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\title {Convergence of the transfer operator for rational maps}
\date{}
\author{Nicolai Haydn \thanks{Mathematics Department, University of Southern California,
Los Angeles, 90089-1113. Email:$<$nhaydn@mtha.usc.edu$>$.}}
\maketitle
\noindent{\bf Abstract:}
We prove that the transfer operator for a general class of
rational maps converges exponentially fast in the supremum norm
and in H\"{o}lder norms for small enough H\"{o}lder exponents
to its principal eigendirection.
\section{Introduction}
Let $T: \mbox{\bf C} \rightarrow \mbox{\bf C}$ be a rational map of degree
$d \geq 2$, and denote by $J$ its Julia set. If $f: J \rightarrow \mbox{\bf R}$
is a continuous function, then we would like to consider the action of the
associated transfer operator ${\cal L}_f$. It is well known that for real $f$ the
operator ${\cal L}_f$ has a largest simple eigenvalue whose associated
eigenfunction and eigenfunctional define an invariant measure $\mu$ on $J$
which is conformal with respect to $P(f)-f$, where $P(f)$ is the pressure of
$f$. If the function $f$ is H\"{o}lder continuous and satisfies the condition
$P(f) - f > 0$, then it was shown \cite{DU1} that $\mu$ is in 'fact the
equilibrium state for $f$, that is it realises the maximum in the variational
principle
$$
P(f) = \sup_{\nu} (h(\nu) + \mu(f)),
$$
where the supremum is over all $T$-invariant probability measures
$\nu$ on $J$, and $h(\nu)$ denotes the metric entropy of $\nu$.
For $f: J \rightarrow \mbox{\bf R}$, one defines the transfer operator
${\cal L}$ by
$$
{\cal L}\phi(x) = \sum_{y \in T^{-1}x} e^{f(y)}\phi(y),
$$
where $\phi$ are functions on $J$ and $x \in J$. In order to use the
euclidean metric on $\mbox{\bf C}$ (rather than the spherical metric on
$\bar{\mbox{\bf C}}$) let us assume that
$\infty \not\in J$, and denote by $C^{\alpha}(J),
\alpha > 0$, the H\"{o}lder continuous functions on $J$ with
H\"{o}lder exponent $\alpha$, that is, if $f \in C^{\alpha}(J)$ then
there exists a smallest constant $|f|_{\alpha}$ so that
$|f(x) - f(y)| \leq |f|_{\alpha} |x-y|^{\alpha}$, for all $x,y \in J$.
If we denote by $|f|_{\infty}$ the supremum norm on $J$, then
the natural norm on $C^{\alpha}(J)$ is given by
$\|\cdot\|_{\alpha} = |\cdot|_{\alpha} + |\cdot|_{\infty}$.
The main result in this paper is to show that ${\cal L}$ contracts to the
eigenspace spanned by $\mu$ exponentially fast in the supremum norm
(theorem \ref{contraction}) and in the H\"{o}lder norm for small enough
H\"{o}lder exponents $\kappa<\alpha$ (theorem \ref{holder.contraction}).
It was previously shown in \cite{DPU} that the contraction is subexponential
at the rate $\vartheta^{\sqrt{n}}$ for some $\vartheta < 1$,
where $n$ is the number of times the
operator ${\cal L}$ is iterated. Clearly, in general we cannot expect the
convergence to be exponential in the $C^{\alpha}$-norm, since this
remains the privilege of the case when the map $T$ is (uniformly)
hyperbolic in which case the operator
${\cal L}: C^{\alpha} \rightarrow C^{\alpha}$ has a
spectral gap which is essential to effect exponential convergence
in $C^{\alpha}$.
Quasicompactness of the transfer operator acting on Sobolev spaces
has been shown by Smirnov \cite{S}.
These results were made possible by many converstations I had with
M. Urbanski. Work on this paper was begun while visiting the SFB 170 at
the University of G\"{o}ttingen.
\section{Inverse branches} %%%%%%%%%%%%%%%%%%%% Section 2
\noindent We shall need the following theorem (where $|{\cal S}|$ denotes the cardinality
of the finite set ${\cal S}$).
\begin{theorem} \label{Branches} Let $J$ not contain any critical periodic point
of $T$ and let $0 < \lambda < 1$.
Then there exist $\varepsilon>0$, $\eta \in (0,1)$, a sequence of
simply connected regions $\Omega_n, n \in \mbox{\bf N}$, and
a disjoint decomposition of the inverse branches of $T^n$ on $\Omega_n$ into
two subsets ${\cal S}'_n = {\cal S}'_n(\lambda)$ and
${\cal S}''_n = {\cal S}''_n(\lambda)$ so that
\noindent (a) $|{\cal S}''_n| \leq r \lambda^{-n}, n \in \mbox{\bf N}$, for
some constant $r$.
\noindent (b) $|\varphi'(z)|\leq\eta^n$ for $z\in\Omega_n$ and
in particular $\mbox{\rm diam}(\varphi(\Omega_n)) \leq \eta^n,
\forall \varphi \in {\cal S}'_n, n \in \mbox{\bf N}$.
\noindent (c) $\mbox{\rm dist}(z, \Omega_n) \leq C_1 e^{-n\varepsilon}$ for all
$z \in J, n\in\mbox{\bf N}$, for some constant $C_1>0$.
\end{theorem}
\noindent The following lemma serves to obtain in lemma \ref{boundary} an
estimate on the boundary behaviour of the Riemann map for a polygon which then
will be used to get uniform bounds on the K\"{o}be constant of polygonal
regions (lemma \ref{Distortion}).
\begin{lemma} \label{Arithmetic}
Let $g(z)$ be analytic in some simply connected
region $\Omega$ and $x_0 \in \partial\Omega$. Moreover, let
$h(z) = (g(z) - g(x_0))^{1/\gamma}$ (with suitable branch cuts),
where $0 < \gamma < 2$. Then
there exists a constant $C_2 \geq 1$, so that
$$
|g(z) - g(x)| \leq C_2 |h(z) - h(x)|^{\gamma}.
$$
\end{lemma}
\noindent {\bf Proof.} We consider two cases, namely $x = x_0$ and
$x \not= x_0$.
