%% V.~M.~Gundlach, Y. Latushkin
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%% A formula for the essential spectral radius of
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\begin{document}

\title{A formula for the essential
spectral radius of Ruelle's transfer operator on smooth H\"older
spaces}
\author{V.~M.~Gundlach,\, Y. Latushkin}
%\address{Institut fur Dynamische Systeme,
%Universit\"{a}t Bremen, Postfach 330 440, 28334 Bremen, Germany}
%\email{gundlach@@mathematik.uni-bremen.de}
%\author{Y. Latushkin}
\address{Institut fur Dynamische Systeme,
Universit\"{a}t Bremen, Postfach 330 440, 28334 Bremen, Germany;\,
Department of  Mathematics,
University of Missouri-Columbia, Columbia, MO 65211, USA}
%\email{gundlach@@mathematik.uni-bremen.de}
\email{gundlach@@mathematik.uni-bremen.de,\,
mathyl@@mizzou1.missouri.edu}
\thanks{YL was supported by
 the NSF grant DMS-9622105
 and by the Research Board of the University of Missouri}


 \keywords{Transfer operator, random dynamical systems, multiplicative
 ergodic theorem, thermodynamical formalism, wavelets, essential
 spectral radius}
 \subjclass{Primary 58F03, 58F15; Secondary 60J10, 54H20}

 \maketitle

\begin{abstract}
We study the deterministic and random Ruelle transfer operator $\cL$
induced by an expanding map $f$ of a smooth $n$-dimensional
manifold $X$ and a bundle automorphism $\varphi$ of an
$\m$-dimensional vector bundle $E$. We prove the following exact
formula for the essential spectral radius of $\cL$ on the space $\cka$
of $\bk$-times continuously differentiable sections of $E$ with
$\alpha$-H\"{o}lder $\bk$-th derivative:
\[
\ress(\cL;\cka)=\exp\left(\sup_{\nu\in\erg}
\{h_\nu+\lambda_\nu-(\bk+\alpha)\chi_\nu\}
\right).
\]
Here $\erg$ is the set of $f$-ergodic measures, $h_\nu$ is the entropy
of $f$ with respect to $\nu$, $\lambda_\nu$ is the largest
Lyapunov-Oseledets exponent of the cocycle $\varphi^k(x)=\varphi
(f^{k-1}x)\cdot\ldots\cdot\varphi(x)$, and $\chi_\nu$ is the smallest
Lyapunov-Oseledets exponent of the differential $Df^k(x)$, $x\in X$,
$k=1,2,\ldots$. A similar result holds for the random case.
\end{abstract}


\section{Main Results}

\noindent Ruelle's deterministic and random transfer operators
play a significant role in dynamical systems
and statistical physics. A study of their spectral properties is important
in the theory of Gibbs measures \cite{Bo,Si}, hyperbolic
dynamics \cite{PP}, zeta-function \cite{Bal,PP,RuBo,Tan},
thermodynamic formalism \cite{RuComm}, piece-wise monotone
transformations \cite{BK}, Fredholm determinants \cite{RuFD},
wavelets \cite[Sections 3,5]{CD} and \cite{Hols}, and many other
questions.

In the current paper we give exact formulas for the essential spectral
radius of the deterministic and random matrix coefficient transfer
operator $\cL$ induced by an expanding map on the space $\cka$ of
differentiable H\"{o}lder vector-functions. For the scalar case the
spectrum of the transfer operator $\cL$ on the space $C^0$ of
continuous functions is a disc centered at zero with the radius $e^P$,
where $P$ is the topological pressure (see \cite{Bo,PP,Si,Wal}).
Since $\cL$ preserves the cone of positive functions, $\cL$ has a
positive eigenfunction that corresponds to $e^P$, and there is a
corresponding eigenvector (which is the Gibbs measure) for $\cL^*$
(see \cite{Bo}).  When one passes from the space of continuous
to the space of smooth H\"{o}lder functions, the essential spectrum of
$\cL$ ``shrinks'', showing finitely many isolated eigenvalues.  There
are quite a few reasons why one wants to obtain a formula for the
radius of the essential spectrum of the Ruelle transfer operator on
the space $\cka$. We mention the following: (1) the essential spectral
radius $\ress(\cL;\cka)$ gives the exponential rate of decay of
correlations for all but finitely many linearly independent test
functions from $\cka$, see \cite{Liv} for the literature and recent
advances on this topic; (2) the reciprocal to $\ress(\cL;\cka)$ gives
the radius of meromorphy for the weighted $\zeta$-function associated
with the system, see \cite{Bal,Hyadn,PP,RuBo}.

In this paper we consider the deterministic transfer operator in the
following setting that is due to D.~Ruelle \cite{RuComm}.  Let $f$ be
an expanding map (small distances are increased by a factor
$\theta>1$) of a smooth compact $n$-dimensional manifold $X$. We
assume that $f$ is not one-to-one, that is
$N=\text{card}\{f^{-1}\{x\}\}>1$.  Let $E$ be a smooth real
$\m$-dimensional vector bundle over $X$, $E_x\simeq \bbR^\m$ and
$\varphi$ be a smooth bundle automorphism over $f$, that is
$\varphi(x):E_x\to E_{fx}$ and $\text{det }\varphi(x)\neq 0$.  For
$\bk=0,1,\ldots$ and $\alpha\in[0,1]$ let $C^{\bk,0}=C^{\bk}$ denote
the space of $\bk$-times continuously differentiable sections $\Phi$
of $E$, and $C^{\bk,\alpha}$, $0< \alpha\le 1$ denote the space of
$\bk$-times continuously differentiable sections $\Phi$ with $\bk$-th
derivative that satisfies a (global) H\"{o}lder condition with
exponent $\alpha$.  On the space $C^{\bk,\alpha}$, $\alpha\in[0,1]$
consider the matrix coefficient Ruelle transfer operator, $\cL$,
defined as follows:
\[
(\cL\Phi)(x)=\sum_{y\in f^{-1}\{x\}}\varphi(y)\Phi(y),
\quad x\in X,\quad \Phi\in C^{\bk,\alpha}.
\]
Let $\erg=\erg(f,X)$ denote the set of all $f$-invariant ergodic Borel
probability measures on $X$, and $h_\nu$ denote the entropy of $f$
with respect to $\nu\in\erg$.

