# Statistical multi-moment bifurcations in random delay coupled swarms

Luis Mier-y-Teran-Romero(1), Brandon Lindley(2), and Ira B. Schwartz(3) (1) US
Naval Research Laboratory, Code 6792, Nonlinear System Dynamics Section,
Plasma Physics Division, Washington, DC 20375

###### Abstract

We study the effects of discrete, randomly distributed time delays on the
dynamics of a coupled system of self-propelling particles. Bifurcation
analysis on a mean field approximation of the system reveals that the system
possesses patterns with certain universal characteristics that depend on
distinguished moments of the time delay distribution. Specifically, we show
both theoretically and numerically that although bifurcations of simple
patterns, such as translations, change stability only as a function of the
first moment of the time delay distribution, more complex patterns arising
from Hopf bifurcations depend on all of the moments.

Recently, much attention has been given to the study of interacting multi-
agent, particle or swarming systems in various natural and engineering fields.
Interestingly, these multi-agent swarms can self-organize and form complex
spatio-temporal patterns even when the coupling between agents is weak. Many
of these investigations have been motivated by a multitude of biological
systems such as schooling fish, swarming locusts, flocking birds, bacterial
colonies, ant movement, etc. Budrene and Berg (1995); Toner and Tu (1995);
Parrish (1999); Topaz and Bertozzi (2004), and have also been applied to the
design of systems of autonomous, communicating robots or agents Leonard and
Fiorelli (2002); Morgan and Schwartz (2005); Chuang et al. (2007) and mobile
sensor networks Lynch et al. (2008).

Many studies describe the swarm system at the individual, or particle, level
via models constructed with ordinary differential equations (ODEs) or delay
differential equations (DDEs) to describe the trajectories Vicsek et al.
(1995); Flierl et al. (1999). When there are a large number of densely-
distributed particles, authors have employed partial differential equations
(PDEs) to describe the average agent density and velocity Toner and Tu (1995,
1998); Edelstein-Keshet et al. (1998); Topaz and Bertozzi (2004). Recently,
the inclusion of noise in such particle-based studies has revealed
interesting, noise-induced transitions between different coherent patterns
Erdmann and Ebeling (2005); Forgoston and Schwartz (2008). Such noise driven
systems have led to the discovery of first and second order phase transitions
in swarm models Aldana et al. (2007).

A topic of intense ongoing research in interacting particle systems, and in
particular in the dynamics of swarms, is the effect of time delays. It is well
known that time delays can have profound dynamical consequences, such as
destabilization and synchronization Englert et al. (2011); Zuo et al. (2010),
and delays have been effectively used for purposes of control Konishi et al.
(2010). Initially, such studies focused on the case of one or a few discrete
time delays. More recently, however, the complex situation of several and
random time delays has been researched Ahlborn and Parlitz (2007); Wu et al.
(2009); Marti et al. (2006). An additional important case is that of
distributed time delays, when the dynamics of the system depends on a
continuous interval in its past instead of on a discrete instant Omi and
Shinomoto (2008); Dykman and Schwartz (2012).

There exists a complex interplay between the attractive coupling, time delay,
and noise intensity that produces transitions between different spatio-
temporal patterns Forgoston and Schwartz (2008); Mier-y Teran-Romero et al.
(2011) in the case of a single, discrete delay. Here, we consider a more
general swarming model where coupling information between particles occurs
with randomly distributed time delays. We perform a bifurcation analysis of a
mean field approximation and reveal the patterns that are possible at
different values of the coupling strength and parameters of the time delay
distribution.

We model the dynamics of a 2D system of $N$ identical self-propelling agents
that are attracted to each other in a symmetric manner. We consider the
effects of finite communication speeds and information-processing times so
that the attraction between agents occurs in a time delayed fashion. The time
delays are non-uniform but they are symmetric among agents
$\tau_{ij}(=\tau_{ji})$, for particles $i$ and $j$, as well as constant in
time. The dynamics of the particles is described by the following governing
equations:

$\displaystyle\ddot{\mathbf{r}}_{i}=$
$\displaystyle\left(1-|\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{r}}_{i}-\frac{a}{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}(\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t-\tau_{ij})),$ (1)

for $i=1,2\ldots,N$. The vector $\mathbf{r}_{i}$ denotes the position of the
$i$th agent at time $t$. The term
$\left(1-|\dot{\mathbf{r}}_{i}|^{2}\right)\dot{\mathbf{r}}_{i}$ represents
self-propulsion and frictional drag forces that act on each agent. The
coupling constant $a$ measures the strength of the attraction between agents
and the time delay between particles $i$ and $j$ is given by $\tau_{ij}$. When
$a=0$ the agents tend to move in a straight line with unit speed as time tends
to infinity. The $N(N-1)/2$ different time delays $0<\tau_{ij}(=\tau_{ji})$
are drawn from a distribution $\rho(\tau)$ whose mean and standard deviation
are denoted by $\mu_{\tau}$ and $\sigma_{\tau}$, respectively.

We obtain a mean field approximation of the swarming system by measuring the
particle’s coordinates relative to the center of mass
$\mathbf{r}_{i}=\mathbf{R}+\delta\mathbf{r}_{i}$, for $i=1,2\ldots,N$, where
$\mathbf{R}(t)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}_{i}(t)$. Following the
approximations from Lindley et al. (2012), we obtain a mean field description
of the swarm:

$\displaystyle\ddot{\mathbf{R}}=$
$\displaystyle\left(1-|\dot{\mathbf{R}}|^{2}\right)\dot{\mathbf{R}}-a\left(\mathbf{\
R}(t)-\int_{0}^{\infty}\mathbf{R}(t-\tau)\rho(\tau)d\tau\right).$ (2)

The approximations necessary to obtain Eq. (2) require that $N$ be
sufficiently large so that
$\frac{1}{N(N-1)}\sum_{i=1}^{N}\mathop{\sum_{j=1}^{N}}_{i\neq
j}\mathbf{R}(t-\tau_{ij})\approx\int_{0}^{\infty}\mathbf{R}(t-\tau)\rho(\tau)d\tau$
and that the swarm particles remain close together.

Figure 1: Bifurcation structure of the translating state of the mean field
Eqs. (2) for the exponential distribution described in the text
($\sigma_{\tau}=0.2$). The translating state merges with: (_i_) the stationary
state along the continuous red curve $a\mu_{\tau}=1$; and (_ii_) the
circularly rotating state along the dashed black curve
$a\langle\tau^{2}\rangle=2$. The component of the translating state parallel
to the motion undergoes a Hopf bifurcation along the green, dotted curve.
(Color online.)

Equations (2) admit a uniformly translating solution
$\mathbf{R}(t)=\mathbf{R}_{0}+\mathbf{V}_{0}\cdot t$ ($\mathbf{R}_{0}$ and
$\mathbf{V}_{0}$ are any constant 2D vectors). The speed $|\mathbf{V}_{0}|$
must satisfy

$\displaystyle|\mathbf{V}_{0}|^{2}=1-a\int_{0}^{\infty}\tau\rho(\tau)d\tau=1-a\mu_{\tau},$
(3)

which shows that this solution is possible as long as the system parameters
lie below the hyperbola $a\mu_{\tau}=1$ in the $(a,\ \mu_{\tau})$ plane.
Remarkably, the speed of the of the translating state depends exclusively on
the mean of the distribution $\rho(\tau)$ and not on any of the higher
moments.

The linear stability of the translating state is examined by taking
$X(t)=\sqrt{1-a\mu_{\tau}}\cdot t+\delta X(t)$ and $Y(t)=\delta Y(t)$. The two
linearized equations decouple and the stability of motions parallel and
perpendicular to the translating direction are determined by the
characteristic equations $\mathcal{D}_{\parallel}(\lambda)$ and
$\mathcal{D}_{\perp}(\lambda)$, respectively:

$\displaystyle\mathcal{D}_{\parallel}(\lambda)=\mathcal{F}(\lambda)-(3a\mu_{\tau}-2)\lambda,\quad\mathcal{D}_{\perp}(\lambda)=\mathcal{F}(\lambda)-a\mu_{\tau}\lambda,$
(4)

where $\mathcal{F}(\lambda)=a\left(1-\langle
e^{-\lambda\tau}\rangle\right)+\lambda^{2}$. The function $\langle
e^{-\lambda\tau}\rangle$ is the moment generating function of $\rho(\tau)$
since the $n$-th moment is equal to
$\langle\tau^{n}\rangle=(-1)^{n}\frac{d^{n}}{d\lambda^{n}}\langle
e^{-\lambda\tau}\rangle|_{\lambda=0}$. Regardless of the choice of $a$ and
$\rho(\tau)$, the characteristic functions $\mathcal{D}_{\parallel}$ and
$\mathcal{D}_{\perp}$ have a zero eigenvalue arising from the translation
invariance of Eq. (2) not . There is a fold bifurcation as an eigenvalue of
$\mathcal{D}_{\parallel}$ crosses the origin when $a\mu_{\tau}=1$, which marks
the disappearance of the translating state as seen from Eq. (3). Numerical
analysis Engelborghs (2000) reveals an additional curve on the $(a,\
\mu_{\tau})$ plane (below the curve $a\mu_{\tau}=1$) along which perturbations
parallel to the translation undergo a Hopf bifurcation as a complex pair of
eigenvalues of $\mathcal{D}_{\parallel}$ cross the imaginary axis.

As for perturbations perpendicular to the translational motion, there is
another fold bifurcation as an eigenvalue of $\mathcal{D}_{\perp}$ crosses the
origin along the curve $a\langle\tau^{2}\rangle=2$, which represents a
bifurcation in which the translating state merges with a circularly rotating
state of infinite radius, as discussed below.

Considering a fixed $\sigma_{\tau}$, the overall stability picture of the
translating state of Eqs. (2) is as follows (see Fig. 1). For values of $(a,\
\mu_{\tau})$ below the curves $a\langle\tau\rangle=1$ and
$a\langle\tau^{2}\rangle=2$ (region A) the translating state is linearly
stable . These two curves may cross at a point that we call the ‘zero
frequency Hopf point’ (ZFH). The transverse direction of the translating state
becomes unstable along the curve $a\langle\tau^{2}\rangle=2$ where this state
merges with the circularly rotating state (along the mentioned curve the
rotating state has an infinite radius); transverse perturbations of the
translating state will thus produce a transition to the rotating state in
regions B and C. From the ZFH point, there emanates a Hopf bifurcation curve
where the parallel component of the translating state becomes unstable so that
in region C there is a transition from the translating state to oscillations
along a straight line. Finally, the translating state ceases to exist along
the curve $a\langle\tau\rangle=1$ where there is a pitchfork type bifurcation
with the stationary steady state solution. The possible behaviors in region D
are discussed below.

Figure 2: Speed of the translating state of Eqs. (1) and (2) as a function of
$\mu_{\tau}$; here $N=150$, $a=0.2$ (a) and $a=1.2$ (b). The red line (color
online) represents the mean field result from Eq. (3); the continuous segment
marks where the translating state is linearly stable and the dashed segment
where it is unstable. The symbols represent numerical simulations of Eqs. (1)
for different time delay distributions: sliding exponential and sliding
uniform $\sigma_{\tau}$ = 0.5 (a), $\sigma_{\tau}$ = 0.05 (b); widening
exponential $\mu_{\tau}=\sigma_{\tau}$ (a) and (b); widening uniform
$\mu_{\tau}=\sqrt{3}\sigma_{\tau}$ (a) and (b).

We compare the mean field bifurcation results with the full swarm system via
numerical simulations. Here, we make use of two different time delay
distributions with mean $\mu_{\tau}$ and standard deviation $\sigma_{\tau}$ to
test our findings. The first is an exponential distribution
$\rho(\tau)=e^{\frac{\tau-\mu_{\tau}+\sigma_{\tau}}{\sigma_{\tau}}}/\sigma_{\tau}$
for $\tau\geq\mu_{\tau}-\sigma_{\tau}$ and zero otherwise; we require
$\sigma_{\tau}\leq\mu_{\tau}$ for proper normalization. The second
distribution is a uniform $\rho(\tau)=\frac{1}{2\sqrt{3}\cdot\sigma_{\tau}}$
for
$\mu_{\tau}-\sqrt{3}\sigma_{\tau}\leq\tau\leq\mu_{\tau}+\sqrt{3}\sigma_{\tau}$
and zero otherwise; here, we require $\sqrt{3}\sigma_{\tau}\leq\mu_{\tau}$.
Moreover, we employ two versions of the mentioned distributions: a ‘sliding’
one ($\sigma_{\tau}$ = const.) and a ‘widening’ version
($\mu_{\tau}\propto\sigma_{\tau}$).

Fig. 2 compares the speed of the swarm obtained from Eq. (3) with the time-
averaged speed of the center of mass obtained from simulations (after the
decay of transients). The swarm particles are all located at the origin at
time zero and move with the speed obtained from Eq. (3) along the $x$ axis. In
these simulations, we use both ‘sliding’ ($\sigma_{\tau}$ fixed) and
‘widening’ (both $\mu_{\tau}$ and $\sigma_{\tau}$ vary) versions of the
exponential and uniform distributions described above. Fig. 2 shows that the
swarm converges to the translating state up to a value of $\mu_{\tau}$ beyond
which the swarm converges to a state in which it oscillates back and forth
along a line with a near zero time-averaged speed (the average is taken over
an interval much longer than the period). The transition to the oscillatory
regime occurs earlier than the mean field prediction. The full simulation
results show that in this deviation from the mean field, the swarm particles
become spread out too far apart and render the approximations leading to Eq.
(2) invalid.