\noindent (I) Let us assume that $x = x_0$. Then we have
$|g(z) - g(x)| = |h(z)|^{\gamma} = |h(z) - h(x)|^{\gamma}$,
as $h(x) = h(x_0) = 0$.
\noindent (II) Let us now assume that $x \not= x_0$ and put
$d = |h(x) - h(x_0)| = |h(x)|$. Let us first consider the case when
$|h(z) - h(x)| \leq d/2$, that is when $h(z)| \geq d/2$. We then
obtain that
\begin{eqnarray}
|g(z) - g(x)| & \leq & \int_{h(z)}^{h(x)} |\frac{dh^{\gamma}}{dh}| \, |dh|
\nonumber \\
& \leq & \gamma \max_{\zeta\in [z,x]} |h(\zeta)|^{\gamma - 1} |h(z) - h(x)|
\nonumber \\
& \leq & \gamma 3^{\gamma} 2^{1-\gamma} |h(z) - h(x)|^{\gamma}. \nonumber
\end{eqnarray}
In the case $|h(z) - h(x)| > d/2$ we get
\begin{eqnarray}
|g(z) - g(x)| &
\leq & |h(z)^{\gamma} - h(x_0)^{\gamma}| + |h(x_0)^{\gamma} - h(x)^{\gamma}|
\nonumber \\
& \leq & |h(z)|^{\gamma} + d^{\gamma} \nonumber \\
& \leq & 5^{\gamma} |h(z) - h(x)|^{\gamma}, \nonumber
\end{eqnarray}
where we used that $|h(z)| \leq d + |h(z) - h(x)| \leq 3 |h(z) - h(x)|$.
Thus the lemma is proven, if we put
$C_2 = \max(5^{\gamma}, \gamma 3^{\gamma} 2^{1-\gamma})$. \hfill$\Box$
\begin{lemma} \label{boundary}
Let $\gamma > 0$ and $N \geq 3$. Then there exists a constant $C_3$
(depending on $\gamma$ and $N$) so that for every polygon
$P$ with at most $N$ vertices and interior
angles $\geq \pi \gamma > 0$ the following estimate holds:
$$
\mbox{\rm dist} (f(z), \partial P) \leq C_3 (1 - |z|)^{\gamma^N}.
$$
where $f: D \rightarrow P$ is the
Riemann map from the unit disc $D$ to $P$.
\end{lemma}
\noindent {\bf Proof.} Let $x_j \in \partial D, j = 1,2,\dots,N,$ be the
prevertices of the polygon $P$ (assume $0 \in P$) under the map $f$. We can assume that
$\mbox{Arg}(x_j) < \mbox{Arg}(x_{j+1})$ for all $j = 1,\dots,N$
($x_{N+1} = x_1$). For every $j$, the map $f$ has by the Schwarz reflection
principle an analytic continuation to the slit plane $\mbox{\bf C} \setminus S_j$,
which it maps conformally to $P$ and its reflection along the edge
$f(S_j)$ which lie in a branched cover of the complex plane,
where $S_j \subset \partial D$ is the arcsegment whose endpoints are the
prevertices $x_j, x_{j+1}$ and which does not contain any other prevertices.
Let $x \in \partial D$ and $B_{\delta}(x)$ be the ball of radius centred at
$x$. Then $f(B_{\delta}(x)) \subset B_{2p}(0)$, $\delta < 1$, where
$p = \mbox{diam} (P)$.
Without loss of generality we can assume that $x_j, j = 1,\dots,M, M < N$ are
those prevertices of $P$ which lie in the ball $B_{\delta}(x)$.
Put $g_0(z) = f(z)$ and define inductively
$$
g_j(z) = (g_{j-1}(z) - g_{j-1}(x_j))^{1/\gamma_j}, j = 1,\dots,M,
$$
where $\pi\gamma_j$ is the interior angle at the vertex $f(x_j)$.
These maps are analytic in $B_{\delta}(x) \cap D$ and $g_M$ can by the
Schwarz reflection principle holomorphically be extended to the entire
ball $B_{\delta}(x)$. From the estimates
$$
\sup_{B_{\delta}(x) \cap D} |g_j(z)| \leq
2^{1/\gamma_j} \sup_{B_{\delta}(x) \cap D} |g_{j-1}(z)|^{1/\gamma_j}
$$
$j = 1,\dots,M$, and the fact that
$\sup_{B_{\delta}(x) \cap D} |g_0(z)| \leq p$ we conclude that
$$
\sup_{B_{\delta}(x) \cap D} |g_M(z)| \leq (2^M p)^{1/\gamma^M}
\leq (2^N p)^{\gamma^{-N}}.
$$
Since by the reflection principle $g_M$ can analytically be extended to
the entire ball $B_{\delta}(x)$ we obtain
$\sup_{B_{\delta}(x)} |g_M(z)| \leq 2 (2^N p)^{\gamma^{-N}}$. For
$z \in B_{\delta/2}(x)$ a Cauchy estimate provides $|g_M'(z)| \leq c_1$,
where $c_1 = 4 (2^N p)^{\gamma^{-N}}$,
and therefore $|g_M(z) - g_M(x)| \leq c_1 |x - z|$.
A repeated application of lemma \ref{Arithmetic} yields the estimate
$$
|g_0(z) - g_0(x)| \leq (C_2^N c_1)^{-\gamma'} |x - z|^{\gamma'},
$$
where ${\gamma'} = \gamma_1\gamma_2\cdots\gamma_M \geq \gamma^N$.
This proves the lemma with $C_3 = (C_2^N c_1)^{-\gamma^N}$.\hfill$\Box$
\begin{lemma} \label{Kobe} (K\"{o}be distortion theorem \cite{Hille})
A univalent function $g$ on the unitdisc satisfies:
$$
\left(\frac{\delta'}{2 - \delta'} \right)^4
\leq \left| \frac{g'(w)}{g'(w')} \right|
\leq \left(\frac{2 - \delta'}{\delta'}\right)^4
$$
for all $|w|, |w'| < 1 - \delta'$, where $\delta' > 0$.