D.~Ruelle in his celebrated paper \cite{RuComm} proved, in particular,
the following estimate for the essential spectral radius
$\ress(\cL;\cka)$ of the operator $\cL$ on $\cka$:
\begin{equation*}\label{RuEst}
\ress(\cL;\cka)\le\exp\left(\sup_{\nu\in\erg}\{h_\nu+\int\limits_X\log
  \|\varphi(x)\|\,d\nu-(\bk+\alpha)\log\theta\}\right),
\end{equation*}
where $\|\cdot\|$ is the matrix norm, and $\theta$ denotes the
corresponding expanding constant. This result was obtained by showing that
$\log\ress(\cL;\cka)\le P(\log\|\varphi\|)-(\bk+\alpha)\log\theta$
and using the variational principle \cite{Bo,Wal} for the topological pressure
$P(\cdot)$.

In the current paper we prove the following sharp formula for the
essential spectral radius of the matrix transfer operator. For each
$\nu\in\erg$, let $\lambda_\nu$ denote the largest Lyapunov-Oseledets
exponent of the cocycle
$\varphi^k(x)=\varphi(f^{k-1}x)\cdot\ldots\cdot\varphi(x)$, and
$\chi_\nu$ denote the smallest Lyapunov-Oseledets exponent of the
differential $Df^k(x)$, $x\in X$, $k=1,2,\ldots$.
\begin{thm}\label{t1} Assume $f$ is expanding. Then
\begin{equation}\label{ft1}
\ress(\cL;\cka)=\exp\left(\sup_{\nu\in\erg}
\{h_\nu+\lambda_\nu-(\bk+\alpha)\chi_\nu\}
\right).
\end{equation}
\end{thm}
\noindent The proof of Theorem~\ref{t1} involves a refinement of
Ruelle's technique from \cite{RuComm} as well as a further development
of an idea due to J.~Mather \cite{Mather} on localization
along trajectories of $f$ of
``almost-eigenfunctions'' for an associated evolution operator,
see \cite{CL} and \cite{CLMS1}. We sketch the proof in the next
section.

 Theorem~\ref{t1} gives a new formula even for the scalar  case
$\m=1$, $\varphi: X\to\bbR$:
\[\ress(\cL;\cka)=\exp\left(\sup_{\nu\in\erg}\{h_\nu+
\int\limits_X\log|\varphi(x)|\,d\nu-(\bk+\alpha)\chi_\nu\}\right),\]
and for the one-dimensional case $n=1$, $X=S^1$:
\begin{equation}\label{mn1}
\ress(\cL;\cka)=\exp\left(
\sup_{\nu\in\erg}
\{h_{\nu}+\int\limits_X\log|\varphi(x)|\,d\nu
-(\bk+\alpha) \int\limits_X \log|f'(x)|\, d\nu \}\right).
\end{equation}
A formula similar to \eqref{ft1} holds
for the evolution operator
\[(T\Phi)(x)=\varphi(f^{-1}x)\Phi(f^{-1}x),\quad
\Phi\in\cka, \quad f\quad\text{is a diffeomorphism},\]
that is,
\begin{equation}\label{evop}
\ress(T;\cka)=\exp\left(\sup_{\nu\in\erg}
\{\lambda_\nu-(\bk+\alpha)\chi_\nu\}
\right).\end{equation}

For $\bk=\alpha=0$, that is, on the space of continuous functions
$C^0$, and for the {\it spectral radius} $\rsp(\cdot)$ the formula
\begin{equation}\label{sprad}
\rsp(\cL,C^0)=\exp\left(\sup_{\nu\in\erg}
\{h_\nu+\lambda_\nu\}
\right)
\end{equation}
 was obtained in \cite[Theorem~3]{CL} (see also \cite[Theorem~4.15]{LS},
 where another type of transfer operators was considered).
For $\alpha=0$, that is, for the space $C^{\bk}$,
the inequality ``$\le$''  in \eqref{ft1}
for the  essential spectral radius
was proved in \cite[Theorem~2]{CL}; also, see \cite{Tan} for a study
of the transfer operator on $C^{\bk}$.
Among other results, an inequality ``$\le$'',
similar to \eqref{mn1}, was obtained in \cite{CI} for $\m=n=1$.
Formula \eqref{evop} completes the line of research in \cite{AL,Kit,LS} where
various formulas for the spectral radius of $T$ were given; in particular,
it was proved in \cite{LS} that $\rsp(T,C^0)=\exp\left(\sup_{\nu\in\erg}
\{\lambda_\nu\}\right)$.