Figure 3: Bifurcation curves of the mean field Eqs. (2) at fixed
$\sigma_{\tau}$ for the two time delay distributions $\rho(\tau)$ described in
the text: exponential (top) and uniform (bottom). The translational state
disappearance curve $a\mu_{\tau}=1$ (red), bifurcation of the translational
state with circularly rotating state curve $a\langle\tau^{2}\rangle=2$
(black). The first four members of the stationary state Hopf bifurcation
curves are also shown (blue, dashed green, dotted-dashed cyan and dotted
magenta). In each panel, $\sigma_{\tau}$ has the value (a) 0.2, (b) 0.95, (c)
0.2 and (d) 0.3, respectively. (Color online.)

In addition to the translating state, Eqs. (2) always possess a stationary
state solution $\mathbf{R}(t)=\mathbf{R}_{0}=$ const. In the full system, Eq.
(1), the stationary state for the center of mass manifests itself in a swarm
‘ring state’, where some particles rotate clockwise and others counter-
clockwise on a circle around a static center of mass. The characteristic
equation that governs the linear stability of the stationary state has the
form $\big{(}\mathcal{D}(\lambda)\big{)}^{2}=0$, where
$\mathcal{D}(\lambda)=\mathcal{F}(\lambda)-\lambda.$ Once more there is a zero
eigenvalue for all choices of $a$ and $\rho(\tau)$ that arises from the
translation invariance of Eqs. (2). Also, since $\mathcal{D}(0)=0$,
$\mathcal{D}^{\prime}(0)=a\mu_{\tau}-1$ and
$\lim_{\lambda\rightarrow\infty}\mathcal{D}(\lambda)=\infty$, the condition
$a\mu_{\tau}-1<0$ guarantees the existence of at least one real and positive
eigenvalue which renders the stationary state linearly unstable. Thus,
$a\mu_{\tau}=1$ is a bifurcation curve on the $(a,\mu_{\tau})$ plane along
which the uniformly translating state bifurcates with the stationary state.

The stationary state undergoes Hopf bifurcations when the equation
$\mathcal{D}(i\omega)=a\left(1-\langle
e^{-i\omega\tau}\rangle\right)-\omega^{2}-i\omega=0$ for $\omega\neq 0$ is
satisfied. The function $\langle e^{-i\omega\tau}\rangle$ is called the
characteristic function of $\rho(\tau)$ and is related to the moment
generating function of the distribution; its Taylor series contains all of the
moments of $\rho(\tau)$. This shows that the location of the Hopf bifurcations
depends on the values of all moments of the time delay distribution. This is
in contrast to the region where the translating state exists $a\mu_{\tau}<1$,
which involves the first moment only.

At the location of the Hopf bifurcations, circular orbits bifurcate from the
stationary state. This may be seen by changing Eq. (2) from the Cartesian
$(X,\ Y)$ to polar coordinates $(R,\ \phi)$, noticing that circular orbits
$R=R_{0}$, $\phi=\omega t$ are possible as long as

$\displaystyle\omega^{2}=a\left(1-\langle\cos\omega\tau\rangle\right),\qquad
R_{0}=\frac{1}{\omega}\sqrt{1-\frac{a}{\omega}\langle\sin\omega\tau\rangle},$
(5)

and then realizing that the first of (5) and the condition $R_{0}=0$ are
precisely the real and imaginary parts of the Hopf bifurcation conditions
$\mathcal{D}(i\omega)=0$.

Figure 4: Center of mass speed as a function of $\mu_{\tau}$ for the ‘sliding’
exponential time delay distribution (left) and the ‘sliding’ uniform
distribution (right). Here $N=150$, $a=2$. The continuous red curve represents
the mean field result from Eq. (5), while the symbols represent the results
from numerical simulations of Eqs. (1). (color online)

Generically, the Hopf conditions for the stationary state
$\mathcal{D}(i\omega)=0$ yield a family of curves in the $(a,\ \mu_{\tau})$
plane (Fig. 3). The first member of the Hopf family emanates from the crossing
of the curves $a\mu_{\tau}=1$ and $a\langle\tau^{2}\rangle=2$; the former
curve is where the translating state bifurcates from the stationary state in a
pitchfork-like bifurcation. Hence the name ‘Zero Frequency Hopf’ for the Hopf-
fold point (Fig. 3a). The first Hopf curve is supercritical and gives rise to
a circularly rotating orbit with radius and frequency given by the first
solution of Eqs. (5). Below this first Hopf curve and $a\mu_{\tau}=1$, in
region A the stationary state is stable. From Eqs. (5) it follows that this
circularly rotating orbit collides with the translating state along the curve
$a\langle\tau^{2}\rangle=2$ where its radius tends to infinity and its speed
to that of the translating state, $\sqrt{1-a\mu_{\tau}}$. Thus, in regions B
and C the system converges to the circularly rotating orbit. The different
regions change shape for the other panels of Fig. 3, but the dynamics remain
as described above.

We compare the mean field prediction for the location of the birth of the
circularly rotating state (first Hopf curve) with the full system. Fig. 4
shows the results from numerical simulations of Eqs. (1) at a fixed value of
$\sigma_{\tau}$ for increasing $\mu_{\tau}$. We plot the speed of the center
of mass averaged over a long time interval, after the decay of transients. In
that plot, the near-zero values of the mean speed (in the interval
$0\lesssim\mu_{\tau}\lesssim 1.4$) indicate that the particles have converged
to the ring state, while for all higher values of $\mu_{\tau}$ the swarm
converges to the rotating state. Remarkably, for the highest values of
$\mu_{\tau}$, the center of mass of the swarm moves faster than unit speed,
the asymptotic speed of uncoupled particles. The reason is that while in its
rotating orbit, the mean time-delayed position of the swarm is actually
“ahead” of the center of mass at the current time, causing the particles to
accelerate forward along the circular orbit.

In summary, we have considered a randomly delay distributed coupled swarm
model, and analyzed the bifurcations of various patterns as a function of
delay characteristics and coupling strength. In particular, we have shown that
the location and shape of the Hopf bifurcation curves is strongly-dependent on
all the moments of $\rho(\tau)$. This dependence, in addition to the fact that
the succeeding Hopf curves in Fig. 3 exhibit higher frequencies of rotation,
makes the higher-order patterns equally sensitive to all the moments of the
delay distribution. In the single delay case with distribution
$\rho=\delta(\tau-\tau_{0})$, where $\delta(\tau)$ is a Dirac delta function,
all of the succeeding Hopf bifurcations are all subcritical and continuous. In
contrast, when all moments are present, the bifurcations may not even be
continuous, presenting their structure as isolated closed curves bounded by
fold bifurcations, as seen in Fig. 3d for a uniform distribution. Finally, we
expect that in other globally delay-coupled systems Choi et al. (2000); Marti
and Masoller (2003); Kozyreff et al. (2001); Masoller et al. (2009), generic
types of behavior involving bifurcations including all moments of the
distribution should be present.

The authors gratefully acknowledge the Office of Naval Research for their
support. LMR is an NIH post doctoral fellow and BL is a post doctoral fellow
of the NRC.

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The string $\gamma_{n(x)}^{-1}\cdots \gamma_1^{-1}$ gives an open subset $\gamma_1\cdots \gamma_{n(x)}\Delta_{p_1}$, which is an open subset for the coding, and on this subset, $T$ is given by $\gamma_{n(x)}^{-1}\cdots \gamma_1^{-1}$.

To prove (1), (3) and (4) in Proposition <ref>, we start with a lemma similar to Lemma <ref>. Define

\begin{equation*}
U_i=g_i^{-1}B(g_i\Delta_{p_i},1/(2\eta))^c\,\,\,\text{for}\,\,\,1\leq i\leq j.
\end{equation*}

If $\gamma$ is an inverse branch in $\calH_0$ which maps $\Delta_{p_i}$ into $\Delta_{p_l}$, then $\gamma^{-1}U_l\subset U_i$.

As $\gamma p_i\in \Delta_{p_l}$, the definition of $U_l$ implies

\begin{equation*}
g_lU_l\subset B(g_l\gamma p_i,1/(2\eta))^c.
\end{equation*}

Because the maps $g_i$'s are bi-Lipschitz (Lemma <ref>), we obtain

\begin{equation}
\label{invcon1}
g_i U_l\subset B(g_i \gamma p_i,1)^c=B(g_i\gamma g_i^{-1}\infty,1)^c.
\end{equation}

By Lemma <ref>, we have

\begin{align}
\label{invcon2}
(g_i\gamma g_i^{-1})^{-1} B(g_i\gamma g_{i}^{-1}\infty, 1)^c=B((g_i\gamma g_i^{-1})^{-1}\infty, h_{g_i\gamma g_{i}^{-1}\infty})%\nonumber\\
\subseteq B((g_i\gamma g_i^{-1})^{-1}\infty,1).
\end{align}

By (<ref>), we obtain

\begin{equation}
\label{invcon3}
d(g_i\Delta_{p_i}, (g_i\gamma g_i^{-1})^{-1}\infty)\geq 1/\eta.
\end{equation}

Combining (<ref>)-(<ref>) together, we conclude that

\begin{equation*}
g_i \gamma^{-1}U_l \subset B((g_i\gamma g_i^{-1})^{-1}\infty,1)\subset B(g_i\Delta_{p_i},1/(2\eta))^c.\qedhere
\end{equation*}

We prove Proposition <ref> (1) and (<ref>).

The proof is to consider an induced map and reduce the number of cusps by $1$ at a time.

Let $q=p_j$.  Denote $\cup_{1\leq i\leq j-1}\Delta_{p_i}=\Delta-\Delta_{q}$ by $X_1$ and for $x\in X_1$, define

\[n_1(x)=\inf\{n\in\N:\, T_0^n(x)\in X_1 \}.\]
The map $T_1$ is given by  $T_1(x)=T_0^{n_1(x)}(x)$ for $x$ such that $n_1(x)<\infty$. Since $T_0$ is irreducible, this induced system is also irreducible on $X_1$.

\begin{align*}
&\mathcal{H}_q:=\text{the set of the inverse branches of}\,\,\,T_0\,\,\,\text{which are from}\,\,\,\Delta_q\,\,\,\text{to}\,\,\,\Delta_q,\\
&\mathcal{H}_p:=\text{the set of the inverse branches of}\,\,\,T_0\,\,\,\text{which are from}\,\,\,X_1\,\,\,\text{to}\,\,\,X_1,\\
&\mathcal{H}_{pq}:=\text{the set of the inverse branches of}\,\,\,T_0\,\,\,\text{which are from}\,\,\,X_1\,\,\,\text{to}\,\,\,\Delta_q,\\
&\mathcal{H}_{qp}:=\text{the set of the inverse branches of}\,\,\,T_0\,\,\,\text{which are from}\,\,\,\Delta_q\,\,\,\text{to}\,\,\,X_1.
\end{align*}

As $T_1$ is a composition of multiples of $T_0$'s, we have

* either $T_1(x)=\gamma^{-1}x$ with $\gamma\in \mathcal{H}_p$,

* or $T_1(x)=\gamma_{n_1(x)}^{-1}\cdots \gamma_1^{-1} x$ with $\gamma_1\in \mathcal{H}_{qp}$, $\gamma_{n_1(x)}\in \mathcal{H}_{pq}$ and $\gamma_l\in \mathcal{H}_q$ for $l=2,\ldots, n_1(x)-1$.

The string $\gamma_{1}\cdots \gamma_{n_1(x)}$ is an inverse branch of $T_1$. Set

\begin{align*}
&\calH_1:=\text{the set of all inverse branches of}\,\,\, T_1,\\
&\mathcal{H}_q^n:=\{\gamma_1\cdots \gamma_n:\,\gamma_i\in\mathcal{H}_q\,\,\,\text{for}\,\,\,1\leq i\leq n\}\,\,\,\text{for every}\,\,\,n\in \mathbb{N}.
\end{align*}

There exists $C>0$ such that for every $n\in \mathbb{N}$ and for every $\gamma\in\calH_q^n$, we have

\begin{equation*}
|\gamma' (x)|\geq |\gamma'|_\infty /C \,\,\,\text{for any}\,\,\,x\in \Delta_q.
\end{equation*}

We first notice that $|\gamma'(x)|\approx |(g_j\gamma g_j^{-1})'(g_jx)|$.
Write $p=\gamma p_j$. By Lemma <ref>, we have

\[| (g_j\gamma g_j^{-1})' (y)|=\frac{h_{g_jp}}{d(y,g_j\gamma^{-1}p_j)^2}.\]

By Lemma <ref>, we have  $d(g_j\Delta_q,g_j\gamma^{-1}p_j)>1/(2\eta)$. Then for every $y\in g_j\Delta_{p_j}=g_j\Delta_q$, the distance
$d(y,g_j\gamma^{-1}p_j)\in [d(g_j\Delta_q,g_j\gamma^{-1}p_j)\pm \operatorname{diam}(g_j\Delta_q)]$, which implies the lemma.