\end{lemma}
\begin{lemma} \label{Distortion} Let $P$ (as above) be a polygon with
$\leq N$ vertices
whose interior angles are $\geq \pi \gamma > 0$, and let $\varphi$ be a
univalent function on $P$. Then for every (small) $\delta > 0$ we
have that
$$
\frac{1}{256}\left(\frac{\delta}{C_3}\right)^{8/\gamma^N}
\leq \left|\frac{\varphi'(z)}{\varphi'(z')}\right|
\leq 256 \left(\frac{C_3}{\delta}\right)^{8/\gamma^N}
$$
for all $z, z' \in P$ for which
$\mbox{\rm dist}(z,\partial P), \mbox{\rm dist}(z',\partial P) \geq \delta$,
where $C_3$ is a constant (determined by lemma \ref{boundary}).
\end{lemma}
\noindent {\bf Proof.} Let $f: D \rightarrow P$ be the Riemann map and
put $\delta' = (\delta/C_3)^{1/\gamma^N}$. This ensures that
$$
P \setminus B_{\delta}(\partial P) \subset f(B_{1-\delta'}(0)).
$$
As $g = \varphi \circ f$ is univalent on the unitdisc $D$ it satisfies
the K\"{o}be distortion inequality
$$
\frac{\delta'^4}{16} \leq \left|\frac{g'(w)}{g'(w')}\right|
\leq \frac{16}{\delta'^4}
$$
for all $w, w' \in B_{1-\delta'}(0)$. Since $g'(w) = \varphi'(z) f'(w)$,
where $z = f(w) \in P$, we obtain
$$\left|\frac{\varphi'(z)}{\varphi'(z')}\right|
= \left|\frac{g'(w)}{g'(w')}\right|
\times \left|\frac{f'(w')}{f'(w)}\right|,
$$
and, since also $f$ satisfies the distortion estimate of
lemma \ref{Kobe}, we obtain
$$
\frac{\delta'^8}{256} \leq \left|\frac{\varphi'(z)}{\varphi'(z')}\right|
\leq \frac{256}{\delta'^8},
$$
for all $z, z' \in P \setminus B_{\delta}(\partial P)$. The estimate in the lemma
now follows readily. \hfill$\Box$
\vspace{3mm}
\noindent {\bf Proof of theorem \ref{Branches}.} Denote by $\Lambda_n$ the critical
values of $T^n$ (critical points of the inverse maps) and let $U$ be a
topological disc in $\mbox{\bf C}$ which avoids all periodic critical
points of $T$. We shall assume that $U$ is a finite region. Denote
by $\pi_x$ the projection of $\mbox{\bf C}$ onto the real axis ($x$-axis)
and by $\pi_y$ the projection onto the imaginary axis ($y$-axis).
Let (without counting muliplicities) $\{x_j: j = 1,2,\dots \}$
be the (finite) set $\pi_x\Lambda_n$,
labelled in such a way that $x_1 < x_2 < \cdots$.
Put $a_j = x_{j+1} - x_j, j = 1,2,\dots$.
Let $\lambda \in (0,1)$ put $\delta = \frac{-\log \lambda}{1536}$
and choose $\varepsilon \in(0,\delta)$. Put
$\varepsilon_n = e^{-\varepsilon n}$ and $\delta_n = e^{-\delta n}$.
Let us assume that $\sum_j a_j > dn^2\varepsilon_n$ (the points in
$\pi_x\Lambda_n$ are spread out). In this case we can find a gap $\tilde{a} > 0$
so that $|\tilde{a} - a_j| > \varepsilon_n \forall j$. Since
$|\Lambda_n| \leq dn$, we can choose $\tilde{a} \leq n\varepsilon_n$.
We say two points
$x_j, x_{j+1}$ are close if
$a_j = x_{j+1} - x_J < \tilde{a} - \varepsilon_n$ and are apart
if $a_j > \tilde{a} + \varepsilon_n$. A set of points
$\{x_j: j = k, k+1, \dots, k+k_0\} \subset \pi_x\Lambda_n$ (some $k, k_0$)
form a {\em cluster}
if $a_j < \tilde{a} - \varepsilon_n$ for $j = k,\dots,k+k_0-1$,
and $a_{j-s}, a{k+k_0} > \tilde{a} + \varepsilon_n$.
The set $\pi_x\Lambda_n$ consists of a finite number of (disjoint)
clusters. In the case when $\sum_j a_j \leq dn^2\varepsilon_n$ (the points in
$\pi_x\Lambda_n$ are closely packed) the entire set $\pi_x\Lambda_n$
might form a single cluster.
We shall now do `branch cuts' to remove all the critical values
$\Lambda_n$ in $U$ by cutting out a cluster at a time.
For large enough $n$ there exists a number $\tilde{b} \in (0,1)$
so that $|\pi_y(z) - \tilde{b}| > \varepsilon_n$ for every point
$z \in \Lambda_n$. Let $\Gamma$ be a cluster of critical values.
If $\pi_y\Gamma \cap (-\infty, \tilde{b}) \not= \emptyset$ then
we do a rectangular cut whose short side is parallel to the $x$-axis
at hight $\max_{z \in \Gamma, \pi_y(z) < \tilde{b}} \pi_y(z)
\leq \tilde{b} - \varepsilon_n$ and whose long sides are parallel
to the $y$-axis, extend all the way in negative $y$-direction
to the boundary of $U$ and are made to enclose the portion of the cluster
$\Gamma$ which lies in the halfplane $\{(x,y): y < \tilde{b}\}$.
To remove the portion of the cluster $\Gamma$ which lies in the
halfplane $\{(x,y): y > \tilde{b}\}$ we do a similar rectangular
cut whose short side has the $y$-coordinate
$\min_{z \in \Gamma, \pi_y(z) > \tilde{b}} \pi_y(z)
\geq \tilde{b} + \varepsilon_n$ and whose long sides reach in the
positive $y$-direction to the boundary $\partial U$.
Notice that the widths of these `cuts' are
$\leq |\Lambda_n|\tilde{a} \leq dn^2\varepsilon_n$
(the absolute values denote the cardinality of the finite set).