We also consider the {\it random } Ruelle transfer operator in the
following setting.  Let $(\Omega ,\cF ,\bP ,\vartheta )$ be an
invertible ergodic abstract dynamical system as a model for noise, and
$f$ be a non-degenerate expanding bundle random dynamical system
\cite{Ar,BogTh} on $\cY\subset\Omega\times X$ over $\vartheta$
consisting of random compact $n$-manifolds $\cY_{\omega}\subset X$,
where $X$ is a topological space with Borel $\sigma$-algebra $\cB$. In
particular $f$ is a smooth cocycle over $\vartheta$ with generator
$f(\omega ):\cY_{\omega} \rightarrow \cY_{\vartheta\omega}$,
$\omega\in\Omega$, as it satisfies
$
f^{k+m}(\omega )=f^k(\vartheta^m\omega )\circ f^m(\omega )$
  for all  $m,k\in\Bbb{N}_+$, $\omega\in\Omega$.
It induces a skew-product transformation $\Theta$ on $\cY$ by $\Theta
(\omega ,x)=(\vartheta\omega ,f(\omega )x)$.  We assume that $f$
admits a symbolic description (cf. \cite{BG}), which need not be true
for general expanding random systems (see \cite{Kif}). Moreover we
consider a smooth random $\m$-dimensional real vector bundle $E$ over
$\cY$, i.e.\ the fibres $E_{\omega ,x}$ depend measurably on $(\omega
,x)\in\cY$ and, for fixed $\omega\in\Omega$, $C^\bk$-smoothly on
$x\in\cY_{\omega}$ and they satisfy $E_{\omega ,x}\simeq E_{\Theta
  (\omega ,x)}\simeq\bbR^\m$. We consider smooth random bundle maps
$\varphi$ on $E$, i.e.\ $\varphi (\omega ,x):E_{\omega ,x}\rightarrow
E_{\Theta (\omega ,x)}$ shows the same dependence on $(\omega
,x)\in\cY$ as $E_{\omega ,x}$. We denote the space of such
$C^{\bk,\alpha}$-smooth random bundle maps by $\bL^{0,\bk,\alpha}(\cY
,\cL (E))$. We are only interested in those $\varphi$ satisfying
$\det\varphi (\omega ,x)\neq0$ and
\[
\varphi\in\bL^{1,\bk,\alpha}(\cY ,\cL (E)):=\{ h\in\bL^{0,\bk,\alpha}(\cY
,\cL (E)): \int\| h(\omega ,.)\|_{C^{\bk,\alpha}}
  d\bP (\omega )<\infty\} ,
\]
where we used $\|.\|_{C^{\bk,\alpha}}$ for the $C^{\bk,\alpha}$ operator
norm. Such a $\varphi$ gives rise to a cocycle over
$\Theta$, namely
$\varphi^k(\omega ,x)=\varphi(\Theta^{k-1}(\omega ,x))\cdot \ldots \cdot
\varphi(\omega ,x)$.
Analogous to the spaces of smooth random bundle maps
$\bL^{i,\bk,\alpha}(\cY ,\cL (E))$, $i=0,1$, one can define the spaces
of smooth random sections of $E$. We denote them by
$\bL^{i,\bk,\alpha}(\cY ,E)$ and introduce for
$\varphi\in\bL^{1,\bk,\alpha}(\cY ,\cL (E))$ random matrix coefficient
transfer operators $\cL=\cL (\varphi )$ on $\bL^{0,k,\alpha}(\cY ,E)$
which act $\omega$-wise as $\cL (\omega )$, $\omega\in\Omega$ by
\[
(\cL (\omega )\Phi )(x)=\sum_{y\in f(\omega )^{-1}\{x\}}\varphi
(\omega ,y)\Phi (y),
\]
where $\Phi$ is a smooth section of $E_{\omega}$. Here we used
$f(\omega )^{-1}$ to specify the preimages of $f(\omega )$.
 Note that such a transfer operator $\cL (\omega )$
maps $C^{\bk,\alpha}$ sections of $E_{\omega}$ to $C^{\bk,\alpha}$
sections of $E_{\vartheta\omega}$, as long as $\varphi (\omega )$ is
at least $C^{\bk,\alpha}$. Moreover, it generates a cocycle
$\cL^n(\omega )=\cL (\vartheta^{n-1}\omega )\cdot\ldots \cdot\cL
(\omega )$ for $n\in\bN_+$.
It gives rise to a mapping on $\bL^{0,\bk,\alpha}(\cY ,E)$ by
\[
(\cL\Phi)(\omega ,x)=(\cL (\vartheta^{-1}\omega )\Phi
(\vartheta^{-1}\omega )) (x).
\]
Note that further integrability conditions are needed to guarantee
that this operator is also acting on $\bL^{1,\bk,\alpha}(\cY ,E)$. If,
for example, $N$ denotes the random variable counting the number of
preimages under $f$ and $N(\omega )\|\varphi
(\omega ,.)\|_{C^{\bk,\alpha}}$ is in $\bL^{\infty}$, then it is clear
that $\cL$ acts on $\bL^{1,\bk,\alpha}(\cY ,E)$. In the following we
will make this assumption.