There exist $C>0$, $\epsilon>0$ such that for every $l\in\N$

\begin{equation*}
%	\mu\{x\in \Delta_{p_1}|\ n(x)>l \}\approx
\sum_{1\leq k\leq l,\ \gamma_k\in\calH_q}|(\gamma_1\cdots \gamma_l)'|_\infty^{\delta}<C(1-\epsilon)^l.
\end{equation*}

Claim: there exists $\epsilon>0$ such that for every $n\in \mathbb{N}$ and for any $h\in \calH_q^n$, we have

\begin{equation}
\label{measure decrease}
\sum_{\gamma\in \calH_q}\mu(h\gamma\Delta_q)\leq (1-\epsilon)\mu(h\Delta_q).
\end{equation}

Proof of the claim:  for measurable set $E\subset \Delta_q$, by Lemma <ref>

\begin{equation}\label{equ:hE}
\mu(hE)=\int_E |h'(x)|_{\S^d}^\delta\dd\mu(x)\approx \int_E |h'(x)|^\delta\dd\mu(x) \in \mu(E) |h'|^{\delta}_\infty[1/C,1].
\end{equation}

Write $F=\cup_{\gamma\in\calH_q} \gamma\Delta_q$. Since $T_0$ is irreducible, we have $\mu(F)<\mu(\Delta_q)$. By (<ref>)

\[\sum_{\gamma\in\calH_q}\mu(h\gamma\Delta_q)=\mu(hF)\leq |h'|_\infty \mu(F)=\frac{\mu(F)}{\mu(\Delta_q-F)}|h'|_\infty \mu(\Delta_q-F)\leq C' \mu(h(\Delta_q-F)) .\]

So we have

\begin{equation*}
(1+1/C')\sum_{\gamma\in \mathcal{H}_q}\mu(h\gamma \Delta_q)\leq \mu(hF)+\mu (h(\Delta_q-F))\leq \mu(h\Delta_q).
\end{equation*}

Using (<ref>), Lemma <ref> and (<ref>) with $E=\Delta_q$, we obtain

\begin{align*}
&\sum_{1\leq k\leq l,\ \gamma_k\in\calH_q}|(\gamma_1\cdots \gamma_l)'|_\infty^{\delta}\leq C \sum_{1\leq k\leq l,\ \gamma_k\in\calH_q}\mu(\gamma_1\cdots \gamma_l\Delta_q)\\
\leq &C \sum_{1\leq k\leq l-1,\ \gamma_k\in\calH_q}(1-\epsilon)\mu(\gamma_1\cdots\gamma_{l-1}\Delta_q) \leq C (1-\epsilon)^l.\qedhere
\end{align*}

We first use Proposition <ref>, (<ref>), Lemma <ref> and <ref> to prove that for the expanding map $T_1$, we have

* There exists $\epsilon_1>0$ such that

\begin{equation}
\label{induce map 1}
\sum_{\gamma\in\calH_1}|\gamma'|_\infty^{\delta-\epsilon_1}<\infty,
\end{equation}

where $|\gamma'|_\infty$ is defined as in (<ref>).

* There exists $\epsilon>0$ such that for every $n\in \mathbb{N}$,

\begin{equation}
\label{induce map 2}
\mu(\{x\in X_1:\,n_1(x)>n+1\})\ll (1-\epsilon)^n.
\end{equation}

The second statement in particular implies that the map $T_1$ is defined almost everywhere in $\Delta-\Delta_q$.

Due to Lemma <ref>, we can find a large $l_0$ such that $\sum_{1\leq k\leq l_0,\ \gamma_k\in\calH_q}|(\gamma_1\cdots \gamma_{l_0})'|_\infty^{\delta}<1$. Then using (<ref>) and submultiplicativity $|(\gamma_1\gamma_2)'|_\infty\leq |(\gamma_1)'|_\infty|(\gamma_2)'|_\infty$, we obtain

$$\sum_{1\leq k\leq l_0,\ \gamma_k\in\calH_q}|(\gamma_1\cdots \gamma_{l_0})'|_\infty^{\delta-\epsilon}<\infty,$$

where $\epsilon>0$ is the constant given by (<ref>). Hence we can find $0<\epsilon_1<\epsilon$ small such that

\begin{equation*}
\sum_{1\leq k\leq l_0,\ \gamma_k\in\calH_q}|(\gamma_1\cdots \gamma_{l_0})'|_\infty^{\delta-\epsilon_1}<1.
\end{equation*}

Using submultiplicativity, we obtain constants $C>0,\rho<1$ such that for $l\in\N$

\begin{equation}\label{equ:1kl}
\sum_{1\leq k\leq l,\ \gamma_k\in\calH_q}|(\gamma_1\cdots \gamma_{l})'|_\infty^{\delta-\epsilon_1}\leq C\rho^l.
\end{equation}

Denote $\sum_{\gamma\in \mathcal{H}_1}|\gamma'|_{\infty}^{\delta-\epsilon_1}$ by $E_q$. For every inverse branch of $T_1$, it can be uniquely decomposed as  $\gamma_0\gamma_1\cdots\gamma_{l}\gamma_{l+1}$ with $\gamma_{l+1}\in\calH_{pq}$, $\gamma_{i}\in\calH_q$ with $i=1,\cdots l$ and $\gamma_0\in\calH_{qp}$.
Using this expression and  submultiplicativity,
we obtain

\begin{equation}\label{equ:ep}
E_q\leq \sum_{\gamma\in\calH_p}|\gamma'
|_\infty^{\delta-\epsilon_1}+\sum_{l\geq 1} (\sum_{\gamma_0\in\calH_{qp}} |\gamma_0'|_\infty^{\delta-\epsilon_1})(\sum_{\gamma_{l+1}\in\calH_{pq}} |\gamma_{l+1}'|_\infty^{\delta-\epsilon_1})
(\sum_{\gamma_i\in\calH_q,1\leq i\leq l}|(\gamma_1\cdots \gamma_l)'|_\infty^{\delta-\epsilon_1}).
\end{equation}

Therefore $E_p$ is also finite due to (<ref>), (<ref>) and (<ref>).

The set of $x$ such that $T_0^n(x)$ is outside of domain of definition of $T_0$ for some $n$ has zero PS measure by Proposition <ref>. We only need to consider the set of $x$ such that $T_0^n(x)$ is  in the domain of definition for every $n$.
If $x\in X_1$ with $n_1(x)>n+1$, then $x$ must be in $\gamma_0\gamma_1\cdots \gamma_n\Delta_q$ with $\gamma_0\in\calH_{qp}$ and $\gamma_i\in\calH_q$ for $1\leq i\leq n$. Therefore
Lemma <ref> implies

\begin{align*}
&\mu(\{x\in X_1:\,n_1(x)>n+1\})\leq \sum_{\gamma_0\in\calH_{qp}}\sum_{\gamma_i\in\calH_q,1\leq i\leq n}\mu( \gamma_0\gamma_1\cdots\gamma_n\Delta_q)\\
\leq& (\sum_{\gamma_0\in\calH_{qp}} |\gamma_0'|_\infty^{\delta})(\sum_{\gamma_i\in\calH_q,1\leq i\leq n}|(\gamma_1\cdots \gamma_n)'|_\infty^{\delta})\ll (1-\epsilon)^n.
\end{align*}

We keep reducing the number of cusps by considering the set $X_2:=X_1-\Delta_{p_{j-1}}$ and the induced map $T_2:X_2\to X_2$ which is constructed similar to $T_1$: in particular, the inverse branches of $T_2$ are compositions of elements in $\mathcal{H}_1$.  Analogs of Lemma <ref> and <ref> for $T_2$ and the replacements of Proposition <ref> and (<ref>)  are (<ref>) and (<ref>) respectively. Using these three ingredients, we can show the properties like (<ref>) and (<ref>) also hold for $T_2$. The proof of Proposition <ref>(1) and (<ref>) will be finished by repeating this.

Now, we will finish proving the rest of results for the coding except Lemma <ref> (UNI).

\begin{equation}\label{equ:Lambda-}
\Lambda_{-}=\Lambda_\Gamma\cap \{|x|>1/(2\eta) \}=\Lambda_\Gamma\cap U_1.
\end{equation}

The contracting map $\gamma$ from $\Delta_0$ to $\Delta_0$ is a composition of maps in $\calH_0$, so the inclusion follows directly from Lemma <ref>.
Write $p=\gamma \infty$. By Lemma <ref>,

\[|(\gamma^{-1})'(x)|_{\S^d}=\frac{h_p}{d(x,p)^2}\frac{1+|x|^2}{1+|\gamma^{-1}x|^2}. \]

For $x\in\Lambda_{-}$,
as $p=\gamma\infty\in\Delta_0$, we have

\begin{equation*}
\frac{1+|x|^2}{d(x,p)^2}\leq \frac{1+|x|^2}{(|x|-\operatorname{diam}(\Delta_0))^2}.
\end{equation*}

The right hand side of the inequality is around $1$ as $|x|\geq 1/(2\eta)$.
For $\gamma^{-1}x\in \Lambda_{ -}$, we have $|\gamma^{-1}x|\geq 1/(2\eta)$. Hence $|(\gamma^{-1})'(x)|_{\S^d}\leq\lambda$ for some $\lambda$ independent of $\gamma$.

By Lemma <ref>, we have

\[|\gamma'(x)|=\frac{h_p}{d(x,\gamma^{-1}\infty)^2}. \]

By Lemma <ref>, we have $\gamma^{-1}\infty\in\Lambda_{-}$, which implies $d(\gamma^{-1}\infty,x)\geq 1/(2\eta)$. Hence

\[
|\gamma'(x)|\leq (2\eta)^2h_p\leq 4\eta^2.\qedhere\]

Proof of Proposition <ref> (4). We need the following lemma, which will also be needed in later sections.

Let $\gamma$ be any element  in $\Gamma$ which does not fix $\infty$. For any $x\in \Delta_0$ and any unit vector $e\in\R^d$, we have

\[\partial_e\log | \gamma'(x)|=-\frac{2\langle x-\xi,e \rangle}{|x-\xi|^2} \]

where $\xi=\gamma^{-1}\infty$.

It can be shown using Lemma <ref> and elementary computation.

For $\gamma\in\calH$, Proposition <ref> (4) can be deduced using Lemma <ref> and the observation that $|\gamma^{-1}\infty|\geq 1/(2\eta)$ (Lemma <ref> and (<ref>)) .

§.§ Verifying UNI

We prove Lemma <ref> in this part.
Let $\Gamma_f$ be the semigroup generated by $\gamma$ in $\calH$ and $\Gamma_b$ be the semigroup generated by $\gamma^{-1}$ with $\gamma\in\calH$. Let $\Lambda_f$ and $\Lambda_b$ be the limit set of $\Gamma_f$ and $\Gamma_b$ on $\partial\H^{d+1}$, that is the set of accumulation points of orbit $\Gamma_f o$ and $\Gamma_b o$ for some $o\in\H^{d+1}$ respectively. It follows from the definition that the limit set $\Lambda_f$ is $\Gamma_f$-invariant and  $\Lambda_b$  is $\Gamma_b$-invariant. Due to <cit.> (convergence property of Möbius transformation), we have that $\Lambda_f$ is a $\Gamma_f$-minimal set and $\Lambda_{b}$ is a $\Gamma_b$-minimal set.

The limit set $\Lambda_b$ is not contained in an affine subspace in $\R^d\cup\{\infty \}$ or a sphere in $\R^d$.

Let $A$ be an affine subspace or a sphere with minimal dimension which contains $\Lambda_b$. Because $\Lambda_b$ is $\Gamma_b$ invariant, the semigroup $\Gamma_b$ must preserve $A$, so does the Zariski closure of $\Gamma_b$. The Zariski closure of a semigroup is a group (see for example <cit.>). The Zariski closures of $\Gamma_f$ and $\Gamma_b$ are the same. Hence $\Gamma_f$ also preserves $A$ and $\Lambda_f$ is in $A$. We claim: $\mu(\Lambda_f)=\mu(\Lambda_{\Gamma}\cap \Delta_0)>0$. Then  because $\Gamma$ is Zariski dense, by <cit.>, we conclude that $\mu(A)$ is non zero if and only if $A=\R^d$, finishing  the proof.

Proof of the claim:  Let $x$ be any point in $\Lambda_{\Gamma}\cap \Delta_0$ such that  $T^nx\in\Lambda_{\Gamma}\cap \Delta_0$ for every $n\in\N$. We can write $x=\gamma_n T^n(x)\in \gamma_n\Delta_0$ for some $\gamma_n\in\calH^n$. Fix any $y\in \Lambda_f$, it follows from Proposition <ref> (3) that $d(\gamma_n y, \gamma_n T^n(x))\to 0$. So $\gamma_n y\rightarrow x$ and $x\in\Lambda_f$. Due to Proposition <ref> (1), the set of $x$'s such that $T^nx\in\Lambda_{\Gamma}\cap \Delta_0$ for every $n\in\N$ is a conull  set in $\Lambda_{\Gamma}\cap \Delta_0$. Hence $\mu(\Lambda_f)=\mu(\Lambda_{\Gamma}\cap \Delta_0)$.