In this way we remove every cluster of critical values and denote
by $U_n$ the region $U$ minus the rectangular cuts. It has
the property that any two points $z, z' \in U_n$ which satisfy
$\mbox{dist}(z,\partial U_n), \mbox{dist}(z',\partial U_n)
\geq \delta_n$ can be connected by a polygonal path $\cal C$
with at most three straight lines parallel to the coordinate axes
and which lies inside a polygonal region $P \subset U_n$
whose edges are parallel to the coordinate axies and satisfies
$\mbox{dist}({\cal C}, \partial P) \geq \frac{1}{\sqrt{2}}\delta_n$.
Therefore, if
$\varphi$ is a univalent function of $U_n$, then by lemma \ref{Distortion}
$$
K_n^{-1} \leq \left|\frac{\varphi'(z)}{\varphi'(z')}\right|
\leq K_n,
$$
where $K_n = 2^{200}\, C_2^{384} \, \delta_n^{-384}$
(with $N \leq 6$ and $\gamma \geq 1/2$), for
$z, z' \in U_n \setminus B_{\delta_n}(\partial U_n)$.
Put
$\Omega_n = T(U_n \setminus B_{\delta_n}(\partial U_n))$.
To construct the inverse branches of $T^n$ on $\Omega_n$ we shall follow
\cite{DU1}. Let ${\cal S}_n$ be the (univalent)
inverse branches of $T^n$ on $\Omega_n$ and denote by $|V|$ the
normalised spherical measure of measurable set $V \subset \mbox{\bf C}$.
Define
${\cal S}''_n = \{\varphi \in {\cal S}_n: |\varphi(\Omega_n)| > \lambda^n\}$
and ${\cal S}'_n = {\cal S}_n \setminus {\cal S}''_n$ (for the set of `good'
branches). Since the multiplicity of the family
$\{\varphi(\Omega_n): \varphi \in {\cal S}_n\}$ is uniformly in $n$ bounded
by some number $r$ \cite{DU} (that is for every point $z \in \mbox{\bf C}$
there are at most $r$ branches $\varphi \in {\cal S}_n$ so that
$z \in \varphi(\Omega_n)$), we obtain $|{\cal S}''_n| \leq r\lambda^{-n}$.
This proves part (a) of the theorem.
To estimate the diameter of the sets $\varphi(\Omega_n)$ for
$\varphi \in {\cal S}'_n$ from their spherical measures we proceed
as follows:
\begin{eqnarray}
|\varphi(\Omega_n)| & \nonumber
\geq & |\Omega_n| \inf_{z\in \Omega_n} |\varphi'(z)|^2 \\
& \geq & K_n^{-2}|\Omega_n|
\sup_{z\in \Omega_n} |\varphi'(z)|^2. \nonumber
\end{eqnarray}
Together with the assumption $|\varphi(\Omega_n)| \leq \lambda^n$,
this yields ($c_1 > 0$)
\begin{eqnarray}
\sup_{z\in \Omega_n} |\varphi'(z)| \nonumber
& \leq & K_n \, \sqrt{\frac{|\varphi(\Omega_n)|}{|\Omega_n|}} \\
&\leq & c_1 \, \lambda^{n/2} \delta_n^{- 384} \nonumber.
\end{eqnarray}
Since $\delta = \frac{-\log \lambda}{1536}$ we obtain
$\mbox{diam}(\varphi(\Omega_n)) \leq \eta^n$, where
$\eta = \sqrt[4]{\lambda}$. This concludes the proof of part (b) of the
theorem.
To prove the last part of the theorem, let us observe that by
construction of the inverse branches
$\mbox{dist}(z,U_n\setminus B_{\delta_n}(U_n)) \leq \varepsilon_n + \delta_n$,
which immediately leads to the estimate
$\mbox{dist}(z, \Omega_n) \leq \sqrt{2}(\varepsilon_n + \delta_n)
\leq C_1 e^{-\varepsilon n}$,
for some $C_1$. \hfill$\Box$
\vspace{3mm}
\noindent {\bf Remark.} In the proof of theorem \ref{Branches} we describe
doing the thickened branch cuts parallel to the imaginary axis.
Naturally the branch cuts can of course be done at any angle with
respect to the imaginary axis. In particular, given any two
points $x,x'\in\Omega$ which lie outside the set
$B_{C_1e^{-\varepsilon n}}(\Lambda_n)$ then we can arrange the
branch cuts in such a way that they avoid $x,x'$---that is
we can assume that $x,x'\in\Omega_n$ whenever
$x,x'\not\in B_{C_1e^{-\varepsilon n}}(\Lambda_n)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section 3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main result}
\begin{theorem} \label{cuts}
Let $f \in C^{\alpha}$ satisfy $P(f) > \sup f$,
and $\mu$ be its equilibrium state on $J$.
Let $\Omega_n$ be as in constructed in Theorem \ref{Branches}
and $\lambda \in (\rho,1)$, where $\rho = e^{\sup f - P(f)}$.
\noindent (a) $|\mu(B_n)| \leq (\rho/\lambda)^n$, where
$B_n = \bigcup_{\varphi \in {\cal S}''_n(\lambda)} \varphi(\Omega_n)$.
\noindent (b) $\mu(J\setminus \Omega_n) \leq (\rho/\lambda)^n$.
\end{theorem}
\noindent {\bf Proof.} The
equilibrium state $\mu$ of a function $f \in C^{\alpha}$ is of the form
$h \nu$, where $h$ is H\"{o}lder continuous \cite{DPU} and
an eigenfunction to the largest eigenvalue $e^{P}$ ($P = P(f)$) of the
transferoperator ${\cal L}_f$. The measure $\nu$ is an eigenfunctional
to the same eigenvalue of the dual operator, or, equivalently,
an $e^{f-P}$-conformal measure \cite{DU}, which means that
$$
\nu(TA) = \int_A e^{P-f} \,d\nu,
$$
for measurable sets $A$ on which $T$ is injective. Hence
$$
\nu(\varphi(\Omega_n)) = \int_{\Omega_n} e^{f^n-nP} \, d\nu
\leq \rho^n,
$$
for all $\varphi \in {\cal S}$, where
$f^n = f + f\circ T + \cdots + f \circ T^{n-1}$ is the $n$th ergodic sum of $f$.