The problem arising in the investigation of the random Ruelle operator
lies in the point spectrum. It is known by the Ruelle-Perron-Frobenius
Theorem (see \cite{PP}) that in the deterministic case there exists a
positive eigenvalue of largest absolute value, while this is in
general not true for the random case (\cite{BG2}). Nevertheless this
special case can be achieved by renormalization such that $1$ becomes
the dominating eigenvalue of $\cL$ (\cite{Gun}). In general one should
look at the limits of the cocycles $\cL^n(\omega )$ to keep the theory
- the formalism and the interpretations - alive. Thus $\omega$-wise
asymptotic growth rates have to be considered. In general these do not
coincide with the asymptotic growth rates for the operator $\cL$ on
$\bL^{1,\bk,\alpha}(\cY ,E)$, but they play the same role in the
random theory as the spectral quantities in the deterministic one. In
particular the role of the essential spectral radius is taken over by
the exponential of the index of compactness (see, e.g. \cite{BG})
which is denoted by $\indc$.
\begin{thm}\label{t2}
Assume that $f$ defines an expanding bundle random dynamical system
that allows a symbolic description and guarantees that $\cL$ defines
 an operator on
$\bL^{1,\bk,\alpha}(\cY ,E)$ provided
 $\varphi\in\bL^{1,\bk,\alpha}(\cY ,E)$. If the additional integrability
condition (\ref{IC}) is satisfied, then
\[
\indc(\cL; \bL^{1,\bk,\alpha}(\cY ,E))=\sup_{\nu\in\erg_{\bP}}
\{h_\nu+\lambda_\nu-(\bk+\alpha)\chi_\nu\} .
\]
\end{thm}
In this theorem $\erg_{\bP}=\erg_{\bP}(f,\cY )$ denotes the set of all
$\Theta$-ergodic probability measures on $(\cY ,\cY\cap \cF\otimes \cB
)$ with marginal $\bP$ on $\Omega$, $\lambda_\nu$ the  largest
Lyapunov-Oseledets exponent of the cocycle $\varphi^k$, $\chi_\nu$
the smallest Lyapunov-Oseledets exponent of the cocycle induced by the
differential of $f^k$ with respect to $x$, and $h_{\nu}$ the (fibre-)
entropy of the random mapping $f$ (see \cite{BogTh}).

{\bf Acknowledgement.} We would like to express our warm thanks to
 Ludwig Arnold for suggestions and discussions. In fact, he suggested
the argument used to prove  equality \eqref{lyapexp} below, which was
crucial for this paper to be written.


\section{Sketch of Proofs}

\noindent The main idea of the proof of
Theorem~\ref{t1} is to relate to $\cL$ an
another transfer operator, $\cK$, induced by a certain
``projectivization'' $F$ of the map $f$ and by a cocycle $\{\psi^k\}$
that depends on $\{\varphi^k\}$, $\bk$ and $\alpha$; see
Theorem~\ref{mainth} below and \cite{Kit2} for the case $N=\m=1$.  The
map $F$ acts on an extension $\cX$ of the manifold $X$ that also
depends on $\bk$ and $\alpha$. The operator $\cK$ is considered on the
space of {\it continuous} sections of a bundle $\cE$ over $\cX$. We
prove that the {\it essential} spectral radius of $\cL$ on the space
of {\it smooth H\"{o}lder} continuous sections of $E$ coincides with
the {\it spectral} radius of $\cK$ on the space of {\it continuous}
sections of $\cE$. We apply formula \eqref{sprad} to compute the
spectral radius of $\cK$.  Then we relate the entropy of $F$ and
the largest Lyapunov-Oseledets exponents of the cocycle $\{\psi^k\}$ to
that of $f$ and the cocycles $\{\varphi^k\}$ and $\{Df^k\}$ to get
\eqref{ft1}. A similar strategy works for the random Ruelle transfer
operator under the additional assumptions which are mainly used to
apply the Pointwise and the Subadditive Ergodic Theorem for
investigations of convergence properties.

We will describe the constructions mentioned above in some more
details. For the sake of simplicity we restrict ourself to the
deterministic case and remark where additional work has to be done in
the random case. Also, we ``pretend'' that $X$ is a subset of
$\bbR^n$, and $E$ is a subset of $X\times\bbR^\m$. Since our
constructions are local, a generalization for the case of nontrivial
bundles is obvious.

For $\bk=0, 1,\ldots$ and $\alpha\in[0,1]$ we define, locally,
$\cX=X\times S^{n-1}\times\ldots\times S^{n-1}$, where we take $\bk$
copies of $S^{n-1}$ if $\alpha=0$ and $\bk+1$ copies if $\alpha>0$.
We denote $\bv=(v_1,\ldots,v_\bk)$ if $\alpha=0$ and
$\bv=(v_0,v_1,\ldots,v_\bk)$ if $\alpha>0$, $\|v_j\|=1$, and let
$(x,\bv)$ denote points in $\cX$.  Globally, one might want to think
of $\cX$ as a bundle over $X$ whose fibers are direct products of
$\bk$ or $\bk+1$ copies of the fibers of the unit tangent bundle at
the corresponding points of $X$.  Using the differential $Df(\cdot)$,
let us define a map $F:\cX\to\cX$ as
\[
F(x,\bv)=\left(fx,\frac{Df(x)v_0}{\|Df(x)v_0\|},
  \frac{Df(x)v_1}{\|Df(x)v_1\|},\ldots,
  \frac{Df(x)v_\bk}{\|Df(x)v_\bk\|}\right)
\]
for $\alpha>0$ and similarly (that is, with no term
$Df(x)v_0/\|Df(x)v_0\|$) for $\alpha=0$. It can be shown that $F$ is
expanding. Let $\cE$ be the
$\m$-dimensional bundle over $\cX$ with fibers $\cE_{(x,\bv)}=E_x$.
Define a bundle automorphism $\psi$ of $\cE$ over $F$ as
\[
\psi(x,\bv)=\frac{\varphi(x)}{\|Df(x)v_0\|^\alpha\cdot
  \prod_{j=1}^\bk \|Df(x)v_j\|}
\]
for $\alpha>0$ and similarly (that
is, with no term $\|Df(x)v_0\|^\alpha$) for $\alpha=0$.  Let $\cC^0$
denote the space of {\it continuous} sections $\Psi$ of $\cE$. On
$\cC^0$ consider the following Ruelle transfer operator $\cK$ induced
by $F$ and $\psi$:
\[
(\cK\Psi)(x,\bv)=\sum_{(y,\bu)\in F^{-1}\{(x,\bv)\}}
\psi(y,\bu)\Psi(y,\bu),\quad (x,\bv)\in\cX.
\]
Let us remark that in the random case this operator becomes
\[
(\cK (\omega )\Psi)(x,\bv)=\sum_{(y,\bu)\in F(\omega )^{-1}\{(x,\bv)\}}
\psi(\omega ,y,\bu)\Psi(y,\bu),\quad (x,\bv)\in\cX (\omega ),
\]
where the $\omega$-versions of the notions in use are obvious. The
integrability condition
\begin{equation}\label{IC}
\log\|\cK (\omega )\|_{\infty}\in\bL^1(\Omega )
\end{equation}
is needed in order to estimate spectral properties with the help of
the Subadditive Ergodic Theorem.

