixed $\psi \in \BB$, $\varphi \in \FF$, $n \ge 1$
$$
\lim_{\epsilon \to 0}
\|\widehat \LL^n_\epsilon \psi -\widehat \LL^n_0 \psi\|_\BB =0\, ,\quad
\lim_{\epsilon \to 0}\|\LL^n_\epsilon \varphi -\LL^n_0 \varphi\|_\FF =0\, .
\tag{3.25}
$$
\item
Let $\bar \gamma < \gamma$ and $\bar \alpha > \alpha$.
Then there is a constant $C > 0$
and an integer $N \ge 0$, so that for all $n \ge N$ there
is $\epsilon(n)$ such that for all $\epsilon < \epsilon(n)$
$$
\eqalign
{
\| \widehat \LL^n_\epsilon - \widehat \LL^n_0 \|_\BB
&\le C R^n \max(\bar \gamma^{-rn}, \bar \alpha^n)\, , \cr
\| \LL^n_\epsilon - \LL^n_0 \|_\FF &\le C R^n \max
(\bar \gamma^{-rn}, \bar\alpha^n) \, .
}\tag{3.26}
$$
\endroster
\endproclaim
\demo{Proof of Proposition 3.4}
\roster
\item
By the triangle inequality it suffices to prove the claims for $n=1$.
To do this, use that each $\theta_\epsilon$ is a probability distribution
and observe that
$$
\eqalign
{
&\lim_{\epsilon \to 0}
\sup_{\xi\omega \in
\text{support} \theta_\epsilon^\natural}
\| \LL_{\xi} \psi_{\xi\omega}- \LL \psi_{\xi\omega}\|_\FF=0 \, ,\cr
&\lim_{\epsilon \to 0}
\sup_{\xi\omega \in
\text{support} \theta_\epsilon^\natural}
\text{Lip}_\omega
D^j( \LL_{\xi} \psi_{\xi\omega}- \LL \psi_{\xi\omega})=0 \, ,
0 \le j \le r \, , \cr
}
\tag{3.27}
$$
(simply apply the Leibniz formula to each term in the finite sums
over inverse branches of the $f_{\xi}$
and use the definition of a small random perturbation).
\item
The argument follows the lines of the proof of Lemma 5 in Baladi-Young [1993]
or Lemma A.1 in Baladi et al. [1995] and is left to the reader.
(We may use that
$\widehat \LL_\epsilon^n\varphi(x,\omega)$ can be written as an integral
over $\theta_\epsilon(d\xi_1) \cdots \theta_\epsilon(d\xi_n)$
of random operators
where the weights $g$ are evaluated at points which depend
on $x$ and $\xi_1, \ldots, \xi_n$ but not on $\omega$.)
\qed
\endroster
\enddemo
\noindent
Theorem 3 \therosteritem{1} is an immediate consequence of Proposition 3.5:
\proclaim{Proposition 3.5}
Consider
a weighted small random perturbation of
$f_0\in C^r_\gamma(M,M)$, $g_0\in C^r(M,\complex)$
given by a family
$\theta_\epsilon$ ($\epsilon \ge 0$),
and write $\widehat \LL_\epsilon$, $\LL_\epsilon$
($\epsilon \ge 0$) for the corresponding transfer operators
acting on $\BB(\alpha)$, respectively $\FF=C^r(M,\complex)$.
Write $R=\exp P(\log |g_0|)$ as usual.
Let $\bar \gamma < \gamma$ and assume that $\alpha < 1/\bar\gamma^r$.
The spectrum of
$\LL_\epsilon$ and that of $\widehat \LL_\epsilon$
outside of
the disc of radius $R /{\bar\gamma}^r$
contains only isolated eigenvalues of finite multiplicity
for small enough $\epsilon$, and both spectra
converge to the spectrum of $\LL_0$ acting on $\FF$
(outside of this disc)
as $\epsilon \to 0$. The corresponding generalised
eigenspaces of $\LL_\epsilon$,
respectively $\widehat \LL_\epsilon$
converge in the $\FF$, respectively $\BB(\alpha)$, topology
to those of $\LL_0$, respectively $\widehat \LL_0$,
and the dual eigenspaces converge in the weak topology.
In particular, for positive weights, the maximal
eigenmeasure $\hat \nu_\epsilon$ of $\widehat \LL_\epsilon$
converges to
$\nu_0 \times \delta^\natural$ with $\delta$ the
Dirac mass at $(f_0,g_0)$.
\endproclaim
\demo{Proof of Proposition 3.5}
The stability of the spectrum and the convergence of the eigenfunctions
(in particular
$\lim_{\epsilon \to 0} \|\rho_\epsilon -\rho_0 \|_r=0$
for positive weights)
follows from Lemma 3
in Baladi-Young [1993] applied to the operators $\LL_\epsilon$, using
the statements in Proposition 3.4 about the operators $\LL_\epsilon$.
Indeed we get from Baladi-Young [1993] that
the spectrum of $\LL_\epsilon$ acting on
$\FF$ and of $\widehat \LL_\epsilon$ acting
on $\BB$ outside
of the disc of radius $R/{\bar \gamma}^r$
both converge to the spectrum of
$\LL_0$ acting on $\FF$ outside of this disc
as $\epsilon$ goes to zero.
The eigenfunctions converge in the $\FF$, respectively
$\BB$ norm.
To get the weak convergence of the eigenfunctionals
it suffices to observe that
the bounds for $\widehat \LL_\epsilon$ in Proposition 3.4
also apply to $\widehat \LL^*_\epsilon$ by
definition of the dual norm. Therefore, Lemma 3 in Baladi-Young [1993] may
also be applied to the family $\widehat \LL^*_\epsilon$,
yielding the desired convergence.
For the final claim,
use $\widehat \LL^*_0 (\nu \times \delta^\natural)
= R (\nu \times \delta^\natural)$ for any maximal
eigenmeasure $\nu$ of $\LL_0$ and the fact that
the multiplicity of the
maximal eigenvalue is constant for small enough $\epsilon$
from Proposition 3.4.