As $h$ is H\"{o}lder continuous
$\mu(\varphi(\Omega_n)) \leq c_1 \, \rho^n$, for some $c_1 > 0$,
and therefore by theorem \ref{Branches} (a)
$$
\mu(B_n) \leq \sum_{\varphi \in {\cal S}''(\lambda)_n} \mu(\varphi(\Omega_n))
\leq c_1\, |{\cal S}''(\lambda)_n|\rho^n \leq c_1\, r \, (\rho/\lambda)^n,
$$
where $\rho/\lambda$ is less than $1$. This proves the theorem.
\hfill$\Box$
\vspace{3mm}
\noindent Let
$\tilde{\cal S}_n$ denote univalent extensions of the inverse branches
${\cal S}_n$ to simply connected regions $\tilde{\Omega}_n$
($\Omega_n \subset \tilde{\Omega}_n$) which are the topological disk
$U$ minus a finite number of regular branch cuts.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% expansion lemma from DPU
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \label{expansion} (\cite{DPU})
There exists a constant $C_4>1$ and $\gamma_0, \xi > 0$, so that
$\mbox{\rm diam}(\varphi(B_{\gamma}(x)))\leq C_4^n\gamma^{\xi}$
for all $x\in J$, $\varphi\in\tilde{\cal S}_n$, $n\geq0$ and
$\gamma \leq\gamma_0$, provided $B_{\delta}(x)\subset\tilde\Omega_n$.
\end{lemma}
\noindent We shall need the following result which is proven in \cite{P} and
\cite{DPU} and which is based on the previous lemma.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DPU Rule 2 part I
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \label{rule2} (\cite{P} and \cite{DPU} (3.1))
There exist constants $C_5$ and $\beta>0$
(depending on $\alpha$ and $T$) so that
$\|{\cal L}_f^n\psi\|_{\beta} \leq C_5\|\psi\|_{\alpha}$, for all
$n\in\mbox{\bf N}$, $\psi \in C^{\alpha}$.
\end{lemma}
\noindent A consequence of this result is that the eigenfunction $h$ to
the largest eigenvalue $e^P$ of the transfer operator
${\cal L}_f: C^{\alpha}\rightarrow C^{\beta}$,
for real $f \in C^{\alpha}$, is H\"{o}lder continuous with exponent $\beta$
which is given by lemma \ref{rule2}.
As $h$ is bounded away from $0$ and $\infty$, we can introduce a normalised
transfer operator $\hat{\cal L}: C^{\alpha}\rightarrow C^{\gamma}$ by
$\hat{\cal L} = e^{-P(f)}{\cal L}_{f+\log h - \log h\circ T}$, where the
H\"{o}lder exponent $\gamma$ depends on $\beta$ and $T$ and is given
by lemma \ref{rule2}. The normalised
transfer operator has the property that the principal eigenvalue has been
rescaled to $1$ with the associated eigenfunctions being the constants,
that is $\hat{\cal L}{\bf 1} = {\bf 1}$.
Consider $\chi(x,x')$ as a function of the first variable
$x\in J$ where the second entry $x'\in J$ is assumed to be a parameter,
and let
$$
\|\chi(x')\|_{\gamma} = \sup_x |\chi(x,x')|
+ \sup_{x\not= y}\frac{|\chi(x,x')-\chi(y,x')|}{|x-y|^{\gamma}},
$$
be its $\gamma$-H\"{o}lder norm ($\gamma>0$) for the parameter value $x'$.
Then we denote by $\|\chi\|_{\gamma} = \sup_{x'}\|\chi(x')\|_{\gamma}$
the norm of $\chi$.
The transfer operator $\hat{\cal L}$ acts on the function $\chi$ as:
$$
\hat{\cal L}\chi(x,x') = \sum_{\varphi\in\tilde{\cal S}_1}
g_1(\varphi x)\chi(\varphi x,\varphi x'),
$$
where $g_1 = \frac{h}{h\circ T}e^{f-P(f)}$.
The following statement is a slight generalisation of lemma \ref{rule2}
and is proven in the same way. (The supremum norm estimate is straighforward
and the estimates on the variation do not depend on the `parameter' $x'$.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DPU Rule 2 part II
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \label{rule3} (\cite{P}, \cite{DPU})
There exist constants $C_6$ and $\delta\in(0,\gamma)$
(depending on $\gamma$ and $T$) so that
$$
\|\hat{\cal L}_f^n\chi\|_{\delta} \leq C_5\|\chi\|_{\gamma},
$$
for all
$n\in\mbox{\bf N}$, where $\chi(x,x')$ is as above a function on $J\times J$
and $\|\chi\|_{\gamma}$ its $\gamma$-H\"{o}lder norm with respect to the
variable $x$.
\end{lemma}
\noindent Let us now prove the main result of this paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Main contraction theorem
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{contraction}
Let $\psi\in C^{\alpha}$ and $\mu$ its equilibrium state.
There exists a $\sigma<1$ and a constant $C_7$,
such that for all $k\geq1$:
$$
|\hat{\cal L}^k\psi-\mu(\psi)|_{\infty} \leq C_7\sigma^k\|\psi\|_{\alpha}.
$$
\end{theorem}
\noindent {\bf Proof.}
Without loss of generality we can assume that $\mu(\psi)=0$.
Let $\rho=e^{\sup f -P(f)}$ and choose $\lambda\in(\rho,1)$.
Let $\eta$ be the contraction rate of the `good' inverse branches
of theorem \ref{Branches}.
Let $C_4$ be as in lemma \ref{expansion}, $\beta$ (depending on
$\alpha$ and $T$) as given by lemma \ref{rule2} and choose
$\eta'\in(\eta,1)$. Moreover choose $\sigma<1$ so that
$\sigma> \max(\vartheta, \eta^{\alpha})$, where
$\vartheta = \max(\eta', e^{-\delta'\varepsilon}, \frac{\rho}{\lambda})$,
where $\delta'<\delta$ is an arbitrarily positive number and
$\delta<\gamma$ is given by lemma \ref{rule3} with $\gamma=\xi\beta$
($\xi$ as in lemma \ref{expansion}. Finally, choose
a number $p>1$ close enough to $1$ so that
$p< 1+\frac{\log(\eta'/\eta)}{\beta\log |T'|_{\infty}}$ and
$C_4^{\beta\frac{p-1}{p}} < \sigma/\vartheta$.
Let $N$ be an integer which will be specified below.