Though the definition of $F$, $\psi$ and $\cK$ are rather cumbersome, their
appearance
in the current context is quite natural. Indeed, assume $\alpha=0$ and consider
the operator ``of differentiation'' $\cD: C^\bk\to \cC^0$ defined as
$(\cD\Phi)(x,\bv)=D\Phi(x)(\bv)$, where $D\Phi$ is the differential.
The following fact was proved in \cite[Lemma 3]{CL}
(see also \cite{Kit2} for the case $N=\m=1$):
\begin{equation}\label{intertv}
\cD\cL\Phi=\cK\cD\Phi+R\Phi,\quad \Phi\in C^\bk,
\end{equation}
where $R$ is a compact operator from $C^\bk$ to $\cC^0$.
To see this fact, one just differentiates $\cL\Phi$, and uses product rule
and chain rule.  In fact, $R\Phi$ contains all terms with low order
derivatives. For example, for $\bk=1$ one has:
\begin{eqnarray*}
(\cD\cL\Phi)(x,v)& = & \sum_{y\in f^{-1}\{x\}}
\phi(y)(D\Phi)(y)[(Df(y))^{-1}(v)]+(R\Phi)(x,v)\\
&=& \sum_{y\in f^{-1}(x)}\phi(y)\cdot\|(Df(y))^{-1}(v)\|
\cdot (D\Phi)(y)\left[
  \frac{(Df(y))^{-1}(v)}{\|(Df(y))^{-1}(v)\|}
  \right]+(R\Phi)(x,v),
  \end{eqnarray*}
which gives the desired result \eqref{intertv}.
\begin{thm}\label{mainth}
Assume $f$ is expanding, $\bk=0,1,\ldots$ and $\alpha\in[0,1]$.
Then
\begin{equation}\label{mf}
\ress(\cL,\cka)=\rsp(\cK,\cC^0).
\end{equation}
\end{thm}
\noindent {\it Sketch of the proof.}
For $\alpha=0$ the proof of the estimate ``$\le$''
in \eqref{mf}  follows (see \cite{CL})
from the intertwining \eqref{intertv}
and general facts from operator theory.
Indeed, $\cD$ has finite dimensional kernel and closed range. Hence,
$\cD$ has a left regularizer $\cD^{-1}$, that is, an operator from
$\cC^0$ to $C^{\bk}$ such that $\cD^{-1}\cD-I$ is a compact operator.
Thus, $\cL-\cD^{-1}\cK\cD$ is compact, and the general formula
\begin{equation}\label{ess}
\ress(\cL)=\lim_{m\to\infty}\inf\{\|\cL^m-R\|^{1/m}: R\quad\text{is
compact}\}
\end{equation}
for the essential spectral radius
does the job. This argument, essentially, was used in \cite{CL}.

Unfortunately, for  $\alpha\in (0,1]$ we do not see such a simple
argument. To prove the estimate ``$\le$''
in \eqref{mf} for any $\alpha\in[0,1]$
 we adopted the technique used by D.~Ruelle in
\cite{RuComm}. The main idea of the proof, as in \cite[p.~249]{RuComm}
is to estimate the norm of $\cL^m-K_{(m)}$, where $K_{(m)}$
are operators of finite rank related to the Taylor expansion of
order $\bk$.

We use symbolic dynamics and adopt notations from \cite{RuComm}. In
particular, fix open sets $U_i$ (see \cite[Proposition 2.3]{RuComm})
and let $ix\in U_i\cap f^{-1}\{x\}$ denote the the $i$-th preimage of
$x$
under $f$. Similarly, we denote $i_1\ldots i_mx\in U_{i_1\ldots i_m}$
a preimage of $x\in X$ under $f^m$ that corresponds to an admissible
$m$-tuple $i^{(m)}:=i_1\ldots i_m$.