\qed
\enddemo
\medskip
\subhead {3.5. The annealed zeta functions}
\endsubhead
\smallskip
\noindent
Theorem 6 will be a consequence of Proposition 3.5
and the following result of Ruelle:
\proclaim{Theorem (Ruelle [1990, Theorem 1.1, Theorem 1.3])}
Consider a $C^r$ (complex) weighted
$\gamma$-expanding system $(\tau^+, g, \theta)$,
write $\LL$ for the corresponding transfer operator
acting on $\FF=C^r(M,\complex)$, and
let $R=\exp Q^{(a)}(\log |g|)$.
\roster
\item
The zeta function $\zeta^{(a)}(z)$ is analytic in the disc
of radius $R^{-1}$ and admits a zero-free
meromorphic extension
to the disc of radius $R^{-1} \gamma$ where its
poles coincide (including multiplicity) with the
inverses of the eigenvalues of modulus
larger than $R /\gamma$ of $\LL$ acting on $\FF$.
More precisely, if $E$ is fixed
then for any $\delta >0$ and $\bar \gamma < \gamma$
there is a
constant $C(\bar \gamma, \delta)> 0$ which does
not depend on the probability distribution $\theta$ on $E$, so that
if $\lambda_1, \ldots, \lambda_N$
are the eigenvalues of $\LL$ of modulus larger than
$R/\bar \gamma$ then the coefficients $a_n$ in the
expansion
$$
\sum_{n=0}^\infty a_n z^n
:=\log (\zeta^{(a)}(z) \cdot \prod_{i=1}^N (1-\lambda_i^{-1} z))\tag{3.28}
$$
satisfy the uniform bounds
$$
|a_n | \le C \exp (n (Q^{(a)}(\log |g|)+\delta))/ \bar \gamma^{n} \, .
\tag{3.29}
$$
\item
The generalised Fredholm determinant $d^{(a)}(z)$ admits an analytic extension
to the disc of radius $R^{-1}\gamma^r$ where its
zeroes coincide (including multiplicity) with the
inverses of the eigenvalues
of modulus larger than $R /\gamma^r$
of $\LL$.
More precisely, if $E$ is fixed
then for any $\delta >0$ and $\bar \gamma < \gamma$
there is a
constant $C(\bar \gamma, \delta)> 0$ which does
not depend on the probability distribution $\theta$
on $E$, so that
if $\lambda_1, \ldots, \lambda_M$
are the eigenvalues of $\LL$ of modulus larger than
$R /\bar \gamma^r$ then the coefficients $b_n$ in the
expansion
$$
\sum_{n=0}^\infty b_n z^n:=
\log (d^{(a)}(z) / \prod_{i=1}^M (1-\lambda_i^{-1} z))\tag{3.30}
$$
satisfy the uniform bounds
$$
|b_n | \le C
\exp (n (Q^{(a)}(\log |g|)+\delta))/ \bar \gamma^{rn} \, .\tag{3.31}
$$
\endroster
\endproclaim
(Ruelle does not state explicitly the $\theta$-uniform bounds
\thetag{3.29},\thetag{3.31} but they are
easily obtained from his proofs.)
\demo{Proof of Theorem 6}
For each fixed $m \ge 1$ we get by definition
of a small random perturbation that $\zeta^{(a)}_\epsilon(m)$ converges
to $\zeta^{(a)}_0(m)$ and $d^{(a)}_\epsilon(m)$ converges to
$d^{(a)}_0(m)$ as $\epsilon \to 0$.
Moreover, the eigenvalues $\lambda_{i,\epsilon}$ of $\LL_\epsilon$
of modulus larger than $R/\bar \gamma^r$ converge to
the corresponding eigenvalues of $\LL_0$ by Proposition 3.5.
The result is therefore an easy exercise on convergent power series
using the uniform bounds in the theorem of Ruelle stated above.
\qed
\enddemo
\medskip
\subhead 3.6. Integrated annealed correlation functions
\endsubhead
\smallskip
For $\varphi_1, \varphi_2 \in \BB$,
and $\mu=\rho \hat\nu$ the annealed equilibrium state
of a positively weighted i.i.d. expanding map $(\tau^+, g, \theta)$,
writing $R$ for the spectral radius of $\widehat \LL$
on $\BB$,
we get
$$
\int (\varphi_1 \circ (\tau^+)^{n})(x,\omega)
\varphi_2(x,\omega) \rho(x,\omega)
\hat \nu(dx,d\omega)
=
{1\over R^n} \int \varphi_1
\widehat \LL^n (\rho \varphi_2) (x,\omega) \hat \nu(dx,d\omega) \, .
\tag{3.32}
$$
(Just use the fact that $\hat \nu$ is an eigenfunctional
for the dual of $\widehat \LL$ and the eigenvalue
$R$.)
If $C_{\varphi_1 \varphi_2}(n)$ denotes
the correlation function \thetag{2.20}, it follows formally that
$$
\sum_{n\ge 0} e^{in\eta} C_{\varphi_1 \varphi_2}(n) =
\int \biggl [
\bigl (1-({e^{i\eta} \over R})\widehat \LL\bigr )^{-1}(\rho\varphi_2)\biggr ]
(x, \omega)
\varphi_1(x, \omega) \hat \nu(dx,d\omega)\, . \tag{3.33}
$$
Our results on the spectrum of $\widehat \LL$
in Proposition 3.1 give the desired meaning
to \thetag{3.33}.
This proves Theorem 4 \therosteritem{1}
(For $\varphi_1, \varphi_2 \in C^r(M,\complex)$, we may
in fact replace $\widehat \LL$ by $\LL$
and $(\tau^+)^n$ by
$f^{(n)}_\omega$ in \thetag{3.32}
and \thetag{3.33}.)