Then we define a sequence of integers $n_j=[n(1-p^{-j})]$ ($[\cdot]$ denotes
integer part), $j=1,\dots,M$, where $M$ is so that $N\leq np^{-M}\leq pN$.
Put $m_j=n_j-n_{j-1}$, $\psi_0=\psi$ and $\psi_j =\hat{\cal L}^{n_j}\psi
=\hat{\cal L}^{m_j}\psi_{j-1}$, $j=1,\dots,M$. Observe that
$np^{-j}\leq n-n_j\leq np^{-j}+1$ and moreover $m_j\leq(p-1)(n-n_j)+1$.
We also have the lower bound $m_j\geq(p-1)np^{-j}=(p-1)np^{-M}p^{M-j}
\geq(p-1)Np^{M-j}$
The theorem will be proven by induction, where the induction hypothesis
shall be
$$
|\hat{\cal L}^k\psi|_{\infty} \leq C_7\sigma^k\|\psi\|_{\alpha},
$$
for $k=0,1,\dots,n-1$. We shall show that the estimate also holds
for $k=n$. In particular we have
$|\psi_j|_{\infty} \leq C_7\sigma^{n_j}\|\psi\|_{\alpha}$ for $j=0,\dots,M$.
Let $\Omega_n$ be the regions for the inverse branches of $T^n$ given
by theorem \ref{Branches} and let $x, x'\in J\cap\Omega_n$. Then
\begin{eqnarray}
\hat{\cal L}^n\psi(x) - \hat{\cal L}^n\psi(x')
&=& \sum_{\varphi \in {\cal S}_n}
\left( g_n(\varphi x)\psi(\varphi x) - g_n(\varphi x')\psi(\varphi x')
\right) \nonumber\\
&=& I + II, \nonumber
\end{eqnarray}
where $g_n = e^{f^n+h-h\circ T^n-nP}$. In the first term,
$$
I= \sum_{\varphi \in {\cal S}_n}
g_n(\varphi x)(\psi(\varphi x) - \psi(\varphi x')),
$$
we get by theorem \ref{Branches}(b) for the contracting branches
$\varphi \in {\cal S}'_n$ :
$$
|\psi(\varphi x) - \psi(\varphi x')|
\leq |\psi|_{\alpha}|\varphi x -\varphi x'|^{\alpha}
\leq |\psi|_{\alpha}\eta^{n\alpha}|x-x'|^{\alpha}.
$$
On the other hand, since $\sup_{y \in \varphi(\Omega_n)} g_n(y)
\leq c_1\, \mu(\varphi(\Omega_n))$, for some $c_1$, we obtain
by theorem \ref{cuts} for the non-contracting branches that
$$
\sum_{\varphi \in {\cal S}''_n} g_n(\varphi x)|\psi(\varphi x) - \psi(\varphi x')|
\leq 2|\psi|_{\infty}c_1\mu(B_n)
\leq 2|\psi|_{\infty}c_1\left(\frac{\rho}{\lambda}\right)^n,
$$
which implies (with some $c_2$):
\begin{eqnarray}
I&\leq& |\psi|_{\alpha}\eta^{n\alpha}|x-x'|^{\alpha}
+2|\psi|_{\infty}c_1\left(\frac{\rho}{\lambda}\right)^n\nonumber\\
&\leq&c_2\|\psi\|_{\alpha}\left(
\max(\eta^{\alpha},\frac{\rho}{\lambda})\right)^n\nonumber\\
&\leq& \frac{C_7}{8}\|\psi\|_{\alpha}\sigma^n,\nonumber
\end{eqnarray}
provided $n$ is large enough.
Let us now consider the second term for which we use the following
telescoping sums:
\begin{eqnarray}
II &=& \sum_{\varphi \in {\cal S}_n}
\psi(\varphi x')\left(g_n(\varphi x) - g_n(\varphi x')\right) \nonumber\\
& =& \sum_{j=1}^M \sum_{\varphi\in\tilde{\cal S}_{n-n_{j-1}}}
\hat{\cal L}^{n_{j-1}}\psi(\varphi x')
\left(g_{n-n_{j-1}}(\varphi x)
-g_{m_j}(\varphi x')g_{n-n_j}(T^{m_j}\varphi x)\right)
\nonumber\\
& =& \sum_{j=1}^M \sum_{\varphi\in\tilde{\cal S}_{n-n_{j-1}}}
\hat{\cal L}^{n_{j-1}}\psi(\varphi x') g_{n-n_j}(T^{m_j}\varphi x)
\left(1-\frac{g_{m_j}(\varphi x')}{g_{m_j}(\varphi x)}\right)\nonumber.
\end{eqnarray}
In order to estimate the second sum for a given value of $j$, let us
consider two cases:
(i) If the point $x$ lies in $J\setminus (\Omega_{n-n_{j-1}}
\setminus B_{C_1e^{-\varepsilon(n-n_{j-1})}}(\Lambda_{n-n_{j-1}}))$ then
following the remark made at the end of section 2, we can
execute the branch cuts at some angle to the imaginary
axis rather than parallel to it so that
$x$ and $x'$ come to lie outside the cut in the region
$\Omega_{n-n_{j-1}}$. Put $y=x$.
(ii) If on the other hand $x\in (J \setminus\Omega_{n-n_{j-1}})
\cap B_{C_1e^{-\varepsilon(n-n_{j-1})}}(\Lambda_{n-n_{j-1}})$,
then let us use the fact the Julia set is uniformly perfect \cite{Hink}
which means that there exists a constant $c_3$ which only deponds on the
map $T$ so that for every $r>0$ and $x\in J$ the distance
$\mbox{dist}(x, J\setminus B_r(x))$ is bounded by $c_3r$. Since the
number of critical values for $T^{n-n_{j-1}}$ is bounded by
$c_4(n-n_{j-1})$ (for some constant $c_4$),
we can find a point $y\in J\cap \Omega_{n-n_{j-1}}$
so that $|x-y| \leq 2c_3c_4(n-n_{j-1})e^{-\varepsilon(n-n_{j-1})}.