Define an operator $\cM^{(m)}\approx \cL^m$ as in formula (4.4) of
\cite{RuComm}.  Note, that $\cM^{(m)}$ is ``almost equivalent'' to the
transfer operator $\cL^m$, that is, $\ress(\cL,\cka)$ can be computed
as on the right-hand side of \eqref{ess} with $\cL^m$ replaced by
$\cM^{(m)}$.  Choose $\bar{x}(i_0\ldots i_m)\in U(i_0\ldots i_m)$,
define $K_{(m)}=\cM^{(m)}{\cal{F}}_\bk$, where ${\cal{F}}_\bk\Phi$ is
the Taylor expansion of $\Phi$ of order $\bk$ at $\bar{x}(i_0\ldots
i_m)$, see \cite[p.  249]{RuComm}. If $\cR$ denotes the remainder of
the Taylor expansion, we need to estimate (cf. formula (4.7) in
\cite{RuComm}) the $C^{\bk,\alpha}$-norm of the function
\begin{equation}
(\cM^{(m)}-K_{(m)})\Phi(x)=
\sum_{i_0\ldots i_{m-1}}\varphi^m(i_0\ldots
i_{m-1}x)\cR(i_0\ldots i_{m-1}x),\quad
x\in U_{i_m}.\label{16.1}
\end{equation}
We note that the function on the right-hand side of \eqref{16.1}
is, in fact, $\cL^m\cR$. We will estimate the norm of this function
in terms of $\rsp(\cK,\cC^0)$.

One can prove (see \cite{CL}) that
\begin{eqnarray*}
\rsp(\cK,\cC^0) & = &\lim_{m\to\infty}\max_{(x,\bv)\in\cX}\left(
\sum_{(y,\bu)\in F^{-m}\{(x,\bv)\}}\|\psi^m(y,\bu)\|\right)^{1/m}\\
&= & \lim_{m\to\infty}\max_{(x,\bv)\in\cX}\left(
\sum_{i^{(m)}}\|\varphi^m(i^{(m)}x)\|\prod_j
\|[Df^m(i^{(m)}x)]^{-1}v_j\|\right)^{1/m}.
\end{eqnarray*}
The main technical estimate that we
need is as follows (we skip its proof).
\begin{lem}\label{MainLemmak} Fix
$k=0,1,\ldots ,\bk$. For each $\epsilon
>$ there exists $C=C(\epsilon)$ such
that for all $m\in \Bbb{N}$ and
$x_1\neq x_2$ one has:
\begin{equation*}\begin{split}
\max_{(v_1,\ldots ,v_k), \|v_j\|=1}\sum_{i^{(m)}}\|
\varphi^m(i^{(m)}x_1)\| & \prod^k_{j=1}
\|[Df^m(i^{(m)}x_1)]^{-1}v_j\|\cdot %\\ &\qquad
\left[\frac{\dist (i^{(m)}x_1,i^{(m)}x_2)}{\dist
(x_1,x_2)}\right]^{\bk-k+\alpha } \\
& \leq C\left(\rsp(\cK,\cC^0)+\epsilon\right)^m,\label{14.1}\end{split}
\end{equation*}
where the sum is taken over all admissible $m$-tuples.\end{lem}

An argument, based on
\eqref{intertv} and Lemma~\ref{MainLemmak}, gives the estimate
\begin{equation}\label{theest}
\|\cL^m\cR\|_{\cka}\le p(m) \const
\|\Phi\|_{\cka}\left(\rsp(\cK,\cC^0)+\epsilon\right)^m,\end{equation}
where $p(m)$ is a polynomial of degree $m$
(cf. formula (4.8) in \cite{RuComm}).
Indeed, for $k=0, 1, \ldots, \bk$
the intertwining \eqref{intertv} gives
$D^k\cL^m\cR=\cK^mD^k\cR+\cR_k$, where
the expression $\cR_k$
contains a variety of terms with the derivatives of $\cR$ up to
the order $k-1$, but does not contain any
$k$-derivatives of ${\cal{R}}$. Now, using Lemma~\ref{MainLemmak},
one can show that
\[\|\cK^mD^k\cR\|_\infty\le \const \|\Phi\|_{\cka}
\left(\rsp(\cK,\cC^0)+\epsilon\right)^m.\]
Since $\cR_k$ contains $p(m)$ terms that satisfy similar estimates,
one has \eqref{theest} for $C^\bk$-norm. Similarly for the H\"{o}lder
norm, and the inequality ``$\le$'' in \eqref{mf} follows.

The main difficulty in the proof of Theorem~\ref{mainth} is the
inequality ``$\ge$'' in \eqref{mf}. To prove this inequality, we
developed an idea of J.~Mather \cite{Mather} on localization of
``almost-eigenfunctions'' of the evolution operator $T$ along
trajectories of $f$, see also the proof of Theorem 4 in \cite{CL}.
Note, that the approximate point spectrum
$\sigma_{\text{ap}}(\cK,\cC^0)$ of the operator $\cK$ on $\cC^0$
contains the whole circle with the radius $\rsp(\cK,\cC^0)$ (see
\cite[Theorem 4]{CL}).  By a simple rescaling, one could assume that
$\rsp(\cK,\cC^0)=1$ and $1\in\sigma_{\text{ap}}(\cK,\cC^0)$.  Thus,
for any small $\epsilon>0$ and big $N\in{\Bbb N}$ there exists
$\Psi\in\cC^0$ of unit norm such that
$\|\cK^N\Psi-\Psi\|_{\cC^0}\le\epsilon$. Using this $\Psi$, we will
construct an ``almost-eigenfunction'' $g_\epsilon$ for $\cL$ on
$\cka$, supported in a finite piece of the $f$-trajectory of a small
ball in $X$.  That is, we will construct $g_\epsilon\in\cka$ such that
\begin{equation}
\|\cL g_\epsilon
-g_\epsilon \|_{\cka}\leq \const\cdot
\epsilon\cdot \|g_\epsilon
\|_{\cka}.\label{32.2}
\end{equation}
This shows that $1\in \sigma_{\text{ap}}(\cL,\cka)$, and the
inequality ``$\ge$'' in \eqref{mf} follows.