For every $x\in\Lambda_\Gamma\cap\overline\Delta_0$, there exist pairs of points $(\xi_{1m},\xi_{2m}),\ m=1,\cdots, k_x$ in the limit set $\Lambda_b$ and $\epsilon_x'>0$ such that for every unit vector $e\in \R^d$ there exists $m$,

\[|\langle \frac{x-\xi_{1m}}{|x-\xi_{1m}|^2}-\frac{x-\xi_{2m}}{|x-\xi_{2m}|^2},e \rangle|>2\epsilon_x'>0. \]

The map $inv_x:\xi \mapsto \frac{x-\xi}{|x-\xi|^2}$ is an inversion and this map is injective. If there exists a unit vector $e\in\R^d$ such that

\[\langle \frac{x-\xi_{1}}{|x-\xi_{1}|^2}-\frac{x-\xi_{2}}{|x-\xi_{2}|^2},e \rangle=0 \]

for all $\xi_1,\xi_2$ in $\Lambda_b$, then $inv_x(\Lambda_b)$ is contained in an affine subspace parallel to $e^\perp$. Hence $\Lambda_b$ itself is contained in an affine subspace in $\R^d\cup\{\infty \}$ or a sphere in $\R^d$, which contradicts Lemma <ref>. Therefore, for every unit vector $e\in\R^d$, there exist $\xi_1,\xi_2$ in $\Lambda_b$ such that

\[\langle \frac{x-\xi_{1}}{|x-\xi_{1}|^2}-\frac{x-\xi_{2}}{|x-\xi_{2}|^2},e \rangle\neq0. \]

We use continuity and compactness to finish the proof.

Let $\xi$ be any point in $\Lambda_b$. For any $\epsilon_2,\epsilon_3>0$, there exists $n_\xi\in\N$ such that for any $n\geq n_\xi$, there exists $\gamma$ in $\calH^n$ satisfying

\[ d_{\S^d}(\gamma^{-1}\infty,\xi)\leq \epsilon_2,\ |\gamma'|_\infty\leq \epsilon_3. \]

Since $\Lambda_b$ is $\Gamma_b$ minimal, for any point $\xi'\in\Lambda_2$, there exists a sequence $\{\gamma_n^{-1}\}$ in $\Gamma_b$ such that $\gamma_n^{-1}\xi'$ converges to $\xi$ and $|\gamma_n'|_\infty$ tends to zero.
By Lemma <ref>, we know that $\gamma_n^{-1}\Lambda_{-}$ also converges to $\xi$. Hence we can always find a $\gamma$ in $\Gamma_f$ with $|\gamma'|_\infty\leq \epsilon_3$ and $d_{\S^d}(\gamma^{-1}\Lambda_{-},\xi)\leq \epsilon_2$. Let $n_\xi$ be the unique number such that $\gamma\in\calH^{n_\xi}$.

For any $\gamma_1\in \cup_{n\geq 1}\calH^{n}$, we have $|(\gamma_1\gamma)'|_\infty\leq |\gamma_1'|_\infty|\gamma'|_\infty\leq |\gamma'|_\infty$ and

\[d_{\S^d}((\gamma_1\gamma)^{-1}\infty,\xi)=d_{\S^d}(\gamma^{-1}(\gamma_1^{-1}\infty),\xi)\leq d_{\S^d}(\gamma^{-1}\Lambda_{-},\xi)\leq\epsilon_2. \]

Therefore, for any $m= n_\xi+n$, choose any $\gamma_1\in\calH^n$ and then $\gamma_1\gamma\in\calH^m$ and it satisfies Lemma <ref>.

Combining the above two lemmas, by Lemma <ref> and the formula $R_n(\gamma x)=-\log|\gamma'(x)|$ for $\gamma\in\calH^n$, we have

\[|\partial_e(R_n\circ \gamma_{1m}-R_n\circ \gamma_{2m})(y)|=|\langle \frac{y-\gamma_{1m}^{-1}\infty}{|y-\gamma_{1m}^{-1}\infty|^2}-\frac{y-\gamma_{2m}^{-1}\infty}{|y-\gamma_{2m}^{-1}\infty|^2},e \rangle|. \]

Using this expression, Lemma <ref>, <ref> and continuity, we obtain

For every $x\in\Lambda_\Gamma\cap\overline\Delta_0$, there exist $\epsilon_x, \epsilon_x'>0$ such that for any $\epsilon_3>0$, there exists $n_x\in\N$ such that the following holds for any $n\geq n_x$. There exist $k_x\in\N$, $\gamma_{im}\in\calH^n$ with $i=1,2$ and $m=1,\ldots, k_x$ satisfying

* $|\gamma'_{im}|<\epsilon_3$ for every $i=1,2$ and $m=1,\ldots,k_x$.

* for any unit vector $e\in\R^d$, there exists $m\in\{1,\ldots,k_x\}$ such that for any $y\in B(x,\epsilon_x)$,

\[|\partial_e(R_n\circ \gamma_{1m}-R_n\circ \gamma_{2m})(y)|\geq\epsilon_x'>0. \]

For every $x\in \Lambda_\Gamma\cap\overline\Delta_0$, we apply Lemma <ref> to $x$ and get two constants $\epsilon_x, \epsilon_x'>0$. Since $\Lambda_\Gamma\cap\overline\Delta_0$ is compact, we can find a finite set $\{x_1,\cdots,x_l \}$ such that $\cup B(x_j,\epsilon_{x_j}/2)\supset \Lambda_\Gamma\cap\overline\Delta_0$. Let $\epsilon_0=\inf\{\epsilon_{x_j}' \}$ and $\r=\inf\{\epsilon_{x_j}/2 \}$. Take $\epsilon_3=\epsilon_0/C$
and $n_0\geq \sup_{1\leq j\leq l}\{n_{x_j} \}$. Fix any $n\geq n_0$.  Then for every $x_j$, there exists a finite set $\{\gamma_{im}\}$ in $\mathcal{H}^{n}$ satisfying results in Lemma <ref>. We put all these $\gamma_{im}$'s together and this is the finite set in $\mathcal{H}^n$ described in Lemma <ref>.  For any $x\in\Lambda_\Gamma\cap\overline\Delta_0$, it is contained in some $B(x_j,\epsilon_{x_j}/2)$. Then $B(x,\r)\subset B(x_j,\epsilon_{x_j})$. The family $\{\gamma_{im}\}$ for $x_j$ will satisfy nonvanishing condition on $B(x,\r)$, that is for every unit vector $e\in\R^d$ there exists $m$ such that
for any $y\in B(x,\r)$

\[|\partial_e(R_{n_0}\circ \gamma_{1m}-R_{n_0}\circ \gamma_{2m})(y)|\geq\epsilon_0>0. \]

Finally, the inequality $|\D\tau_{im}|_\infty\leq C_2$ is due to (<ref>).

§ SPECTRAL GAP AND DOLGOPYAT-TYPE SPECTRAL ESTIMATE

Let us comment the difference of Dolgopyat's argument between our case and that in [1], [22], [35] and [58].

First difference is cancellation lemma.
In [1] or [22], the measure on the unstable leaf is Lebesgue or absolute continuous. So they only need to find a positive portion of intervals such that the angles of fixed two branches has positive distance to $2\pi\Z$. In [35] and [58], they use the separation property of the limit set and of the coding. They roughly proved that at a suitable scale, for three nearby cylinders, there exists at least one cylinder such that the angles are different. The separation property is the key to obtain different angles.
In our case, the separation property is still true by using an important property of Patterson-Sullivan measure for geometrically finite case, that is Proposition <ref>

Second difference is a boundary issue. Because we use measure property to find separation, our cancellation lemma cannot apply to points near the boundary of $\Delta_0$. Hence we need to carefully treat points near boundary using friendliness of PS measure.

In this section, we prove a Dolgopyat-type spectral estimate and the main result is Proposition <ref>. Our argument is influenced by the one in [1, 2, 5, 22, 35, 58] and there is some technical variation in the current setting. The proof involves  proving a cancellation lemma (Lemma <ref>) and using it to obtain $L^2$ contraction. The rough idea is as follows. Denote the set $\Delta_0\cap \Lambda_{\Gamma}$ by $\Lambda_0$. With the UNI property (Lemma <ref>) available, for each ball $B(y,r)$ with $y\in \Lambda_0$, one uses the doubling property of the PS measure to find a point $x\in B(y,r)\cap \Lambda_0$ such that cancellation happens on $B(x,r')$. Then, to run the classical argument, one needs to find finitely many such pairwise disjoint balls $B(x_i,r')$'s contained in $\Delta_0$ such that  $\sqcup B(x_i,Dr')$ covers $\Delta_0$ for some $D>1$. The difficulty lies in that the balls $B(x_i,r)$'s are produced using PS measure so the position of $B(x_i,r')$'s is in some sense random and  some $B(x_i,r')$ may not be fully contained in $\Delta_0$. To overcome this, we find $B(x_i,r')$'s which only cover a subset of $\Delta_0$ and divide the proof of Proposition <ref> into the cases when the iteration is small and when the iteration is large.

§.§ Twisted transfer operators

For $s\in \mathbb{C}$, let $L_s$ be the twisted transfer operator defined by

\begin{equation}
L_s(u)(x)=\sum_{\gamma\in\calH} |\gamma'(x)|^{\delta+s}u(\gamma x).%=\sum_{\gamma\in\calH}e^{-sR(\gamma x)}|\gamma'(x)|^\delta u(\gamma x).
\end{equation}

For $u:\Delta_0\to \mathbb{C}$, define

\begin{equation*}
\uLip=\max \{|u|_{\infty},\,\ulip\},
\end{equation*}

where $\ulip=\sup_{x\neq y} |u(x)-u(y)|/d(x,y)$, where $d(\cdot, \cdot)$ is the Euclidean distance. Denote by $\text{Lip}(\Delta_0)$ the space of functions $u:\Delta_0\to \mathbb{C}$ with $\uLip <\infty$.  We also introduce a family of equivalent norms on $\text{Lip}(\Delta_0)$:

\begin{equation*}
\ub=\max\{|u|_{\infty},\,\ulip/(1+|b|)\},\,\,\,b\in \mathbb{R}.
\end{equation*}

With  Proposition <ref> available, we obtain the following  lemma by a verbatim of the proof of <cit.>.

Write $s=\sigma+i b$.
The family $s\mapsto L_s$ of operators on $\operatorname{Lip}(\Delta_0)$ is continuous on $\{s\in\C:\ \sigma>-\epsilon_o\}$, where $\epsilon_o$ is given as in Proposition <ref> (4). Moreover, $\sup_{|\sigma|<\epsilon_o} \lVert L_s\rVert_b<\infty$.

Define the PS measure $\mu_E$ on $\Delta_0$ with respect to the Euclidean metric by

\[\dd\mu_E(x)=(1+|x|^2)^\delta\dd \mu(x). \]

Using the quasi-invariance of the PS measure $\mu$, we obtain that the operator $L_0$ preserves the measure $\mu_E$ by a straightforward computation.
Our main result of Dolgopyat argument is the following $L^2$ contracting proposition.

There exist $C>0,\ \beta <1, \epsilon>0$ and $b_0>0$ such that for all $v$ in $\operatorname{Lip}(\Delta_0)$, $m\in\N$ and $s=\sigma+ib$ with $|\sigma|\leq\epsilon$ and $|b|>b_0$, we have

\begin{equation*}
\int|L_s^{m}v|^2\dd\mu_E\leq C\beta^m\|v\|^2_b.
\end{equation*}

The proof will be given at the end of Section <ref>.

For any measurable subset $E$ of $\Delta_0$, we have

\begin{align*}
\tilde\nu(T^{-1}E)=\sum_{j}\tilde\nu(\gamma_j(E))=\int_E\sum_j \frac{f_0(\gamma_jx)}{f_0(x)}|\D \gamma_j'x|^\delta \dd\tilde\nu(x)=\int_E \tilde{L}_01(x)\dd\tilde\nu(x)=\int_E 1(x)\dd\tilde\nu(x)=\tilde\nu(E)
\end{align*}

So in fact the invariant measure $\nu$ is given by the formula (<ref>).

Recall that $\nu$ is the unique $T$-invariant ergodic probability measure on $\Delta_0\cap \Lambda_{\Gamma}$ which is absolutely continuous with respect to the PS measure $\mu$ with a positive Lipschitz density function $\bar f_0$. So $\nu$ is also absolutely continuous with respect to $\mu_{E}$ with a positive Lipschitz density function $f_0$. Based on these, it is a classical result that the operator $L_0$ acts on $\operatorname{Lip}(\Delta_0)$ and  has a spectral gap and a simple isolated eigenvalue at $1$ with $f_0$ the corresponding eigenfunction.

For $\sigma\in \mathbb{R}$ close enough to $0$, $L_{\sigma}$ acting on $\text{Lip}(\Delta_0)$ is a continuous perturbation of $L_0$ (see Lemma <ref>). Hence, it has a unique eigenvalue $\lambda_{\sigma}$ close to $1$, and the corresponding eigenfunction $f_{\sigma}$ (normalized so that $\int f_{\sigma}=1$) belongs to $\text{Lip}(\Delta_0)$, strictly positive, and tends to $f_0$ in $\text{Lip}(\Delta_0)$ as $\sigma\to 0$. Choose a sufficiently small $\epsilon\in (0,\epsilon_o)$ such that for $\sigma\in (-\epsilon,\epsilon)$, $f_{\sigma}$ is well defined and

\begin{equation*}
1/2\leq \lambda_{\sigma}\leq 2,\,\,\, f_{0}/2\leq f_{\sigma}\leq 2 f_{0},\,\,\, |f_0|_{\text{Lip}}/2\leq |f_{\sigma}|_{\text{Lip}}\leq 2|f_0|_{\text{Lip}}.
\end{equation*}

For $s=\sigma+ib$ with $|\sigma|<\epsilon$ and $b\in \mathbb{R}$, define a modified transfer operator $\tilde{L}_{s}$ by

\begin{equation}
\tilde{L}_{s}(u)=(\lambda_{\sigma}f_{\sigma})^{-1}L_{s}(f_{\sigma}u).
\end{equation}

It satisfies $\tilde{L}_{\sigma}1=1$, and $|\tilde{L}_{s}u|\leq \tilde{L}_{\sigma}|u|$.