Finally, Theorem 5 \therosteritem{1} follows from Proposition 3.5,
just as in the proof of Theorem~3.
\medskip
\head {4. The quenched transfer operator $\widehat \MM$}
\endhead
\smallskip
In this section, we restrict again to the case of
positive weights and we
construct
a normalised integrated operator $\widehat \MM$
related to the quenched state.
This will be useful to study the quenched correlation
spectrum and its stability
for small perturbations, in particular to prove
Theorem 3 \therosteritem{2}, Theorem 4 \therosteritem{2},
and Theorem 5 \therosteritem {2}.
Consider first the two-sided situation
$(\tau,g, \theta)$, viewing $g$ as a function on
$M \times \Omega$ depending only on $x$ and $\omega_0$.
Using the notations and
results from Kifer [1992] recalled in \thetag{3.12} above,
i.e., uniquely defined positive
numbers $\lambda^\omega$,
Borel probability measures $\nu^\omega$ on $M$, and functions
$h_\omega : M \to \real$ with $\nu^\omega(h_\omega)=1$, such that
$
\LL_{\omega} h_\omega = \lambda^\omega h_{\sigma\omega}$, and
$\LL_{\omega}^* \nu^{\sigma\omega} = \lambda^\omega \nu^{\omega}$,
we first show:
\proclaim{Proposition 4.1
(Properties of $\lambda^\omega$ and $\nu^\omega$)}
\roster
\item
Since $\LL _\omega = \LL_{\omega_0}$,
the objects $\lambda^\omega$ and $\nu^\omega$ only depend
on $\omega_k$ for $k \ge 0$.
\item
The map $\omega \mapsto \log \lambda^\omega$ is Lipschitz
from $\Omega^+ \to \real^+$ for the metric $d_\alpha$ for any
$\alpha > 1/\gamma$.
\endroster
\endproclaim
\demo{Proof of Proposition 4.1}
The first assertion is a consequence of the proof of
Lemma 2.2 in Kifer [1992].
To prove the second claim,
we use the observations of Kifer [1992, p. 16]
that $\lambda^\omega = \nu^{\sigma\omega} (\LL_\omega (1))$,
and
$$
\nu^\omega(\varphi)=
\lim_{n \to \infty}
\bigl (
{\LL^{n}_\omega \varphi \over \LL^{n}_\omega 1 }
\bigr )\, . \tag{4.1}
$$
Let $1<\bar \gamma < \gamma$ and $\alpha = 1/\bar\gamma$,
we shall prove that there is a constant $C>0$ so that
for any $\omega, \tilde \omega \in \Omega^+$
we have $|\log \lambda^\omega - \log \lambda^{\tilde \omega}|
\le C d_\alpha(\omega, \tilde \omega)$.
We begin with two purely dynamical remarks.
First observe that, by compactness of $E$, there is
$\bar \epsilon(E)$ such that whenever $d_r(f_\omega, f_{\tilde \omega})
< \bar \epsilon$ then $f_\omega$ and $f_{\tilde \omega}$ have the
same degree, and, moreover, for each $x \in M$ the bijection $\Psi$
between $\{ y \mid f_\omega(y)=x\}$ and
$\{ \tilde y \mid f_{\tilde \omega} (\tilde y) = x\}$
can be chosen in such a way that
$$
d_M(y, \Psi(y))\le {d_r(f_\omega, f_{\tilde \omega}) \over \gamma} \, .
\tag{4.2}
$$
Indeed, if $\bar \epsilon$ is small enough we may
chose $\Psi$ so that $y$ and $\Psi(y)$ are in the image
of the same local inverse branch of $f_\omega$, so that
if \thetag{4.2} were violated for some $y$
we would have
$$
\eqalign
{
0=d_M(x,x)&= d_M(f_\omega y, f_{\tilde\omega}(\Psi(y))\cr
&\ge d_M(f_\omega y, f_\omega(\Psi(y)) -
d_M(f_\omega(\Psi(y)),f_{\tilde\omega}(\Psi(y))\cr
&> \gamma \cdot
{d_r(f_\omega, f_{\tilde \omega})\over \gamma} -
d_r(f_\omega, f_{\tilde \omega})=0 \, ,
}\tag{4.3}
$$
a contradiction.
We claim now that
for all $\omega$, $\tilde \omega$,
all $n \ge 1$, and up to exchanging $\omega$ and $\tilde \omega$,
there exists for any point $x \in M$ a surjective map
$$
\Psi_{n,\omega, \tilde \omega,x}
: Y_{n,\omega,x} =\{ y \in M \mid f^{(n)}_\omega (y)= x \}
\to
Y_{n,\tilde \omega,x} =
\{\tilde y\in M \mid
f^{(n)}_{\tilde \omega} (\tilde y) = x \} \, .
\tag{4.4}
$$
(If all $f_\omega$ have the same degree, then the
$\Psi_{n,\omega, \tilde \omega,x}$
are bijections, otherwise, the cardinality of the fibers can
be unbounded as $n \to \infty$.)
Moreover, fixing $\bar \epsilon(E)$ as above,
there is
a constant $C>0$ so that if, in addition
$\delta=d_{1/\bar\gamma} (\omega, \tilde \omega ) \le \bar \epsilon$, then there
is $n(\delta)$ so that for $n \ge n(\delta)$
and any $y\in Y_{n,\omega,x}$, we have
$$
d_M(y,\Psi_{n,\omega, \tilde \omega,x}(y)) < C d_{1/\bar\gamma}
(\omega, \tilde \omega ) \, .\tag{4.5}
$$
To prove \thetag{4.5},
we first note that since
$d_E(\omega_k,\tilde \omega_k) \le \delta \cdot \bar\gamma^k$ for
all $k \ge 0$, there is for any
$\delta\le \bar \epsilon$ an iterate $k_0(\delta)\ge 0$
with $\delta \bar\gamma^{k_0} \le \bar\epsilon
< \delta \bar \gamma^{k_0+1}$
(if $\delta > \bar \epsilon$ we set $k_0=0$). For
any $n\ge n(\delta)= k_0(\delta)$, and any $x \in M$
consider the finite sets $Y=Y_{n-k_0,\sigma^{k_0}\omega,x}$
and $\tilde Y=Y_{n-k_0,\sigma^{k_0}\tilde\omega,x}$,
assuming that $\# Y \ge \#\tilde Y$ (the other case is
symmetric) and choose an arbitrary surjection
$\Psi : Y \to \tilde Y$.