In a similar way if $x'\in (J \setminus\Omega_{n-n_{j-1}})
\cap B_{C_1e^{-\varepsilon(n-n_{j-1})}}(\Lambda_{n-n_{j-1}})$,
then there is a point $y'\in J\cap \Omega_{n-n_{j-1}}$ so that
$|x'-y'| \leq 2c_3c_4(n-n_{j-1}) e^{-\varepsilon(n-n_{j-1})}$
(and otherwise we put $y'=x'$).
Now, since $\hat{\cal L}^{n_{j-1}}\psi = \psi_{j-1}$
and $p>1$ was chosen so that
$|T'|_{\infty}^{\beta m_j}\eta^{n-n_{j-1}}
\leq|T'|_{\infty}^{\beta}\eta'^{n-n_{j-1}}$
we obtain considering only the contracting branches ($c_6\leq c_5C_7$):
\begin{eqnarray}
& &\hspace{-1in}\left|\sum_{\varphi\in{\cal S}'_{n-n_{j-1}}}
\hat{\cal L}^{n_{j-1}}\psi(\varphi y') g_{n-n_j}(T^{m_j}\varphi y)
\left(1-\frac{g_{m_j}(\varphi y')}{g_{m_j}(\varphi y)}\right)\right|\nonumber \\
&\leq& c_5|\psi_{j-1}|_{\infty}|T'|_{\infty}^{\beta m_j}
\eta^{n-n_{j-1}}|y-y'|^{\beta}
\sum_{\varphi\in\tilde{\cal S}_{n-n_j}} g_{n-n_j}(\varphi y)\nonumber\\
&\leq& c_6\|\psi\|_{\alpha}\sigma^{n_{j-1}}|T'|_{\infty}^{\beta}
\eta'^{n-n_{j-1}}\nonumber\\
&\leq& c_6\|\psi\|_{\alpha}\sigma^n|T'|_{\infty}^{\beta}
\left(\frac{\eta'}{\sigma}\right)^{n-n_{j-1}},
\nonumber
\end{eqnarray}
where we used that
\begin{eqnarray}
\left|1 - \frac{ g_m(\varphi y')}{g_m(\varphi y)} \right|
&=&\left|1-e^{f^m(\varphi y')-f^m(\varphi y)
+h(\varphi y')-h(\varphi y)+h(T^m\varphi y)-h(T^m\varphi y')}\right|\nonumber\\
&\leq& c_5\, |T'|_{\infty}^{m\beta} |\varphi y-\varphi y'|^{\beta},\nonumber
\end{eqnarray}
(as $|T^j\varphi y'-T^j\varphi y|\leq |T'|_{\infty}^j|\varphi y'-\varphi y|$)
for some constant $c_5$ which depends on the functions $f$ and $h$.
For the non-contracting branches we proceed as above in estimating
term $I$:
$$
\left|\sum_{\varphi\in{\cal S}''_{n-n_{j-1}}}
\hat{\cal L}^{n_{j-1}}\psi(\varphi y') g_{n-n_j}(T^{m_j}\varphi y)
\left(1-\frac{g_{m_j}(\varphi y')}{g_{m_j}(\varphi y)}\right)\right|
\leq 2c_1|\psi_{j-1}|_{\infty}
\left(\frac{\rho}{\lambda}\right)^{n-n_{j-1}}.
$$
In order to estimate the error we made when replacing
$x$ by $y\in\Omega_{n-n_{j-1}}$, put
$$
\chi_j(x,y) = \sum_{\varphi\in\tilde{\cal S}_{m_j}}
\psi_{j-1}(\varphi y) \left(g_{m_j}(\varphi x)-g_{m_j}(\varphi y)\right),
$$
(note that $\chi_j(y,y) = 0$) and apply lemma \ref{rule3} to obtain
\begin{eqnarray}
& &\hspace{-1in}\left|\sum_{\varphi\in\tilde{\cal S}_{n-n_{j-1}}}
\hat{\cal L}^{n_{j-1}}\psi(\varphi y) g_{n-n_j}(T^{m_j}\varphi x)
\left(1-\frac{g_{m_j}(\varphi y)}{g_{m_j}(\varphi x)}\right)\right|\nonumber \\
&=& \left|\hat{\cal L}^{n-n_j}\chi_j(x,y)\right|\nonumber \\
&\leq& C_6 \|\chi_j\|_{\gamma}|x-y|^{\delta}\nonumber \\
&\leq& C_6 \|\chi_j\|_{\gamma}2c_3c_4(n-n_{j-1})
e^{-\delta\varepsilon(n-n_{j-1})},\nonumber
\end{eqnarray}
where $\gamma=\xi\beta$ and $\delta<\gamma$ was given by lemma \ref{rule3}.
If we let $\delta'<\delta$ then we get with some constant $c_7$:
$$
\left|\hat{\cal L}^{n-n_j}\chi_j(x,y)\right|
\leq c_7 \|\chi_j\|_{\gamma}e^{-\delta'\varepsilon(n-n_{j-1})}.
$$
It remains to
estimate the $\gamma$-norm of $\chi_j$. By the induction hypothesis:
$|\chi_j|_{\infty} \leq 2|\psi_{j-1}|_{\infty}
\leq 2C_7\sigma^{n_{j-1}}\|\psi\|_{\alpha}$.
Since, by lemma \ref{expansion}
$|T^{\ell}\varphi z - T^{\ell}\varphi z'|\leq C_4^{m-\ell}|z-z'|^{\xi}$,
$\ell=0,\dots,m$, $\varphi\in\tilde{\cal S}_m$, for any two points
$z,z'\in J$ (that are close enough), we can write:
\begin{eqnarray}
|\chi_j(z,y)-\chi_j(z',y)| &\leq& |\psi_{j-1}|_{\infty}
\sum_{\varphi\in\tilde{\cal S}_{m_j}}
|g_{m_j}(\varphi z) - g_{m_j}(\varphi z')|\nonumber \\
&\leq& |\psi_{j-1}|_{\infty}
\sum_{\varphi\in\tilde{\cal S}_{m_j}} g_{m_j}(\varphi z)
\left|1 - \frac{g_{m_j}(\varphi z')}{g_{m_j}(\varphi z)}\right|\nonumber \\
&\leq& c_8|\psi_{j-1}|_{\infty}C_4^{\beta m_j}|z-z'|^{\beta\xi},\nonumber
\end{eqnarray}
for some constant $c_8$. One concludes that
$$
\|\chi_j\|_{\gamma}\leq c_8C_4^{\beta m_j}|\psi_{j-1}|_{\infty}
\leq c_8C_7C_4^{\beta m_j}\sigma^{n_{j-1}}\|\psi\|_{\alpha}.