For simplicity of the exposition, we give here
the construction of $g_\epsilon$ only for $\bk=0$, that is, for the
H\"{o}lder space $C^{0,\alpha}$; the case $\bk=1,2,\ldots$
requires more technicalities.

Fix a small $\epsilon>0$. Choose  $N=N(\epsilon)$ big enough.
Note, that by Lemma \ref{MainLemmak} with $\rsp(\cK,\cC^0)=1$ there
exists $C=C(\epsilon^2)$ such that for any $x_1\neq x_2$ one has:
\begin{equation}\label{37.0}
\sum_{i^{(2N+1)}}\|\varphi^{2N+1} (i^{(2N+1)}x_1)\|
\left[\frac{\dist (i^{(2N+1)}x_1,
i^{(2N+1)}x_2)}{\dist (x_1,x_2)}\right]^\alpha
\leq C(\epsilon^2)(1+\epsilon ^2)^{2N+1}.
\end{equation}
Choose $\Psi$ as described above and small
$\delta=\delta(N,\epsilon)$.  Fix $(x_0,v_0)\in {\cal{X}}$ such that
$\|\Psi (x_0,v_0)\|\geq 1/2$. Choose a ball $B\ni x_0$ with radius $d$
so small that the components of $f^{k-N}(B)$, $k=0,\ldots,2N+1$, are
disjoint. Choose a bump function $\beta :X\to [0,\delta ]$ such that
$\supp(\beta )\subset \varphi^{-N}(B)$, $\|\beta \|_{C^{0,\alpha}}\leq
1$, $\beta(y_0)=0$ for all $y_0=i^{(N)}x_0\in f^{-N}(x_0)$ and, in
addition,
\begin{equation*}
\lim_{t\to 0}t^{-\alpha }\beta\circ
f^{-N}_{i^{(N)}}(x_0+tv_0)=\|[Df^N(y_0)]^{-1}
v_0\|^\alpha \label{beta5}
\end{equation*}
for each branch $f^{-N}_{i^{(N)}}$ of the inverse map for $f^N$.
Denote
\[
h(y)=\beta (y)\Psi (y_0,u_0), \quad y\in f^{-N}_{i^{(N)}}(B), \quad
y_0=i^{(N)}x_0, \quad u_0=[Df^N(i^{(N)}x_0)]^{-1}v_0.
\]
We define the desired ``almost-eigenfunction'' for $\cL$ as follows
(cf. \cite{Kit2} and \cite{CL}):
$$g_\epsilon (x)=(1-\epsilon
)^{|k-N|}(\cL^kh)(x),\quad x\in
\varphi^{k-N}(B),\quad k=0,\ldots ,2N+1.$$
 Let us remark that this construction
also works in the random case due to the assumption of the existence
of symbolic dynamics for the expanding random system.

We claim that \eqref{32.2} holds for this $g_\epsilon$.
Indeed, the choice of $g_\epsilon$
leads to the estimate
\[
\|g_\epsilon\|_{C^{0,\alpha}}\ge\sup_{0\le t\le d}t^{-\alpha}
\|g_\epsilon(x_0+tv_0)\|  \ge 1/4.\]
A calculation shows that
\begin{equation*}\begin{split}\|g_\epsilon
-\cL g_\epsilon \|_{C^{0,\alpha}}&\leq
(1-\epsilon )^{N+1}\|h\|_{C^{0,\alpha
}}+2\epsilon \|g_\epsilon \|_{C^{0,\alpha}}%\\&
+(1-\epsilon)^N\|
\cL^{2N+1}h
\|_{C^{0,\alpha
}}.\label{40.1}\end{split}\end{equation*}
We note that
\begin{equation*}\|h\|_{C^{0,\alpha}}\leq
\|\beta \|_{C^{0,\alpha}}\|\Psi (y_0,v_0)\|\leq
1\leq 4\|g_\epsilon
\|_{C^{0,\alpha}}.\label{40.2}\end{equation*}
To give an estimate for $\|\cL^{2N+1}h\|_{C^{0,\alpha}}$, we recall that
$\|\cL^{2N+1}h\|_{C^0}\le \delta\|\cL^{2N+1}\|$ due to the choice of
$h$ and $\|\beta\|_{C^0}\le\delta$. To estimate the H\"{o}lder
seminorm
\[\sup\{\|\cL^{2N+1}h(x_1)-\cL^{2N+1}h(x_2)\|
\cdot[\text{dist} (x_1,x_2)]^{-\alpha} : x_1\neq x_2\},\]
we use \eqref{37.0}.
After some calculations one has
\[\|\cL^{2N+1}h\|_{C^{0,\alpha}}\le \delta (C+\|\cL^{2N+1}\|)
+C(\epsilon^2)(1+\epsilon^2)^{2N+1}\]
with a constant $C$,
 which is known as soon as  $\epsilon$ and $N$ are fixed.
Therefore, for sufficiently small $\delta$, one has
\begin{align*}\|g_\epsilon- \cL
g_\epsilon\|_{C^{0,\alpha}}&\leq
4(1-\epsilon)^{N+1}\|g_\epsilon
\|_{C^{0,\alpha}}+2\epsilon\|g_
\epsilon\|_{C^{0,\alpha}}+%\\&
2(1-\epsilon)^NC(\epsilon^2)
(1+\epsilon^2)^{2N+1}\cdot
4\|g_\epsilon\|_{C^{0,\alpha
}}.\end{align*}
This  gives the desired result \eqref{32.2} provided $N=N(\epsilon)$
was chosen large enough.
$\Box$