If $\delta > \bar \epsilon$, we are done.
Otherwise,
we fix an arbitrary pair $(y, \tilde y=\Psi(y)) \in Y \times \tilde Y$.
Using the fact that for any $j \ge 1$ and any $u, v$ in $M$
the sets $(f^{(j)}_\omega )^{-1} (u)$ and
$(f^{(j)}_\omega )^{-1} (v)$ are in bijection with the distance
between two paired points not larger than $d_M(u,v)/\gamma^j$
(recall that each $f_\omega$ is $\gamma$-expanding),
using the simplified notation
$d((f^{(j)}_\omega)^{-1}(u), (f^{(j)}_\omega)^{-1}(v))$
to represent the maximum distance between two such paired
points, and defining
$d((f^{(j)}_\omega)^{-1}(u), (f^{(j)}_{\tilde \omega})^{-1}(u))$
analogously, we get
by applying sucessively \thetag{4.2} and recalling
the definition of $k_0$
$$
\eqalign
{
d((f^{(k_0)}_{\omega})^{-1} (y),&(f^{(k_0)}_{\tilde \omega})^{-1}
(\tilde y))\cr
&\le
d((f^{(k_0)}_{\omega})^{-1} (y),(f^{(k_0)}_{ \omega})^{-1} (\tilde y))
+
d((f^{(k_0)}_{\omega})^{-1} (\tilde y),
(f^{(k_0)}_{\tilde \omega})^{-1} (\tilde y))
\cr
&\le
{\text{diam}\, M\over \gamma^{k_0}}
+
d((f^{(k_0-1)}_{\omega})^{-1} (f_{\omega_{k_0}})^{-1}(\tilde y),
(f^{(k_0-1)}_{\omega})^{-1 }(f_{\tilde\omega_{k_0}})^{-1} (\tilde y))\cr
&\qquad\qquad\quad+
d((f^{(k_0-1)}_{\omega})^{-1} (f_{\tilde\omega_{k_0}})^{-1}(\tilde y),
(f^{(k_0-1)}_{\tilde\omega})^{-1}(f_{\tilde\omega_{k_0}})^{-1} (\tilde y))
\cr
&\le \cdots\cr
&\le
{\bar \gamma^{k_0+1} \cdot \text{diam}\, M
\over \gamma^{k_0} \bar \gamma^{k_0+1}}
+
\sum_{j=0}^{k_0-1}{ \bar \epsilon\over \gamma^{k_0-1-j} \cdot \bar \gamma^j}
\cr
&\le
\delta \cdot \bar \gamma \cdot
\biggl ({ \bar\gamma^{k_0}\cdot \text{diam}\, M
\over \bar \epsilon \gamma^{k_0}}
+ {1 \over 1-(\bar \gamma/\gamma)} \biggr )\, ,
}
\tag{4.6}
$$
as claimed.
We now write
$$
\eqalign
{
{\lambda^\omega\over\lambda^{\tilde \omega}}&=
\lim_{n \to \infty}
{\LL^{n}_\omega (1) \cdot \LL_{\sigma \tilde \omega}^{n-1}( 1 )
\over \LL_{\sigma\omega}^{n-1}(1)\cdot \LL_{\tilde \omega}^{n} (1) }\cr
&=
\lim_{n \to \infty} \sup_x
{\displaystyle{\sum_{f^{(n)}_\omega u=x}
\prod_{k=0}^{n-1}} g_{\sigma^k(\omega)}(f^{(k)}_\omega u)
\cdot
\displaystyle{\sum_{f^{(n-1)}_{\sigma \tilde \omega} s=x}
\prod_{k=0}^{n-2}}
g_{\sigma^{k+1}(\tilde \omega)}(f^{(k)}_{\sigma \tilde \omega} s)
\over
{\displaystyle{\sum_{f^{(n-1)}_{\sigma\omega} v=x} }
\prod_{k=0}^{n-2}} g_{\sigma^{k+1}(\omega)}(f^{(k)}_{\sigma \omega} v)\cdot
\displaystyle{\sum_{f^{(n)}_{\tilde \omega} t=x}
\prod_{k=0}^{n-1}} g_{\sigma^k(\tilde \omega)}(f^{(k)}_{\tilde \omega} t)}\cr
&=
\lim_{n \to \infty} \sup_x
{\displaystyle{\sum \Sb f^{(n-1)}_{\sigma \omega} y=x \cr
f^{(n-1)}_{\sigma \tilde \omega} s=x \endSb
\prod_{k=0}^{n-2}} g_{\sigma^{k+1}( \omega)}(f^{(k)}_{\sigma\omega} y)
\cdot
g_{\sigma^{k+1}(\tilde \omega)}(f^{(k)}_{\sigma\tilde\omega} s )
\displaystyle{\sum_{f_{\omega_0} u=y}} g_{\omega_0}(u)
\over
{\displaystyle{\sum \Sb f^{(n-1)}_{\sigma \omega} v=x \cr
f^{(n-1)}_{\sigma \tilde \omega} r=x\endSb
\prod_{k=0}^{n-2}} g_{\sigma^{k+1}( \omega)}(f^{(k)}_{\sigma\omega} v)
\cdot
g_{\sigma^{k+1}(\tilde \omega)}(f^{(k)}_{\sigma\tilde\omega} r)
\displaystyle{\sum_{f_{\tilde \omega_0} t=r}}
g_{\tilde \omega_0}(t)}
} \, . \cr
}\tag{4.7}
$$
If $\delta \le \bar \epsilon$
(the case $\delta > \bar \epsilon$ is simpler since we just
need to bound \thetag{4.7} uniformly from above and below),
we consider the right-hand-side
of \thetag{4.7} for a fixed
$n\ge n(\delta)$ and any $x\in M$, and
assume that the surjection $\Psi_{n,\omega,\tilde \omega}$
is as in \thetag{4.4} (the other case is similar).