$$
Therefore
$$
\left|\hat{\cal L}^{n-n_j}\chi_j(x,y)\right|
\leq c_9C_7C_4^{\beta m_j}\|\psi\|_{\alpha} \sigma^n
\left(\frac{e^{-\delta'\varepsilon}}{\sigma}\right)^{n-n_{j-1}},
$$
for some constant $c_9$, and we arrive at the
estimate ($c_{10}>0$ and recall that $m_j\leq(p-1)(n-n_j)+3$ and
$n-n_j\geq np^{-j}\geq Np^{M-j}=Np^k$ if we put $k=M-j$):
\begin{eqnarray}
II & \leq& (c_6|T'|_{\infty}^{\beta}+2c_1+c_9C_7)\|\psi\|_{\alpha}
\sigma^n\sum_{j=1}^M
C_4^{\beta m_j}\left(\frac{\vartheta}{\sigma}\right)^{n-n_{j-1}}\nonumber\\
& \leq& c_{10}C_7\|\psi\|_{\alpha}
\sigma^n\sum_{k=0}^{\infty}
\left(\frac{\vartheta C_4^{\beta(p-1)}}{\sigma}\right)^
{Np^k},\nonumber
\end{eqnarray}
where $\vartheta =
\max(\eta',e^{-\delta'\varepsilon},\frac{\rho}{\lambda})<\sigma$.
Since the number $p>1$ was chosen so that the expression inside
the brackets is less than $1$ we can make
the sum over $k$ arbitrarily small if we only choose $N$ large enough.
In particular we can achieve that
$II\leq \frac{C_7}{8}\|\psi\|_{\alpha}\sigma^n$.
Finally we obtain the estimate :
$$
\left|\hat{\cal L}^n\psi(x) - \hat{\cal L}^n\psi(x') \right|
\leq I + II \leq \frac{C_7}{4}\|\psi\|_{\alpha}\sigma^n,
$$
for $x, x' \in J\cap\Omega_n$.
Now suppose $y, y'\in J\setminus\Omega_n$, then, in the same way as
above (according to the remark), we can assume that in fact $y,y'$ lie in a
$C_1e^{-n\varepsilon}$-neighbourhood of the critical values
$\Lambda_n$. Since $J$ is a uniformly perfect set, we can find, as above,
$x,x' \in J\cap\Omega_n$ so that
$|x-y|,|x'-y'| \leq n2c_4c_3C_1e^{-n\varepsilon}$.
Therefore, as by lemma \ref{rule2}
$|\hat{\cal L}^n\psi|_{\tau}\leq C_5\|\psi\|_{\alpha}$, we obtain that
$$
\left|\hat{\cal L}^n\psi(y) - \hat{\cal L}^n\psi(y') \right|
\leq \frac{C_7}{4}\|\psi\|_{\alpha}\sigma^n
+ 4nC_5c_3c_4C_1\|\psi\|_{\alpha}e^{-n\varepsilon}
\leq \frac{C_7}{2}\|\psi\|_{\alpha}\sigma^n,
$$
for all $n$, provided
$C_7>6C_5c_3c_4C_1\sup_{n>0}ne^{-n\varepsilon}\sigma^{-n}$.
Since $\mu(\psi)=\mu(\hat{\cal L}^n\psi)=0$
and $\hat{\cal L}^n\psi$ is continuous, this implies that
$|\hat{\cal L}^n\psi|_{\infty} \leq C_7\|\psi\|_{\alpha}\sigma^n$.
\hfill$\Box$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Holder contraction theorem
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \label{holder.contraction}
Let $\psi\in C^{\alpha}$ and, as above, $\mu$ the equilibrium state
of the potential $f\in C^{\alpha}$. Let
$\beta<\alpha$ be as in lemma \ref{rule2} and $\xi$ as in lemma
\ref{expansion}. For every (positive) small enough $\kappa<\xi\beta$
there exist a $\varsigma<1$ and a constant $C_8$,
such that for all $n\geq1$:
$$
\|\hat{\cal L}^n\psi-\mu(\psi)\|_{\kappa} \leq C_8\varsigma^n\|\psi\|_{\alpha}.
$$
\end{theorem}
\noindent {\bf Proof.} We have to show that the H\"{o}lder constants
$|\hat{\cal L}^n\psi|_{\kappa}$ goes exponentially fast to zero with
some rate $\varsigma<1$. For simplicity's sake we shall as in the
previous theorem assume that $\mu(\psi)=0$. To estimate the variation
of $\hat{\cal L}^n\psi$ let $x,x'\in J$ and put $\epsilon=|x-x'|$.
Assume $\epsilon>0$. To get bounds on the variation for large $n$ we
shall use theorem \ref{contraction} and for small $n$ we use
lemma \ref{expansion}. Let $\beta$, $\xi$ and $C_4>1$ be as in lemmas
\ref{rule2} and \ref{expansion} and choose $\kappa<\xi\beta$ small
enough so that $\sigma C_4^{\frac{2\kappa\beta}{\xi\beta-\kappa}}$
is less than $1$. Put
$$
P(\epsilon)=\frac{\xi\beta-\kappa}{2\beta}\;\frac{|\log\epsilon|}{\log C_4},
$$ and consider the two cases (I) and (II):
(I) If $n\geq P(\epsilon)$ then, by theorem \ref{contraction},
$$
\left|\hat{\cal L}^n\psi(x) - \hat{\cal L}^n\psi(x') \right|
\leq 2|\hat{\cal L}^n\psi|_{\infty}
\leq 2C_7\|\psi\|_{\alpha}\sigma^n
\leq C_8\|\psi\|_{\alpha}\varsigma^n\epsilon^{\kappa},
$$
where the last inequality holds with $C_8\geq2C_7$ provided
$$
\frac{\varsigma}{\sigma}\geq C_4^{\frac{2\kappa\beta}{\xi\beta-\kappa}}.
$$
(II) If $n