To derive Theorem~\ref{t1} from Theorem~\ref{mainth}, we apply formula
\eqref{sprad} for the operator $\cK$:
\[
\rsp(\cK,\cC^0)=\exp\left(
  \sup_{\tnu\in\erg(F,\cX)} \{h_\tnu(F)+\tlam_\tnu\}\right),
\]
where $h_\tnu(F)$ is the entropy of $F$ and $\tlam_\tnu$ is the
largest Lyapunov-Oseledets exponent for the cocycle $\{\psi^k\}$ over
$F$ with respect to an $F$-ergodic measure $\tnu$ on $\cX$. Recall,
that by the Multiplicative Ergodic Theorem, for each $\nu\in\erg(f,X)$
there exists a full $\nu$-measure subset $X_\nu$ of $X$ such that for
each point $x$ from this subset there exists a filtration of the
tangent space $T_xX=W^1_x\supset\ldots\supset W^{n'}_x\supset
W^{n'+1}_x=\{ 0\}$, $n'\le n$ such that the {\it exact}
Lyapunov-Oseledets exponent $\chi(x,v)=\lim_{k\rightarrow\infty}
k^{-1}\log\|Df^k(x)v\|$ for the cocycle $\{Df^k\}$ exists and is equal
to $\chi_\nu^i$ if and only if $v\in W^i_x\setminus W^{i+1}_x$. (Note
that here we had to use the one-sided version of the Multiplicative
Ergodic Theorem, as the linear cocycle $\{Df^k\}$ is defined only over
a non-invertible skew-product transformation $\Theta$.) Let
$E^i_x=S^{n-1}\cap W^i_x$, $i=1,\ldots,n'$ be the trace of the Oseledets
subbundle $W^i$ on the unit sphere in the tangent space $T_xX$.

To express the Lyapunov-Oseledets exponents $\tlam_\tnu$ in terms of
$\lambda_\nu$ and $\chi_\nu$, we will use a description of all
measures $\tnu$ on $\cX$, that are $F$-ergodic for the
projectivization $F$ of $f$ (see \cite[Chapter~6]{Ar}).
This description says that
every $\tnu\in\erg(F,\cX)$ has a form
$\tnu=\nu\times\mu_0\times\mu_1\times\ldots\times\mu_\bk$ (for
$\alpha>0$, and with no term $\mu_0$ for $\alpha=0$) where $\mu_k$ are
supported on the Oseledets subbundles $E^i$.
We claim, that
\begin{equation}\label{lyapexp}
\sup_{\tnu\in\erg(F,\cX)}\{h_\tnu(F)+\tlam_\tnu\}=
\sup_{\nu\in\erg(f,X)}\{h_\nu(f)+\lambda_\nu-
(\bk+\alpha)\chi_\nu\},
\end{equation}
where $\lambda_\nu$ is the largest Lyapunov-Oseledets exponent
for the cocycle $\{\varphi^k\}$, and $\chi_\nu=\chi^{n'}_\nu$ is the smallest
Lyapunov-Oseledets exponent for the cocycle $\{Df^k\}$.

To prove the inequality ``$\le$'' in \eqref{lyapexp},
 fix any $\tnu\in\erg(F,\cX)$.
Note (see, e.g., \cite{LS}, Lemma~4.14)
that $h_\nu(f)=h_\tnu(F)$ for $\nu=\text{ proj }\tnu$, the projection of the
measure $\tnu$ on $X$. Pick a point $(x,\bv)\in\cX_\tnu$ such
that there exists the exact
Lyapunov-Oseledets exponent $\tlam_\tnu=\lim k^{-1}\log\|\psi^k(x,\bv)\|$
for the cocycle $\{\psi^k\}$, and, at the
same time, $x\in X_\nu$, that is, there exist
 the exact Lyapunov-Oseledets exponents
$\lambda_\nu=\lim k^{-1}\log\|\varphi^k(x)\|$ for $\{\varphi^k\}$
and $\chi_\nu(x,v_j)$ for $\{Df^k\}$ . Then
\begin{eqnarray}
\tlam_\tnu & = & \lim_{k\to\infty}k^{-1}\log\|\varphi^k(x)\|
 % \\ & - &
- \sum_{j=1}^\bk\lim_{k\to\infty}k^{-1}\log\|Df^k(x)v_j\|-
\alpha\lim_{k\to\infty}k^{-1}\log\|Df^k(x)v_0\| \nonumber \\
&\le &\lambda_\nu-(\bk+\alpha)\chi_\nu,
\quad \bv=(v_0,v_1,\ldots,v_\bk), \label{Neq}
\end{eqnarray}
since $\chi_\nu$ is the {\it smallest} Lyapunov-Oseledets exponent
for $\{Df^k\}$.

To prove the inequality ``$\ge$'' in \eqref{lyapexp}, fix any
$\nu\in\erg(f,X)$.  Pick a measure $\mu$ supported on $E^{n'}$, the
trace of the Oseledets subbundle corresponding to the {\it smallest}
exponent $\chi_\nu=\chi^{n'}_\nu$. Form
$\tnu=\nu\times\mu\times\ldots\times\mu$.  Then $h_\tnu(F)=h_\nu(f)$
as before.  Also, for any $(x,\bv)\in\cX_\tnu$, $\tnu(\cX_\tnu)=1$
where the exact limits $\tlam_\tnu$, $\lambda_\nu$, $\chi_\nu(x,v_j)$
exist, one has the {\it equality} in \eqref{Neq}, since each of the
vectors $v_j$ is a vector from $E^{n'}_x$.  This concludes the proof
of Theorem~\ref{t1}.









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\end{document}