It suffices to replace $g_{\omega_0}(u)$
by $g_{\tilde \omega_0} (t) \cdot
(g_{\omega_0}(u)/g_{\tilde \omega_0} (t)) $
(where $t=\Psi_{n,\omega,\tilde \omega}(u)$)
in the numerator, and to use the remark
that $d_M(t,u) < d_{1/\bar\gamma}(\omega, \tilde \omega)$,
the fact that $d_E(\omega_0, \tilde \omega_0) \le \delta$,
as well as
the following trivial inequalities for numbers $a_i, c_i > 0$
with $i\in \II$ finite
$$
\inf_{i\in \II} c_i \le {\sum_{i\in \II} a_i c_i \over \sum_{i\in \II} a_i}
\le \sup_{i \in \II} c_i \, .\qed\tag{4.8}
$$
\enddemo
\remark{Remark 4.1}
Equation \thetag{4.7}
in the proof of Proposition 4.1 also shows
that $\lambda^\omega$ only depends on $\omega_0$ if
$f_{\omega_0}$ is independent of $\omega_0$ (as in the random
Ising model in Section 2.4). In this case, and whenever
$\lambda^\omega= \lambda^{\omega_0}$, it is not difficult
to check by looking
at the proof of Kifer [1992, Proposition 2.5]
that $h_\omega$ only depends on $\omega_i$ for $i < 0$.
\endremark
\remark{Remark 4.2}
\roster
\item
Since
the Lipschitz constant of $\log \lambda^\omega_\epsilon$
is uniform in $\epsilon$ in the case of a small random
perturbation $\theta_\epsilon$, we get
$$
\lim_{\epsilon\to 0}
(\sup_{\omega}
\log \lambda^\omega_\epsilon - \inf_ {\omega} \log \lambda^\omega_\epsilon)
=0\, .
\tag{4.9}
$$
Since $\int\log \lambda^\omega \theta^\natural_\epsilon(d\omega)
= Q^{(q)}_\epsilon (\log g)$
and we know from Kifer [1992, Section 4] that
$ Q^{(q)}_\epsilon (\log g)\to Q^{(q)}_0 (\log g)$ as $\epsilon \to 0$
we find
$$
\lim_{\epsilon \to 0} \sup_\omega
|\log \lambda^\omega_\epsilon- Q^{(q)} (\log g)|=0 \, . \tag{4.10}
$$
\item
In the case of a small random perturbation $\theta_\epsilon$,
we claim that $\log \lambda^\omega_\epsilon$ when viewed as a $C^\beta$
function of $\omega$ (for any $0 < \beta < 1$) has a
H\"older constant which goes to zero as $\epsilon \to 0$.
(This will be useful in Proposition 4.3 below.)
To see this, let $C > 0$ be an upper bound for the Lipschitz constant
of $\log \lambda_\epsilon^\omega$ for $\epsilon < \epsilon_0$, and
observe that for any fixed
$\epsilon$ and any $\omega$, $\tilde \omega$, with
each $\omega_i$, $\tilde \omega_i$ in the support of $\theta_\epsilon$,
we have
$d_\alpha(\omega, \tilde \omega) \le \epsilon/(1-\alpha)$ so that
$$
|\log \lambda_\epsilon^\omega - \log \lambda_\epsilon^{\tilde \omega}|
\le {C \epsilon^{1-\beta} \over (1-\alpha)^{1-\beta}}
\cdot d_\alpha(\omega, \tilde \omega)^\beta \, .\tag{4.11}
$$
\endroster
\endremark
\medskip
We now define the {\it integrated quenched
transfer operator} $\widehat \MM$ acting on bounded functions
$\varphi : M \times \Omega^+\to \complex$ by
$$
(\widehat \MM \varphi) (x,\omega_+) =
\int {1\over \lambda^{\xi \wedge \omega_+}}
( \LL_\xi \varphi_{\xi \wedge \omega_+} ) (x)\, \theta(d\xi) \, .
\tag{4.12}
$$
Observe that the spectral radius of $\widehat \MM$
acting on bounded functions is equal to $1$.
\remark
{Remark 4.3}
There is in general no corresponding operator
$\MM$ acting on functions which only depend on the $x$-variable.
Such a definition exists
if $\lambda^\omega$ only depends on $\omega_0$
(e.g., if the dynamical system
is deterministic but not the weight
see Remark 4.1). This is the case in particular
when $\lambda^\omega$ is constant
(for example in the
SRB case where $\lambda^\omega\equiv 1$,
or for a constant weight whenever the degree of the
$f_\xi$ is constant),
then the operator $\widehat \MM$ is
simply $\widehat \LL$ rescaled by $\lambda=\lambda^\omega$
so that $\mu^{(a)}=\mu^{(q)}$.
\endremark
\smallskip
Since
$\lambda^\omega$ is Lipschitz
for $d_\alpha$ ($\alpha>1/\gamma$)
by Proposition 4.1 \therosteritem{2} we
may consider $\widehat \MM$ as an operator acting on $\BB(\alpha)$
and we find:
\proclaim{Proposition 4.2 (Quasicompacity of $\widehat \MM$)}
The operator $\widehat \MM$ acting
on $\BB(\alpha)$ for $\alpha > 1/\gamma$ has spectral radius equal to $1$
and essential spectral radius strictly smaller than $1$.