There is a constant $C_{\sixt}>1$ such that

\begin{equation}
|\LL^n_s v|_{\operatorname{Lip}} \leq C_{\sixt} (1+|b|) |v|_{\infty} +C_{\sixt} \lambda^n |v|_{\operatorname{Lip}}
\end{equation}

holds for any $s=\sigma+ib$ with $|\sigma|<\epsilon$, and all $n\geq 1,\,v\in \operatorname{Lip}(\Delta_0)$, where $\lambda$ is given as in Proposition <ref>.

The proof of this lemma is a verbatim of proof of <cit.>. The following lemma can be deduced from
Lemma <ref>  by a straightforward computation.

We have $\lVert \LL^n_s\rVert_b\leq 2C_{\sixt}$ for all $s=\sigma+ib$ with $|\sigma|<\epsilon$ and all $n\geq 1$.

§.§ Cancellation lemma

For $b\in\R$, let

\begin{align*}
\mathcal{C}_b=&\{(u,v):u,v\in Lip(\Delta_0),u>0,0\leq |v|\leq u,\\
& |u(x)-u(y)|,|v(x)-v(y)|\leq |b|u(x)|x-y| \}.
\end{align*}

The main result of this subsection is the cancellation lemma (Lemma <ref>) and the proof is inspired by the proof of analogous results in [35] and [58]. We start with detailing all the constants.

Let $C_{\sev}$ be the constant which will be specified in (<ref>). We define the cone

For $b\in \mathbb{R}$, let

\begin{align*}
\mathcal{C}_b=&\{(u,v):\,u,\,v\in \text{Lip}(\Delta_0),\,u>0,\,0\leq |v|\leq u,\,|\log u|_{\text{Lip}}\leq C_{\sev} |b|,\\
&|v(x)-v(y)|\leq C_{\sev} |b| u(y)d(x,y)\,\,\,\text{for all}\,\,\,x,y\in \Delta_0\}.
\end{align*}

Let $r>0$ and $\epsilon_0>0$ be the same constants as the ones in Lemma <ref>. We apply Lemma <ref> with $C=16C_{\sev}$. Let $n_0$ be a sufficiently large integer which satisfies Lemma <ref> and the inequality

\begin{equation}
\label{cone const 1}
\lambda^{n_0}C_{\sev}(1+\operatorname{diam}(\Delta_0))\leq 1.
\end{equation}

Let $\gamma_{mj}$, with $m=1,2$, $j=1,\ldots,j_0$ be the inverse branches given by Lemma <ref>.

Let $k\in\N$ be such that

\begin{equation}
\label{equ:k}
\end{equation}

where $C_2$ is given in (<ref>).

Note that the measure $\nu$ is absolutely continuous with respect to the PS measure $\mu$. Let $D>0$ be such that for all $x\in\Lambda_\Gamma\cap \Delta_0$ and $r'\leq 1/C_{\three}$ with $C_{\three}$ given in  Proposition <ref>

\begin{equation}
\label{equ:D}
\nu(B(x,Dr'))>\nu(B(x,(k+2)r')).
\end{equation}

Let $\eps_2>0$ be such that

\begin{equation}
\label{equ:epsilon3}
(2C_{\sev}\eps_2+1/4)e^{2C_{\sev}\eps_2}\leq 3/4.
\end{equation}

Let $\eps_3>0$ be such that

\begin{equation}\label{equ:epsilon2}
\eps_3(D+2)< \min\{\eps_2,\ r,\ 1/C_{\three}\},\ \eps_3(D+2)(C_2+\epsilon_0)<3\pi/2,\ \eps_3k \epsilon_0<\pi.
\end{equation}

Recall the notation $\tau_{mj}$ introduced in Lemma <ref>. For $s=\sigma+ib\in\C$, define

There exists $0< \eta_0<1$ such that the following holds. For $s=\sigma+ib$ with $|\sigma|\leq \epsilon$, $|b|>1$, for $(u,v)\in C_b$, and for any $y\in\Lambda_0$, there exists $x\in B(y,\eps_3D/|b|)\cap \Lambda_\Gamma$ such that we have the following:
there exists $j\in \{1,\ldots,j_0\}$ such that one of the following  inequalities holds on $B(x,\eps_3/|b|)$:

\begin{align*}
\textbf{type } \gamma_{1j}:\ 	|A_{s,\gamma_{1j}}(v)+A_{s,\gamma_{2j}}(v)|\leq\eta_0A_{\sigma,\gamma_{1j}}(u)+A_{\sigma,\gamma_{2j}}(u),\\
\textbf{type } \gamma_{2j}:\ 	|A_{s,\gamma_{1j}}(v)+A_{s,\gamma_{2j}}(v)|\leq A_{\sigma,\gamma_{1j}}(u)+\eta_0 A_{\sigma,\gamma_{2j}}(u).
\end{align*}

We first prove a quick estimate.

Let $\eps_2$ be the constant defined in (<ref>).
For any $|b|>1$, for $(u,v)\in C_b$ and  for a ball $Z$ of radius $\eps_2/|b|$, we have

* $\inf_Zu\geq e^{-2C_{\sev}\eps_2}\sup_Zu;$

* either $|v|\leq \frac{3}{4}u $
for all $x\in Z$
or $|v|\geq \frac{1}{4}u $
for all $x\in Z$.

The first inequality due to $|\log u(x)-\log u(y)|%\leq |u(x)-u(y)|/|u(x)|
\leq C_{\sev}|b||x-y|$ for every $x,y\in\Delta_0$.

Suppose there exists $x_0\in Z$ such that $|v(x_0)|\leq \frac{1}{4}u(x_0)$. Then

\begin{align*}
|v(x)|&\leq |v(x)-v(x_0)|+\frac{1}{4}u(x_0)\leq C_{\sev}|x-x_0||b|\sup_Zu+\frac{1}{4}\sup_Zu\\
&\leq (2C_{\sev}\eps_2+\frac{1}{4})\sup_Zu\leq (2C_{\sev}\eps_2+\frac{1}{4})e^{2C_{\sev}\eps_2}\inf_Z u\leq \frac{3}{4}u(x).\qedhere
\end{align*}

It follows (<ref>) that there exists $x_0\in (B(y,\eps_3 D/|b|)-B(y,(k+2)\eps_3/|b|))\cap\Lambda_\Gamma$.
Let $B_1=B(y,\eps_3/|b|)$, $B_2=B(x_0,\eps_3/|b|)$ and $\hat{B}$ the smallest ball containing $B_1\cup B_2$. For all $x\in B_1,x'\in B_2$, we have

\begin{equation}\label{equ:xx'}
d(x,x')\in \frac{\eps_3}{b}[k,D+2].
\end{equation}

In view of (<ref>), the radius of $\hat{B}$ is smaller than $\eps_2/|b|$ and it is contained in $B(y,r)$. Let $e_0=(y-x_0)/|y-x_0|$.

By Lemma <ref> for the point $y$ there exists $j$ in $\{1,\cdots,j_0\}$ such that (<ref>) holds for $B(y,r)$ with $e=e_0$. From now on, $j$ is fixed, so we abbreviate $(\gamma_{1j},\gamma_{2j})$ to $(\gamma_1,\gamma_2)$ and $(\tau_{1j},\tau_{2j})$ to $(\tau_1,\tau_2)$.

Due to $|\gamma_m'|_\infty\leq \lambda\leq 1$, the radius of $\gamma_{m}\hat{B}$ is smaller than $\epsilon_{2}/|b|$. So we can apply Lemma <ref> to $\gamma_{m}\hat{B}$ and we have that
$|v(\gamma_mx)|\geq\frac{1}{4} u(\gamma_mx)$ for all $x\in \hat{B}$ or $|v(\gamma_mx)|\leq\frac{3}{4} u(\gamma_mx)$ for all $x\in \hat{B}$. 	Suppose that

\[|v(\gamma_mx)|\leq \frac{3}{4}u(\gamma_mx) \]

holds for some $m\in\{1,2 \}$ for all $x\in \hat B$. Then Lemma <ref> can be proved by a straightforward computation.

Suppose that for $x\in \hat{B}$ and $m=1,2$

\begin{equation}\label{equ:notsmall}
|v(\gamma_mx)|\geq \frac{1}{4}u(\gamma_mx).
\end{equation}

Claim: Under the assumption of (<ref>), there exists $C_{\eig}>0$ independent of $b$ and $(u,v)$ such that for $l\in \{1,2\}$, we have

\begin{equation}\label{claim}
\text{either}\,\,\,\left|\frac{A_{s,\gamma_1}(v)}{A_{s,\gamma_2}(v)}\right|\leq C_{\eig}\text{ for all }x\in B_l\text{ or }\left|\frac{A_{s,\gamma_2}(v)}{A_{s,\gamma_1}(v)}\right|\leq C_{\eig}\text{ for all }x\in B_l.
\end{equation}

Fix any $x_0\in \Delta_0$. Due to $|\tau_m'|_{\infty}\leq C_2$ (see (<ref>)), we have for any $x\in \hat{B}$,
$$|\tau_1(x)-\tau_2(x)|\leq |\tau_1(x_0)-\tau_2(x_0)|+2C_2|x-x_0|.$$
Hence there exists a constant $C({\tau_1,\tau_2})$ depending on $\tau_1,\tau_2$ such that

\[\left|\frac{A_{s,\gamma_1}(v)}{A_{s,\gamma_2}(v)}\right|\leq C(\tau_1,\tau_2)\frac{f_\sigma(\gamma_1x)u(\gamma_1x)}{f_\sigma(\gamma_2x)u(\gamma_2x)}. \]

For the middle term,

\[\frac{f_\sigma(\gamma_1x)}{f_\sigma(\gamma_2x)}\leq 4\frac{\sup f_0}{\inf f_0}. \]

Since the radius of $\gamma_2B_l$ is less than $\eps_2/|b|$, using Lemma <ref>, we have for every $x$ in $B_l$

\[ \frac{u(\gamma_1x)}{u(\gamma_2x)}\leq  \frac{\sup_{B_l}u(\gamma_1)}{\inf_{B_l}u(\gamma_2)}\leq e^{2C_{\sev}\eps_2}\frac{\sup_{B_l}u(\gamma_1)}{\sup_{B_l}u(\gamma_2)}.  \]

Putting these together, we have

\[\left|\frac{A_{s,\gamma_1}(v)}{A_{s,\gamma_2}(v)}\right|\leq C_{\eig}\frac{\sup_{B_l}u(\gamma_1)}{\sup_{B_l}u(\gamma_2)} \]

where $C_{\eig}=4C(\tau_1,\tau_2)e^{2C_{\sev}\eps_2}\frac{\sup f_0}{\inf f_0}$.	We have a similar inequality for $\left|\frac{A_{s,\gamma_2}(v)}{A_{s,\gamma_1}(v)}\right|$.  Note that either $\frac{\sup_{B_l}u(\gamma_1)}{\sup_{B_l}u(\gamma_2)}\leq 1$ or $\frac{\sup_{B_l}u(\gamma_2)}{\sup_{B_l}u(\gamma_1)}\leq 1$. The proof of the claim finishes .

Now we start to compute the angle and our definitions are only for $x\in\hat{B}$. The function $\arg(v(\gamma_mx))$ is well defined because $|v(\gamma_mx)|\geq u(\gamma_mx)/4>0$. Let

\[\Theta(x)=b(\tau_1(x)-\tau_2(x)), \ V(x)=\arg(v(\gamma_1x))-\arg(v(\gamma_2x)), \]

and let

We apply Lemma <ref> to $\hat{B}$ and obtain that for $x\in \hat{B}$,

\[|\partial_{e_0}\Theta(x)|\geq |b|\epsilon_0, \,\,\,|\Theta'(x)|\leq |b|C_2 . \]

For the angle function,

by (<ref>) and (<ref>), we have for $i\in\{1,2\}$ and $x,x'\in\hat{B}$

\begin{align*}
|\arg v(\gamma_ix)-\arg v(\gamma_ix')|&=|\operatorname{Im}(\log v(\gamma_ix)-\log v(\gamma_ix'))|\leq \frac{|v(\gamma_ix)-v(\gamma_ix')|}{|v(\gamma_ix)|}\\
&\leq C_{\sev}|b|\frac{u(\gamma_ix)|\gamma_ix-\gamma_ix'|}{|v(\gamma_ix)|}\leq |b|\epsilon_0|x-x'|/4.
\end{align*}

This implies that for $x,x'\in\hat{B}$

\[ |V(x)-V(x')|\leq |b|\epsilon_0|x-x'|/2. \]

Combining the estimates for $\Theta$ and $V$, we obtain for $x,x'\in\hat{B}$

\begin{equation}\label{equ:phileq}
|\Phi(x)-\Phi(x')|\leq b(C_2+\epsilon_0)|x-x'|,
\end{equation}

and for $x, x+te_0\in \hat{B}$ with $t\in\R^+$,

\[|\Phi(x)-\Phi(x+te_0)|\geq b\epsilon_0 t/2.\]

Hence for $x_1=y,\ x_2=x_0$ which are the centers of $B_1$ and $B_2$ respectively, by (<ref>),

\begin{equation}\label{equ:phixx'}
|\Phi(x_1)-\Phi(x_2)|\in \eps_3[k\epsilon_0/2,(D+2)(C_2+\epsilon_0)].
\end{equation}

We claim that there exists $l\in \{1,2\}$ such that

\begin{equation}\label{equ:phiz}
\end{equation}

If not so, then both the distance from $\Phi(x_1)$ to $2\pi \Z$ and that from $\Phi(x_2)$ to $2\pi\Z$ are less than $\epsilon_4$. By (<ref>) and (<ref>)

\[|\Phi(x_1)-\Phi(x_2)|\leq \eps_3(D+2)(C_2+\epsilon_0) \leq 3\pi/2< 2\pi-2\epsilon_4. \]

Hence $\Phi(x_1),\Phi(x_2)$ are in a ball $(2n\pi-\epsilon_4,2n\pi+\epsilon_4)$ with $n\in\Z$. This implies that

\[|\Phi(x_1)-\Phi(x_2)|\leq 2\epsilon_4=\eps_3k\epsilon_0/4, \]

contradicting with (<ref>).