The spectral radius is a simple eigenvalue
with a corresponding eigenfunction which coincides with
$$
\tilde \rho_{\omega_0\wedge\omega_+} (x)
= \int h_\omega(x) \, \Theta^-(d\omega_-) \, .
\tag{4.13}
$$
where we use the notations
$\Omega = \Omega^- \times E\times \Omega^+$,
$\omega=\omega_- \wedge \omega_0 \wedge \omega_+$,
and $\Theta^-$ for the marginal of the measure
$\Theta$
on $\Omega^-$.
The corresponding eigenfunctional for $\widehat \MM^*$
is the positive measure $\tilde \nu$ on $M\times \Omega^+$ defined
by
$\tilde \nu(\varphi)
= \int \nu^{\omega_+} (\varphi_{\omega_+}) \, \Theta^+(d\omega_+)$.
The
probability measure
$\tilde \nu \tilde \rho/\tilde \nu(\tilde \rho)$
is the unique relativised quenched equilibrium state for
$\tau^+, g, \theta$.
\endproclaim
Note in particular that $\tilde \rho(x,\omega) \in \BB(\alpha)$.
\demo{Proof of Proposition 4.2}
The claims about the spectral radius and essential spectral
radius follow by the same adaptation of the
results of Ruelle [1990] and
Fried [1995] as in Proposition 3.1.
Since we have assumed that the weight $g$ is positive,
$\widehat \MM$ has a simple
maximal fixed point also when
acting on $L^1(\bar \nu)$ for the unique probability
measure such that $\MM^* \bar \nu=\bar \nu$.
Note that $\rho(x,\omega) \in L^\infty(\bar\nu)
\subset L^1(\bar \nu)$.
We have
$$
\eqalign
{
(\widehat \MM \tilde \rho) (x,\omega_+)
&= \int {1\over \lambda^{\xi \wedge \omega_+}}
\LL_\xi \tilde \rho_{\xi\wedge\omega_+} \theta(d\xi)\cr
&= \int \int
{1\over \lambda^{\xi \wedge \omega_+}}
\LL_\xi h_{\omega_-\cdot \wedge\xi\wedge\omega_+} (x)
\, \Theta^-(d\omega_-) \theta(d\xi)\cr
&= \int h_{\omega_-\wedge\xi \cdot \wedge\omega_+} (x)
\, \Theta^-(d\omega_-) \theta(d\xi)= \tilde \rho(x,\omega_+) \, .
}\tag{4.14}
$$
Also, since
$
\LL_\xi^* \nu_{\omega_+} = \lambda^{\xi\wedge \omega} \nu_{\xi\wedge\omega_+}
$,
we get for $\varphi \in L^1(\tilde \nu)$:
$$
\eqalign
{
\widehat \MM^* \tilde \nu (\varphi) &=
\int \nu^{\omega_+}
( \widehat \MM \varphi_{\omega_+}) \, \Theta^+ (d\omega_+) \cr
&= \int \nu^{\omega_+}\bigl (
\int {1\over \lambda^{\xi\wedge\omega_+} }
\LL_\xi \varphi_{\xi \wedge \omega_+} \, \theta(d\xi) \bigr ) \,
\Theta^+ (d\omega_+) \cr
&= \int \nu^{\xi\wedge \omega_+} (\varphi_{\xi\wedge\omega_+}) \,
\theta(d\xi)\, \Theta^+ (d\omega_+)= \tilde \nu(\varphi) \, ,
}\tag{4.15}
$$
so that $\tilde \nu = \bar \nu$.
Since $\widehat \MM$ is just
the transfer operator associated to the weight
$g_{\omega_0}(x)/\lambda^\omega$,
the same computation
as \thetag{3.6} shows that $\mu$ is
$\tau^+$-invariant
(the fact that the weight
now depends on the full sequence $\omega$ and that the eigenfunction
$\tilde \rho$ depends on $\omega$
play no role there).
Finally, we can check along
the lines of the proof of Proposition 3.2 that the measure
$\tilde \nu \tilde \rho/\tilde \nu(\tilde \rho)$ is a
one-sided quenched relativised equilibrium state.
\qed
\enddemo
When $\lambda^\omega=\lambda^{\omega_0}$ we have
that $\tilde \rho_{\omega}(x)$ only depends on $x$ by
Remark 4.1. In that case,
the marginal on $M$ of the quenched state $\mu^{(q)}=
\tilde \rho\tilde \nu/\tilde \nu(\tilde \rho)$
is equal to $\tilde \rho \bar \nu$ where $\tilde \rho$ and $\bar \nu$
are the maximal eigenfunctions of $\MM$ acting on $C^r(M)$
and its dual $\MM^*$.
Finally, we have the following stability result
which implies Proposition 3 \therosteritem{2}:
\proclaim{Proposition 4.3 (Quenched stability)}
Consider
a positively weighted small random perturbation of
$f_0\in C^r_\gamma(M,M)$, $g_0\in C^r(M,\real^+_*)$
given by a family
$\theta_\epsilon$ ($\epsilon \ge 0$),
and write $\widehat \MM_\epsilon$, $\widehat \LL_0$
($\epsilon \ge 0$) for the corresponding transfer operators
acting on $\BB(\alpha)$ ($\alpha > 1/\gamma$).
Write $R_0=\exp P(\log |g_0|)$ as usual.
The spectrum of
$R_0\widehat \MM_\epsilon$ acting
on $\BB(\alpha)$
contains only isolated eigenvalues of finite multiplicity
outside of
the disc of radius $R_0 /\alpha$
for small enough $\epsilon$, and
converges to the spectrum of $\widehat \LL_0$
(outside of this disc) acting on $\BB$
as $\epsilon \to 0$. The corresponding generalised
eigenspaces of $\widehat \MM_\epsilon$
converge in the $\BB(\alpha)$ topology
to those $\widehat \LL_0$,
and the dual eigenspaces converge in the weak topology.
In particular the maximal
eigenmeasure $\tilde \nu_\epsilon$ converges to
$\nu_0 \times \delta^\natural$ with $\delta$ the
Dirac mass at $(f_0,g_0)$.
\endproclaim
\demo{Proof of Proposition 4.3}
We first claim that for any $0 < \beta < 1$
the operators $\widehat \MM_\epsilon$ ($\epsilon \ge 0$)
have the same spectral
radius when acting on $\BB(\alpha)$ or
$\BB(\alpha,\beta)$ (where $\BB(\alpha,\beta)$ is
obtained by replacing Lipschitz by $\beta$-H\"older in
the definition of
$\BB(\alpha)$), that their essential spectral radius acting
on $\BB(\alpha,\beta)$ is
not bigger than $1/\alpha^\beta$ and that their eigenvalues
of modulus larger than $1/\alpha^\beta$
coincide with those of $\widehat \MM_\epsilon$ acting on $\BB(\alpha)$.
(Analogous properties hold for $\widehat \LL_\epsilon$.)
This is obtained again by adapting the results
of Ruelle [1990] or Fried [1995].
It therefore suffices to show the claim for $\widehat \MM_\epsilon$
and $\widehat \LL_0$ acting
on $\BB(\alpha, \beta)$ for all $0 < \beta < 1$.
Writing $R_\epsilon$ for the
spectral radius of $\widehat \LL_\epsilon$,
we have by definition of $\widehat \MM_\epsilon$
and \thetag{4.9},\thetag{4.11}
that for all $n \ge 1$ and small enough $\epsilon$ and all $n \ge 1$
$$
\eqalign
{
\|R_0^n \widehat \MM_\epsilon^n &
- \widehat \LL_\epsilon^n \|_{\BB(\alpha,\beta)}
\cr
&\le \sup_{(\xi_1 \cdots \xi_n \omega)
\in \text{support} \theta_\epsilon^\natural}
\biggl [ {R_0^n \over \lambda^{\xi_n\omega} \cdots
\lambda^{\xi_1 \cdots \xi_n \omega}} -1
\biggr ]
\| \widehat \LL^n _\epsilon\|_{\BB(\alpha,\beta)}\cr
&
+ \sup_{(\xi_1 \cdots \xi_n \omega)
\in \text{support} \theta_\epsilon^\natural}
\biggl [
{
{R_0^n\over \lambda^{\xi_n\omega} \cdots
\lambda^{\xi_1 \cdots \xi_n \omega}}
\cdot \sum_{j=1}^n\left (
{
\text{H${}^\beta$}_\omega (\lambda^{\xi_j \cdots\xi_n\omega})
\over \lambda^{\xi_j \cdots \xi_n \omega}}\right )
}\biggr ]
\| \widehat \LL^n _\epsilon\|_{\BB(\alpha,\beta)}\cr
&\le c_{n,\epsilon}\, R_\epsilon^n \, ,
}\tag{4.16}
$$
with $c_{n,\epsilon}$ a constant tending to zero when
$\epsilon \to 0$ for each fixed $n$,
and $\text{H${}^\beta$}_\omega(\psi(\omega))$ the $\beta$-H\"older
constant of a function $\psi : \omega^+ \to \real$.
It follows from \thetag{4.16} and Proposition 3.4
(which also holds for $\widehat \LL_\epsilon$
when considering $\BB(\alpha,\beta)$, up
to replacing $\alpha$ by $\alpha^\beta$ in \thetag{3.26})
that
the analogue of Proposition 3.4 holds for $\widehat \MM_\epsilon$, i.e.,
for any fixed $\psi \in \BB(\alpha,\beta)$, and $n \ge 1$
$$
\lim_{\epsilon \to 0}
\|R_0 ^n\widehat \MM^n_\epsilon \psi -
\widehat \LL_0^n \psi\|_{\BB(\alpha,\beta)} =0\, ,
\tag{4.17}
$$
and there is a constant $C > 0$
and an integer $N \ge 0$, so that for all $n \ge N$ there
is $\epsilon(n)$ such that for all $\epsilon < \epsilon(n)$
$$
\| R_0^n \widehat \MM^n_\epsilon - \widehat \LL^n_0 \|_{\BB(\alpha,\beta)}
\le C R^n \alpha^{n\beta}\, .
\tag{4.18}
$$
We may thus use Lemma 3 in Baladi-Young [1993] just as
in the proof of Proposition 3.5.
\qed
\enddemo
\smallskip
We may define correlation functions
associated with the quenched Gibbs state
$\mu^{(q)} = \tilde \rho \tilde \nu$
and test functions in $\BB(\alpha)$
or $\BB(\alpha, \beta)$ ($\alpha > 1/\gamma$ and $0 < \beta < 1$)
and proceed as
in Section 3.6. The relevant spectrum is
that of $\widehat \MM$ acting on $\BB(\alpha)$
and we get Theorem 4 \therosteritem {2} by
Proposition 4.2 and Theorem 5 (2) by Proposition 4.3.
Note finally that a quenched zeta function $\zeta^{(q)}$ may be introduced by
normalising $\zeta_m^{(a)}$ in the definition in \thetag{2.24} through
the $\lambda^\omega$ associated with
$m$-periodic sequences $\omega$. The results of Ruelle [1990] apply again,
relating the discrete spectrum of $\widehat \MM$ and the poles of
$\zeta^{(q)}(z)$.
%\medskip
\newpage
\head {Appendix A}
\endhead
\smallskip
\subhead{Proof of \thetag{3.5}}
\endsubhead
\smallskip
We show here how the proof of Lemma 1 in Fried [1995]
can be adapted to our skew-product situation, trying
to keep close to the notation there.