Without loss of generality, we may assume (<ref>) holds for $x_1$. For any $x$ in the ball $B_1$, by (<ref>) and (<ref>)

\[|\Phi(x)-\Phi(x_l)|\leq (C_2+\epsilon_0)\eps_3\leq k\eps_3\epsilon_0/16=\epsilon_4/2. \]

Combined with (<ref>), we have

\begin{equation}
\label{equ:awayinteger}
\end{equation}

In conclusion, there exists $l\in \{1,2\}$ such that for all $x\in B_l$, $d(\Phi(x),2\pi\Z)>\epsilon_4/2$ and (<ref>) holds. Without loss of generality, we may assume $|A_{s,\gamma_1}(v)(x)/A_{s,\gamma_2}(v)(x)|\leq C_{\eig}$ for all $x\in B_l$. By an elementary inequality <cit.>, there exists $0<\eta_0<1$ depending on $\epsilon_4$ and $C_{\eig}$ such that on $B_l$

\begin{equation*}
|A_{s,\gamma_1}(v)+A_{s,\gamma_2}(v)|\leq \eta_0|A_{s,\gamma_1}(v)|+|A_{s,\gamma_2}(v)|\leq\eta_0A_{\sigma,\gamma_1}(u)+A_{\sigma,\gamma_2}(u).\qedhere
%\\\text{or}\,\,\,&|A_{s,\gamma_1}(v)+A_{s,\gamma_2}(v)|\leq A_{\sigma,\gamma_1}(u)+\eta_0 A_{\sigma,\gamma_2}(u).
\end{equation*}

For $b$ with $|b|\geq 1$, let

\begin{equation}\label{equ:deltab}
\Delta_b=\{x\in\Delta_0|\ d(x,\partial\Delta_0)>\frac{\eps_3(D+1)}{|b|} \}.
\end{equation}

For any $(u,v)\in \calC_b$, We can find $\{x_i\}_{1\leq i\leq l_0}\subset\Lambda_0:=\Lambda_{\Gamma}\cap \Delta_0$ such that $B(x_i,\eps_3/|b|)$'s are disjoint balls contained in $\Delta_0$,

\[\Lambda_0\cap\Delta_b\subset \cup_{1\leq i\leq l_0}B(x_i,2\eps_3D/|b|), \]

and on each $B(x_i,\eps_3/|b|)$ one of the $2j_0$ inequalities in Lemma <ref> holds. In fact, suppose we have already found some points $x_i$'s but $\cup B(x_i,2\eps_3D/|b|)$ don't cover the set $\Lambda_0\cap\Delta_b$. Then for a point $y\in \Lambda_0\cap \Delta_b-\cup B(x_i,2\eps_3D/|b|)$, we apply Lemma <ref> to $y$ and obtain a point $x\in B(y,\eps_3 D/|b|)\cap \Lambda_0$ such that Lemma <ref> holds on $B(x,\eps_3/|b|)$. Moreover, the ball $B(x,\eps_3/|b|)$ is contained in $\Delta_0$ and it is disjoint from $\cup B(x_i,\eps_3/|b|)$.

Let $B_i=B(x_i,\eps_3/|b|)$ and $\tilde{B_i}=B(x_i,\eps_3/(3|b|))$  for $i=1,\cdots,l_0$. Let $\eta\in[\eta_0,1)$ and define a $C^1$ function $\chi:\Delta_0\rightarrow[\eta,1]$ as follows:
it equals 1 outside of $\cup_{m,j,i} \gamma_{mj} B_i$; for each $B_i$, if $B_i$ is of type $\gamma_{mj}$, let $\chi(\gamma_{mj}(y))=\eta$ for $y\in \tilde{B_i}$ and $\chi\equiv 1$ on other $\gamma_{m'j'}B_i$. We can choose $\eta$ close to 1 and independent of $b$ such that $|\chi'(x)|\leq |b|$ for all $x\in \Delta_0$.

Under the same assumptions as in Lemma <ref>, for $(u,v)\in \mathcal{C}_b$ and $\chi=\chi(b,u,v)$ a $C^1$ function described as above, we have

\begin{equation*}
\label{equ:con ineq}
|\LL_s^{n_0} v|\leq \LL_\sigma^{n_0}(\chi u).
\end{equation*}

Define  $J_i=B(x_i,2\eps_3D/|b|)$ for $i=1,\cdots,l_0$ and let $\tilde{B}=\cup\tilde{B_j}$.

Suppose that $w$ is a positive Lipschitz function with $|\log w(x)-\log w(y)|\leq K|b||x-y|$ for some $K>0$. Then

\begin{equation}
\label{equ:contr int}
\int_{\tilde{B}}w\dd\nu \geq \epsilon_4\int_{\Delta_b} w\ \dd\nu,
\end{equation}

with $\epsilon_4=\epsilon_5 e^{-4\eps_3DK}$, where $\epsilon_5$ comes from doubling property only depending on $D$ and $\nu$.

Since $\cup_j J_i$ covers $\Delta_b$, it is sufficient to prove for each $i$ we have a similar inequality.
Due to hypothesis, we obtain $\inf_{\tilde{B_i}}w\geq e^{-4\eps_3DK}\sup_{J_i}w$. By doubling property, there exists $\epsilon_5$ depending on $D$ such that

\[\nu(\tilde{B_i})\geq\epsilon_5\nu(J_i). \]

\[\int_{\tilde{B_i}}w\ \dd\nu\geq \nu(\tilde{B_i})\inf_{\tilde{B_i}}w\geq \epsilon_5\nu(J_i) e^{-4\eps_3DK}\sup_{J_i}w\geq \epsilon_4\int_{J_i}w\ \dd\nu.\qedhere \]

§.§ Invariance of Cone Condition

We define the constants

\begin{align}
\label{cone const 2}
&C_{\sev}'=16(\delta+\epsilon)C_2|f_0|_{\infty} |f_0^{-1}|_{\infty}+16 |f_0^{-1}|_{\infty}|f_0|_{\text{Lip}}+4C_2+2,\\
\label{cone const 3}
&C_{\sev}=\max\{8|f_0^{-1}|_{\infty} |f_0|_{\text{Lip}} +(\delta+3)C_2+1+4|f_0|_{\infty}|f_0^{-1}|_{\infty}C_{\sev}', 6C_{\sixt}\}.
\end{align}

Let $C_{\sev}>0$ be the constant defined in (<ref>) and $n_0$ be the constant defined in (<ref>). For $s=\sigma+ib$ with $|\sigma|<\epsilon$ and $|b|>1$, for $(u,v)\in \mathcal{C}_b$,
we have

\begin{equation}
(\LL^{n_0}_{\sigma}(\chi u),\,\LL^{n_0}_s v)\in \mathcal{C}_b,
\end{equation}

where $\chi=\chi (b,u,v)$ is the same as the one in Corollary <ref>.

The proof is a verbatim of the proof of <cit.>.

§.§ $L^2$ contraction for bounded iterations

In this part, we will prove Proposition <ref> for the case when $m$ bounded by $\log |b|$. Compared with [2], where they can finish the proof of an analog of Proposition <ref> at this stage, we have the difficulty about the boundary. More precisely, Proposition <ref> is one of the ingredients to obtain Proposition <ref>. Now the integration region of the right hand side of (<ref>)  is $\Delta_b$, which is smaller than $\Delta_0$, so it just enables us to obtain $L^2$ contraction for bounded iterations. For large iteration, we will use a Lipschitz contraction lemma (Lemma <ref>) to obtain $L^2$ contraction in the next subsection.

For $|b|>1$ and $v\in \operatorname{Lip}(\Delta_0)$, if $|v|_{\operatorname{Lip}}\geq C_{\sev} |b| |v|_{\infty}$, then

\[\|\LL_s^{n_0}v\|_b\leq \frac{9}{10}\|v\|_b. \]

We have

\[|\LL_s^{n_0}v|_\infty\leq |v|_\infty\leq \frac{1}{C_{\sev}|b|}|v|_{Lip}\leq \frac{2}{C_{\sev}}\|v\|_b. \]

By Lemma <ref>, we obtain

\[|\LL_s^{n_0}v|_{\operatorname{Lip}}\leq C_{\sixt}(1+|b|)|v|_\infty+C_{\sixt}\lambda^{n_0}|v|_{\operatorname{Lip}}\leq (1+|b|)(\frac{C_{\sixt}(1+|b|)}{C_{\sev}|b|}+C_{\sixt}\lambda^{n_0})\|v\|_b. \]

The proof is complete by using the inequalities $C_{\sev}\geq 6C_{\sixt}$ and $\lambda^{n_0}C_{\sev}\leq 1$.

There exist $C_{\nin}>0$ and $\beta<1$ such that for all $s=\sigma+ib$ with $|\sigma|<\epsilon$ and $|b|$ large enough and $m\leq [C_{\nin}\log|b|]$

\begin{equation}\label{equ:L2Linfinty}
\int |\LL^{mn_0}_{s} v|^2\dd\nu \leq \beta^m\|v\|_b^2.
\end{equation}

If for all $0\leq p\leq m-1$, we have $|\LL_s^{pn_0}v|_{\text{Lip}}\geq C_{\sev} |b| |\LL_s^{pn_0}v|_{\infty}$, then by Lemma <ref>,

\[\int |\LL_s^{mn_0}v|^2\dd\nu\leq \|\LL_s^{mn_0}v\|_b^2\leq (\frac{9}{10})^m\|v\|_b^2.  \]

Otherwise, suppose $p$ is the smallest integer such that $|\LL_s^{pn_0}v|_{\operatorname{Lip}}\leq C_{\sev}|b||\LL_s^{pn_0}v|_\infty$. We consider $v'=\LL_s^{pn_0}v$. Then Lemma <ref> implies $\|v'\|_b\leq (\frac{9}{10})^p\|v\|_b$. We only need to show that

\[\int|\LL_s^{(m-p)n_0}v'|^2\dd\nu\leq \beta^{m-p}\|v'\|_b^2. \]

We reduce to the case when $p=0$, that is $|v|_{\operatorname{Lip}}\leq C_{\sev}|b||v|_\infty$.
Define $u_0\equiv 1,\,v_0=v/|v|_{\infty}$ and induitively,

\begin{equation*}
u_{m+1}=\LL^{n_0}_{\sigma}(\chi_{m}u_{m}),\,\,\, v_{m+1}=\LL^{n_0}_{s}(v_m),
\end{equation*}

where $\chi_m=\chi (b,u_m,v_m)$. It is immediate that $(u_0,v_0)\in \mathcal{C}_b$, and it follows from Lemma <ref> that $(u_m,v_m)\in \mathcal{C}_b$ for all $m$. Hence in particular the $\chi_m$'s are well defined.

We will show that there exist $\beta_1\in (0,1)$,  $\kappa>0$ and $C>0$ such that for all $m$

\begin{equation}
\label{induction eq}
\int u_{m+1}^2 \dd\nu \leq \beta_1 \int u_m^2 \dd\nu+C|b|^{-\kappa}.
\end{equation}

Then note that

\begin{equation*}
|\LL^{mn_0}_s v|=|v|_{\infty} |\LL^{mn_0}_s v_0|=|v|_{\infty} |v_m|\leq |v|_{\infty} u_m.
\end{equation*}

As a result,

\begin{equation*}
\int |\LL^{mn_0}_s v|^2\dd\nu\leq |v|_{\infty}^2 \int u_m^2 \dd\nu \leq |v|^2_{\infty}(\beta_1^m \int u^2_0 \dd\nu+C|b|^{-\kappa}\sum_{0\leq l\leq m-1}\beta_1^l ) \leq(\beta_1^m+C|b|^{-\kappa}/(1-\beta_1)) |v|_{\infty}^2.
\end{equation*}

We can find $C_{\nin}>0$ and $\beta<1$ such that for any large enough $|b|$, (<ref>) holds for all $m\leq [C_{\nin}\log|b|]$.