For $\varphi \in \BB(\alpha)$ we define bounded
functions $\nu_j(x, \omega)$
($j=0, \ldots, r, r+1$)
on $M \times \Omega^+$ by setting
$$
\nu_j(\varphi)( x, \omega)=
\cases
|\varphi(x,\omega)|&j=0\cr
\| D^j_x \varphi(\cdot, \omega)\|&
1 \le j \le r \, ,\cr
\text{Lip}_\omega D^{j-(r+1)}\varphi(x, \cdot)
&r+1 \le j\le 2r+1 \, ,
\endcases
\tag{A.1}
$$
where $\text{Lip}_\omega \psi(x,\cdot)$, for $\psi$
a complex or matrix-valued function, is the
smallest constant $K(x)$ so that
$$
\| \psi(x,\tilde \omega) - \psi(x, \bar \omega)\|
\le K(x) d_\alpha(\tilde \omega, \bar\omega) \, ,
$$
for all $\tilde \omega$, $\bar \omega$ in $\Omega^+$
(where $\| \cdot\|$ denotes complex modulus or matrix norm).
Just like Fried [1995, p. 1063], we find that for all $j$
there are numbers $F_{jk}$ so that
$$
\nu_j\bigl ((\psi^n)^*\varphi(x,\omega) \bigr )
\le \sum_{k=0}^j F_{jk} \nu_k(\varphi)
(\psi^n(x,\omega)) \, , \tag{A.2}
$$
with $F_{jj} = (\gamma^{-j})^n$ for $0\le j \le r$,
$F_{j, j}= \alpha^n (\gamma^{-j+(r+1)})^n$,
for $r+1 \le j \le 2r+1$
and $F_{j\ell} = 0$ for $\ell=0 < j$. It remains
to estimate
$$
\sup_{(x,\omega)\in\text{Im}\, \psi^n}
\nu_j(\varphi - T \varphi)(x,\omega)\tag{A.3}
$$
for functions $\varphi$ with
$\| \varphi\| \le 1$.
Obviously
$$
\eqalign
{
\|D^j \varphi(x,\omega) -
&D^jT(\tilde x ,\tilde \omega)(\varphi)(x,\omega)\|\cr
&\le
\|D^j\varphi(x,\omega) - D^j T(\tilde x ,\omega)(\varphi)\|
+\|D^jT(\tilde x ,\omega)(\varphi) - D^j T(\tilde x ,\tilde \omega)(\varphi)\|
\, .
}
\tag{A.4}
$$
To bound the first term in the right-hand-side of
\thetag{A.4} it is useful to observe that
$M$ may be embedded in euclidean space
such that for any $x,y$ in $M$ there is a piecewise
linear path between them with length bounded by
a uniform constant times $d_M(x,y)$, and such that
the local inverse branches $(f_{\xi})^{-1}_i$ may be
extended to an open neighbourhood of $M$ with
uniformly bounded derivatives (this is a
weakened but sufficient version of assumptions
\therosteritem{1}--\therosteritem{3} in Fried [1995, p. 1062]).
Therefore
the arguments of Fried [1995, p. 1064] yield
$$
\sup_{x,\omega \in \text{Im} \psi^n}
\nu_j((1-T) \varphi) (x,\omega)\le \gamma^{-n(r-j)}+\alpha^n\, ,
$$
for
$0 \le j \le r$
and
$$
\sup_{x \in \text{Im} \psi^n}
\nu_{j} ((1-T)\varphi(x,\omega) ) \le 1 \, ,
$$
for $r+1 \le j \le 2r+1$
so that the proof of \thetag{3.5} may be completed
just as the proof of Lemma 1 in Fried [1995].
\qed
\medskip
\head {Appendix B}
\endhead
\smallskip
\subhead{The non i.i.d. case}
\endsubhead
\smallskip
We use the setup of Section 2.1, except that we {\it do not} assume
that the $\sigma^+$ invariant and mixing
probability measure $\Theta^+$ on $\Omega^+$
is a product measure, and indicate
how our results could be extended.
Since $\Theta^+$ is
$\sigma^+$ invariant, its decomposition on $E \times \Omega^+$
takes the special form
$$
\Theta^+(d\omega) = \theta^{\sigma^+ \omega} (d\omega_0)
\Theta^+(d\omega) \, . \tag {B.1}
$$
We now {\it assume} further that the functionals
$\theta^{\omega}$ are Lipschitz functions of $\omega \in \Omega^+$
for some metric $d_\alpha$.
We then define the annealed integrated
operator $\widehat \LL$ acting on $\BB(\alpha)$
by
$$
(\widehat \LL \varphi )_\omega(x)
=\int_E (\LL_\xi \varphi_{\xi \omega}) (x) \theta^{\omega} (d\xi) \, . \tag{B.2}
$$
There is in general no operator $\LL$ acting on $\FF$ in the non
i.i.d. setting, but our main results (quasicompactness,
annealed zeta function, stability of spectrum, etc.) should hold as
before (see Remark 2.1, note however that
neither the quenched nor the annealed
one-sided SRB state is a product measure
on $M \times \Omega^+$ in general).
The definition of the annealed Gibbs state
is unchanged, and the definition of the annealed equilibrium
state is \thetag{2.10}, with the following formula
for the specific entropy for site
of a $\sigma^+$ invariant measure
$\upsilon$ with respect to the family $\theta$ of
a priori measures $\theta^{\omega} (d\xi)$ on $E$:
$$
h^{\theta}( \upsilon) =
- \int_{\Omega^+} \log \beta(\xi\omega) \, \upsilon(d(\xi\omega))
\tag{B.3}
$$
if $\upsilon(d(\xi\omega))$
is absolutely continuous with respect to
$\theta^\omega(d\xi) \upsilon(d\omega)$, with Radon-Nikodym
derivative $\beta(\xi\omega)$,
and otherwise, $h^{\theta} (\upsilon)= -\infty$.
The operator $\widehat \MM$ can be defined as in
\thetag{4.12}, replacing $\theta(d\xi)$ by $\theta^\omega(d\xi)$
and the results on annealed and quenched states
should hold in this more general setting.
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\enddocument