Now we prove (<ref>). By definition

\begin{align*}
u_{m+1}(x)&=\lambda_{\sigma}^{-n_0} f_{\sigma}^{-1}(x) \sum_{\gamma\in \calH^{n_0}} |\gamma'(x)|^{\delta+\sigma} f_{\sigma}(\gamma x) \chi_{m}(\gamma x) u_m(\gamma x)\\
&= \lambda_{\sigma}^{-n_0} f_{\sigma}^{-1}(x)\sum_{\gamma\in \calH^{n_0}} \left(|\gamma'(x)|^{\delta/2} f_{\sigma}^{1/2}(\gamma x) u_m(\gamma x)\right) \left(|\gamma'(x)|^{\delta/2+\sigma} f_{\sigma}^{1/2}(\gamma x) \chi_m(\gamma x)\right),
\end{align*}

so by Cauchy-Schwarz

\begin{align*}
u_{m+1}^2(x)\leq& (\lambda_{\sigma}^{-n_0} f_{\sigma}(x))^{-2}\left(\sum_{\gamma\in \calH^{n_0}} |\gamma'(x)|^{\delta} f_{\sigma}(\gamma x) u_{m}^2(\gamma x)\right) \left(\sum_{\gamma\in \calH^{n_0}} |\gamma'(x)|^{\delta+2\sigma} f_{\sigma}(\gamma x) \chi_m^2(\gamma x)\right)\\
\leq & \xi (\sigma) \LL^{n_0}_0 (u^2_m) \LL^{n_0}_{2\sigma} (\chi_m^2)
\end{align*}

where (noting that $\lambda_0=1$)

\begin{equation*}
\xi (\sigma) =(\lambda^{-2}_{\sigma} \lambda_{2\sigma})^{n_0} \left|\frac{f_0}{f_{\sigma}}\right|_{\infty} \left|\frac{f_{2\sigma}}{f_{\sigma}}\right|_{\infty} \left| \frac{f_{\sigma}}{f_0}\right|_{\infty} \left|\frac{f_{\sigma}}{f_{2\sigma}}\right|_{\infty}.
\end{equation*}

As in Proposition <ref>, we write $\Delta_0=\tilde{B}\sqcup \tilde{B}^c$. Let $\mathcal{H}_c$ be the set of inverse branches given by Lemma <ref>. If $y\in \tilde{B}$, then there exists $\gamma_i\in \mathcal{H}_{c}$  such that

\begin{align*}
(\LL^{n_0}_{2\sigma}\chi^2_{m})(y)\leq & \lambda^{-n_0}_{2\sigma} f_{2\sigma} (y)^{-1} \left(\eta^2 |\gamma'_i(y)|^{\delta+2\sigma} f_{2\sigma}(\gamma_iy)+\sum_{\gamma\in \calH^{n_0}\backslash \{\gamma_i\}} |\gamma'(y)|^{\delta+2\sigma} f_{2\sigma}(y)\chi_m^2(\gamma y) \right)\\
=& (\LL^{n_0}_{2\sigma} 1)(y)-(1-\eta^2) \lambda^{-n_0}_{2\sigma} f_{2\sigma}(y)^{-1} |\gamma'_i(y)|^{\delta+2\sigma}f_{2\sigma}(\gamma_iy)\\
\leq & 1-(1-\eta^2) 2^{-(n_0+2)} \inf f_0 \cdot |f_0|_{\infty}^{-1} \cdot \inf_{\{\gamma_i\in \mathcal{H}_c\}} |\gamma'_i|^{\delta+2\sigma}=\eta_1<1,
\end{align*}

In this way we obtain that there exists $\eta_1<1$ such that

\begin{equation*}
\begin{cases}
\xi (\sigma) \eta_1 (\LL^{n_0}_0 u_m^2)(y), & y\in \tilde{B},\\
\xi (\sigma) (\LL^{n_0}_0 u_m^2)(y), & y\in \tilde{B}^c.
\end{cases}
\end{equation*}

Since $(u_m,v_m)\in \mathcal{C}_b$, it follows in particular that $|\log u_m|_{\text{Lip}}\leq C_{\sev} |b|$. Hence by (<ref>),

\begin{equation*}
u_m^2(\gamma x)/u_m^2(\gamma y)\leq \exp (2C_{\sev}\lambda^{n_0} |b|d(x,y))\leq \exp (2|b|d(x,y)).
\end{equation*}

Let $w=\LL^{n_0}_0(u_m^2)$. Then

\begin{align*}
\frac{w(x)}{w(y)}=\frac{f_0(y)\sum_{\gamma\in \calH^{n_0}}|\gamma'(x)|^{\delta} f_0(\gamma x)u_m^2(\gamma x) }{f_0(x) \sum_{\gamma\in \calH^{n_0}} |\gamma'(y)|^{\delta} f_0(\gamma y) u_m^2(\gamma y)}\leq \exp \left(\left(2|f_0^{-1}|_{\infty} |f_0|_{\text{Lip}} +\delta C_2 +2|b|\right) d(x,y)\right).
\end{align*}

Hence $|\log w|_{\text{Lip}}\leq K|b|$ with $K=2|f_0^{-1}|_{\infty} |f_0|_{\text{Lip}} +2C_2+2$. Using Proposition <ref>, we have

\begin{equation*}
(1-\eta_1) \int_{\tilde{B}} w\dd\nu \geq \epsilon_4 (1-\eta_1)\int_{\Delta_b} w \dd\nu.
\end{equation*}

Setting $\beta'=1-\epsilon_4(1-\eta_1)$, we can further write

\begin{equation*}
\eta_1\int_{\tilde{B}} w\dd\nu+\int_{\Delta_b-\tilde{B}} w\dd\nu\leq \beta' \int_{\Delta_b} w\dd\nu\leq \beta'\int_{\Delta_0} w \dd\nu.
\end{equation*}

\begin{align}
\int_{\Delta_b} u^2_{m+1}\dd\nu \leq & \xi (\sigma) \left(\eta_1\int_{\tilde{B}} \LL^{n_0}_0(u^2_m)\dd\nu +\int_{\Delta_b-\tilde{B}} \LL^{n_0}_0 (u^2_m)\dd\nu\right)\nonumber\\
\label{bounded contraction 1}
\leq &\xi(\sigma) \beta' \int_{\Delta_0} \LL^{n_0}_0(u^2_m)\dd\nu =\xi (\sigma) \beta' \int_{\Delta_0} u^2_m\dd\nu.
\end{align}

By (<ref>), (<ref>) and  $|u_{m+1}|\leq 1$,

\begin{equation}
\label{bounded contraction 2}
\int_{\Delta_b-\Delta_0}u_{m+1}^2 \dd\nu\leq \nu(\Delta_0-\Delta_b)\leq C|b|^{-\kappa}.
\end{equation}

Finally we can shrink $\epsilon$ if necessary so that $\xi (\sigma)\beta'<1$ for $|\sigma|<\epsilon$ and then (<ref>) and (<ref>) imply (<ref>).

§.§ Proof of Proposition <ref>

There exist $\epsilon\in (0,1),\, \tau\in (0,1)$ and $C_{\twi}>0$  such that for all $s=\sigma+ib$ with $|\sigma|<\epsilon$, $n\geq 1$ and $v\in \operatorname{Lip}(\Delta_0)$, we have

\begin{equation*}
|\LL^n_s v|^2_{\infty} \leq C_{\twi} (1+|b|)\tau^n |v|_{\infty} \lVert v\rVert_b+C_{\twi}B^n|v|_{\infty} \int |v| \dd\nu
\end{equation*}

where $B>1$ is a constant depending on $\epsilon$ and it tends to $1$ as $\epsilon\to 0$.

We have

\begin{align*}
|\LL^n_s v(x)|\leq & \lambda_{\sigma}^{-n} f_{\sigma}^{-1}(x) \sum_{\gamma\in \calH^n} |\gamma'(x)|^{\delta+\sigma} f_{\sigma}(\gamma x) |v|(\gamma x)\\
=& \lambda^{-n}_{\sigma} f_{\sigma}^{-1}(x) \sum_{\gamma\in \calH^n} \left(|\gamma'(x)|^{\delta/2+\sigma} f_{\sigma}^{1/2}(\gamma x) |v|^{1/2}(\gamma x)\right) \left(|\gamma'(x)|^{\delta/2} f^{1/2}_{\sigma}(\gamma x) |v|^{1/2}(\gamma x)\right).
\end{align*}

Using Cauchy-Schwarz, we obtain

\begin{align*}
|\LL^n_s v(x)|^2 \leq (\lambda_{\sigma}^{-2}\lambda_{2\sigma})^n \xi(\sigma) \LL^n_{2\sigma}(|v|) (x) \cdot \LL^n_0(|v|)(x),
\end{align*}

where $\xi(\sigma)=|f_0/f_{\sigma}|_{\infty} |f_{2\sigma}/f_{\sigma}|_{\infty} |f_{\sigma}/f_0|_{\infty} |f_{\sigma}/f_{2\sigma}|_{\infty}\leq 64$. Hence

\begin{equation}
\label{bound}
|\LL^n_s v|^2_{\infty} \leq 64 B^n |v|_{\infty} |\LL^n_0(|v|)|_{\infty},
\end{equation}

where $B>1$ is a constant depending on $\epsilon$ with $B\to 1$ as $\epsilon\to 0$.

Since $\LL_0$ is a normalized transfer operator for the uniformly expanding map $T$, there exists $\tau_1\in (0,1)$ such that   $|\LL^n_0 v|_{\infty}\leq C\tau_1^n\lVert v\rVert_{\text{Lip}}$ for all $v\in \text{Lip}(\Delta_0)$ with $\int v \dd\nu=0$. (This is a consequence of spectral gap of quasi-compact operator $\LL_0$.) Hence by decomposing $|v|$ into $(|v|-\int |v|\dd\nu)+\int |v|\dd\nu$, we obtain

\begin{equation*}
| \LL^n_0(|v|)|_{\infty} \leq 2C\tau_1^n \lVert v\rVert_{\text{Lip}}+\int |v|\dd\nu.
\end{equation*}

Substituting into (<ref>), we have

\begin{equation*}
|\LL^n_s v|_{\infty}^2\leq 128 C(B\tau_1)^n (1+|b|)|v|_{\infty} \lVert v\rVert_b +64 B^n |v|_{\infty} \int |v| \dd\nu.
\end{equation*}

Finally, shrink $\epsilon$ if necessary so that $\tau=B\tau_1<1$.

There exist $C>0,\ \epsilon \in (0,1),\, A>0$ and $\beta\in (0,1)$ such that

\begin{equation*}
\lVert \LL^{mn_0}_s v\rVert_b\leq C\beta^m \lVert v\rVert_b
\end{equation*}

for all $m\geq A\log |b|,\, s=\sigma+ib$ with $|\sigma|<\epsilon$ and $|b|$ large enough, and all $v\in \operatorname{Lip}(\Delta_0)$.

Let $N=[C_{\nin}\log|b|]n_0$. Using Lemma <ref> for $\LL_s^N v$ and $n=lN$,  Lemma <ref> and (<ref>), we obtain

\begin{align*}
|\LL_s^{(l+1)N}v |_\infty^2&\leq C_{\twi}(1+|b|)\tau^{lN}|\LL_s^N v|_\infty\|\LL_s^Nv\|_b+C_{\twi}B^{lN}|\LL_s^Nv|_\infty (\int |\LL_s^Nv|^2\dd \nu)^{1/2}.
\\
&\leq 2C_{\sixt}C_{\twi}(1+|b|)\tau^{lN}|v|_\infty\|v\|_b+2C_{\sixt}C_{\twi}B^{lN}|v|_\infty \beta^{N/2}\|v\|_b.
\end{align*}

We fix $l$ depending on $\tau, C_{\nin}$ and $n_0$ such that $(1+|b|)\tau^{lN/2}\leq 1$. Then by shrinking $B$ if necessary, there exists $\beta_1<1$, such that

\begin{equation}\label{equ:l+1}
|\LL_s^{(l+1)N}v|_\infty\leq \beta_1^{(l+1)N}\|v\|_b.
\end{equation}

For Lipschitz norm, we have

\begin{align*}
|\LL_s^{(l+2)N}v|_{\operatorname{Lip}}&\leq C_{\sixt}(1+|b|)|\LL_s^{(l+1)N}v|_\infty+C_{\sixt}\lambda^N|\LL_s^{(l+1)N}v|_{\operatorname{Lip}}\\
&\leq C_{\sixt}(1+|b|)\beta_1^{(l+1)N}\|v\|_b+C_{\sixt}^2\lambda^N((1+|b|)|v|_\infty+\lambda^{(l+1)N}|v|_{\operatorname{Lip}})\\
&\leq  C_{\sixt}^2(1+|b|)\|v\|_b(\beta_1^{(l+1)N}+\lambda^N+\lambda^{(l+2)N})\leq 3C_{\sixt}^2(1+|b|)\beta_2^N\|v\|_b,
\end{align*}

for some $\beta_2<1$, where we use Lemma <ref> to get the first inequality and (<ref>) to get the second one. For the infinity norm, by (<ref>) and Lemma <ref>, we obtain

\[|\LL_s^{(l+2)N}v|_\infty\leq 2C_{\sixt}\beta_1^{(l+1)N}\|v\|_b. \]

Combining these two norm estimates, we obtain

\begin{equation}\label{equ:l+2}
\|\LL_s^{(l+2)N}v\|_b\leq C_{\sixt}^2(2\beta_1^{(l+1)N}+3\beta_2^{N})\|v\|_b\leq \beta_3^{(l+2)N/n_0}\|v\|_b,
\end{equation}

for some $\beta_3<1$ if $|b|$ is large enough to absorb the constant $6C_{\sixt}^2$.

Let $A=2(l+2)C_{\nin}$ and $N_1=(l+2)N/n_0=(l+2)[C_{\nin}\log|b|]\leq A\log|b|$. For $m\geq A\log|b|$, we can write $m=dN_1+r$ with $r\in\N$ and $r< N_1$. Therefore by (<ref>) and Lemma <ref>,

\begin{equation*}
\|\LL_s^{mn_0}v\|_b=\| \LL_s^{dN_1n_0}(\LL_s^{rn_0}v)\|_b\leq\beta_3^{dN_1}\|\LL_s^{rn_0}v\|_b\leq 2C_{\sixt}\beta_3^{dN_1}\|v\|_b\leq 2C_{\sixt}(\sqrt{\beta_3})^m\|v\|_b.\qedhere
%\leq \beta_4^m\|b\|_b.
\end{equation*}

It is sufficient to prove that for all $m\in \mathbb{N}$,

\begin{equation}\label{LLs}
\int |\LL_s^{mn_0}v|^2\dd\nu\leq C\beta^m\|v\|^2_b.
\end{equation}

Then for any $k\in N$, suppose $k=mn_0+r$ with $0\leq r<n_0$. We have

\[\int |L_s^kv|^2\dd\mu_E\leq C\lambda_\sigma^k\int |\LL_s^k(f_{\sigma}^{-1}v)|^2\dd\nu\leq C\lambda_\sigma^k \beta^m\|\LL_s^r(f_{\sigma}^{-1}v)\|_b^2\leq C\lambda_\sigma^k\beta^m\|f_{\sigma}^{-1}v\|_b^2\leq C\lambda_{\sigma}^{k}\beta^m \|v\|_b^2.\]

By choosing $\epsilon$ small such that $\lambda_\sigma^{n_0}\beta<1$ for any $|\sigma|<\epsilon$, we obtain Proposition <ref>.

It remains to prove (<ref>). For $m> A\log|b|$, by Lemma <ref>, we obtain

\[\int|\LL_s^{mn_0}v|^2\dd\nu\leq \|\LL_s^{mn_0}v \|_b^2\leq C\beta^m\|v\|_b^2. \]

For $A\log|b|\geq m\geq C_{\nin}\log|b|$, by (<ref>) and Lemma <ref>, we know

\[\int|\LL_s^{mn_0}v|^2\dd\nu\leq \beta^{[C_{\nin}\log|b|]}\|\LL_s^{(m-[C_{\nin}\log|b|])n_0}v \|_b^2\leq 2C_{\sixt}\beta^{[C_{\nin}\log|b|]}\|v\|^2_b\leq 2C_{\sixt}\beta_1^m\|v\|^2_b%\leq \beta_2^m\|v\|_b,
\]

for some $\beta_1=\beta^{C_{\nin}/A}<1$.

The case when $m\leq C_{\nin}\log |b|$ has been verified in Lemma <ref>.

§ EXPONENTIAL MIXING

In this section, we prove Theorem <ref>. As a first step, an analogous result concerning expanding semiflow will be proved. Let $T:\Lambda_+\to \Lambda_+$ be the uniformly expanding map and $R:\Lambda_+\to \mathbb{R}_+$ be the roof function as defined in Proposition <ref>. Set $\Lambda_+^{R}=\{(x,t)\in \Lambda_+\times \mathbb{R}: 0\leq t<R(x)\}$. We define a semi-flow $T_t:\Lambda_+^R\to \Lambda_+^R$ by $T_s(x,t)=(T^nx, t+s-R_n(x))$ where $n$ is the unique integer satisfying $R_n(x)\leq t+s<R_{n+1}(x)$. Recall that $\nu$ is the unique $T$-invariant ergodic probability measure on $\Lambda_+$. Then the flow $T_t$ preserves the probability measure $\nu^{R}=\nu\times \operatorname{Leb}/(\nu\times \operatorname{Leb})(\Lambda_+^R)$. We will also use the probability measure $\mu_E^{R}=\mu_E\times \operatorname{Leb}/(\mu_E\times \operatorname{Leb})(\Lambda_+^R)$ on $\Lambda_+^R$. We show that $T_t$ is exponentially mixing.

For a bounded function on $\Lambda_+^R$, we define two norms. Set

\begin{align*}
&\|U\|_{\calB_0}=|U|_\infty+\sup_{ (x,a)\neq(x',a')\in\Lambda_+^R}\frac{|U(x,a)-U(x',a')|}{d(x,x')+|a-a'|},\\
&\|V\|_{\calB_1}=|V|_\infty+\sup_{x\in \Lambda_+} \frac{\operatorname{Var}_{(0,R(x))}\{t\mapsto V(x,t) \}}{R(x)},
\end{align*}

where $\operatorname{Var}_{(0,R(x))}\{t\mapsto V(x,t) \}$ is the total variation of the function $t\mapsto V(x,t)$ on the interval $(0,R(x))$.

There exist $C>0,\ \epsilon>0$ such that for all $t>0$ and for any two functions $U,\ V$ on $\Lambda_{+}^R$ with $\|U\|_{\calB_0},\ \|V\|_{\calB_1}$ finite, we have

\[\left|\int U\cdot V\circ T_t\dd\mu_E^R-\left(\int U\dd\mu_E^R\right)\left(\int V\dd\nu^R\right)\right|\leq Ce^{-\epsilon t}\|U\|_{\calB_0}\|V\|_{\calB_1}. \]

Applying this theorem to the function $U(x,t)\frac{\dd\nu}{\dd\mu_E}(x)$, we obtain

\begin{equation}
\label{semiflownu}
\left|\int U\cdot V\circ T_t\dd\nu^R-\left(\int U\dd\nu^R\right) \left(\int V\dd\nu^R\right)\right|\leq Ce^{-\epsilon t}\|U\|_{\calB_0}\|V\|_{\calB_1}.
\end{equation}

With Proposition <ref> available, Theorem <ref> can be proved essentially along the same lines as the proof of <cit.> (see also <cit.>). We provide a sketch of the proof here. For a pair of functions $U,V$, let $\rho(t)=\int U\cdot V\circ T_t\dd\mu_E^R$ be the correlation function and the observation is that the Laplace transform of $\rho$, denoted by $\hat{\rho}$, can be expressed as a sum of twisted transfer operators $L_s$ <cit.>. One shows that $\hat{\rho}$ admits an analytic continuation to a neighborhood of each point $s=ib$ and this part of the argument uses the quasi-compactness of the twisted transfer operators <cit.>. When $|b|$ is large, the Dolgopyat-type estimate (Proposition <ref>), which is a replacement of <cit.> in the current setting, is used to imply that $\hat{\rho}$ admits an analytic extension to a strip $\{s=\sigma+ib\in \mathbb{C}: |\sigma|<\sigma_0\}$ for all sufficiently small $\sigma_0$ <cit.>.  The result of exponential mixing then follows from the classical Paley-Wiener theorem <cit.>.

The difference between our result and that in [2] is the classes of functions in concern. The only adjustment we need to make is <cit.>, which is a norm estimate for $C^1$ functions in their paper, but for functions with finite $\calB_0$ norm in the current setting. The precise statement is as follows. For a function $U:\Lambda_+\to \mathbb{R}$ with $\| U\|_{\calB_0}<\infty$ and $s\in \mathbb{C}$, set $\hat{U}_{s}(x)=\int_0^{R(x)} e^{-ts}U(x,t)\dd t $.

There exists $C>0$ such that for $s=\sigma+ib$ with $|\sigma|\leq\epsilon_o/4$ ($\epsilon_o$ is given as in Proposition <ref> (5)), the function $L_s\hat{U}_{-s}$ is Lipschitz on $\Delta_0$ and

\[\|L_s\hat{U}_{-s}\|_b\leq \frac{C\|U\|_{\calB_0}}{\max\{ 1,|b|\}}. \]

We first prove for $x\in\Lambda_{+}$ we have

\begin{equation*}
\label{laplaceU}
\end{equation*}

By definition, we have

\begin{equation*}
\hat{U}_{-s}(x)=\int_0^{R(x)}U(x,t)e^{ts}\dd t.
\end{equation*}

The case when $|b|\leq 1$ is easy. When $|b|>1$, one uses integration by parts and the fact that $U$ is Lipschitz with respect to $t$ to obtain

\[ |\hat{U}_{-s}(x)|\leq (2|U|_\infty e^{\epsilon_oR(x)/4}+|U|_{\operatorname{Lip}}R(x)e^{\epsilon_o R(x)/4})/\max\{1,|b|\}.\]

\[|L_s\hat{U}_{-s}|\leq \frac{C\|U\|_{\calB_0}}{\max\{1,|b|\}}L_\sigma(e^{\epsilon_oR/2}). \]

Observe that by (<ref>)

\begin{equation}
\label{boundedeigen}
L_\sigma(e^{\epsilon_oR/2})=\sum_{\gamma\in\calH}|\gamma'(x)|^{\delta+\sigma}e^{\epsilon_o R(\gamma x)/2}\leq \sum_{\gamma\in\calH}|\gamma'(x)|^{\delta-3\epsilon_o/4}<\infty.
%=\tilde{L}_{\sigma+\epsilon_o/2}(f_{\epsilon_o/2}^{-1})\leq \sup |f_{\epsilon_o/2}^{-1}|<\infty.
\end{equation}

So 	$|L_s\hat{U}_{-s}|\leq \frac{C\|U\|_{\calB_0}}{\max\{1,|b|\}} $.

We estimate the Lipschitz norm of $L_s\hat{U}_{-s}$. We have

\[L_s\hat{U}_{-s}(x)-L_s\hat{U}_{-s}(y)=\sum_{\gamma\in\calH}|\gamma'(x)|^{\delta+s}(\hat{U}_{-s}(\gamma x)-\hat{U}_{-s}(\gamma y))+(|\gamma'(x)|^{\delta+s}-|\gamma'(y)|^{\delta+s})\hat{U}_{-s}(\gamma y). \]

The term $|\gamma'(x)|^{\delta+s}-|\gamma'(y)|^{\delta+s}$ can be estimated using Proposition <ref> (4).
For the term $\hat{U}_{-s}(\gamma x)-\hat{U}_{-s}(\gamma y)$, suppose that $R(\gamma x)\geq R(\gamma y)$, we use Proposition <ref> (4) again and get

\begin{align*}
|\hat{U}_{-s}(\gamma x)-\hat{U}_{-s}(\gamma y)|&\leq |R(\gamma x)-R(\gamma y)||U|e^{\sigma R(\gamma x)}+\int_0^{R(\gamma y)}|U(\gamma x,t)-U(\gamma y,t)|e^{t\sigma}\dd t\\
&\leq (C_1 e^{\sigma R(\gamma x)}+R(\gamma y)e^{\sigma R(\gamma y)})\|U\|_{\calB_0}d(x,y).
\end{align*}

Then we use (<ref>) to conclude that there exists some $C$ (independent of $U$) such that

\[|L_s\hat{U}_{-s}|_{\operatorname{Lip}}\leq C \|U\|_{\calB_0}.\qedhere \]

Now Theorem <ref> can be proved using the same lines as the proof of <cit.> (see also <cit.>). In particular, in the proof of <cit.>, we use (<ref>) to replace <cit.> and Proposition <ref> (2) to relate the measures $\hat{\nu}^{R}$ and $\nu^R$.

§ RESONANCE FREE REGION

Recall that $\Gamma$ is a geometrically finite discrete subgroup in $G=\operatorname{SO}(d+1,1)^{\circ}$. We begin by defining the measures $m^{\operatorname{BR}}$, $m^{\operatorname{BR_*}}$ and $m^{\operatorname{Haar}}$.  Recall the definition of the BMS measure on $\operatorname{T}^1(\mathbb{H}^{d+1})\cong \partial^2(\mathbb{H}^{d+1})\times \mathbb{R}$:

\begin{equation*}
\dd\tilde{m}^{\operatorname{BMS}}(x,x_-,s)=e^{\delta \beta_x(o,x_*)} e^{\delta \beta_{x_-}(o,x_*)} \dd\mu(x) \dd \mu(x_-)\dd s,
\end{equation*}

where $x_*$ is the based point of the unit tangent vector given by $(x,x_-,s)$. We define the measures $\tilde{m}^{\operatorname{BR}}$, $\tilde{m}^{\operatorname{BR_*}}$ and $\tilde{m}^{\operatorname{Haar}}$ on $\operatorname{T}^1(\mathbb{H}^{d+1})\cong \partial^2(\mathbb{H}^{d+1})\times \mathbb{R}$ similarly as follows:

\begin{align*}
\dd\tilde{m}^{\operatorname{BR}}(x,x_-,s)&=e^{d\beta_x(o,x_*)} e^{\delta \beta_{x_-}(o,x_*)} \dd m_o(x) \dd \mu(x_-)\dd s;\\
\dd\tilde{m}^{\operatorname{BR_*}}(x,x_-,s)&=e^{\delta \beta_x(o,x_*)} e^{\delta \beta_{x_-}(o,x_*)} \dd\mu(x) \dd m_o(x_-)\dd s;\\
\dd\tilde{m}^{\operatorname{Haar}}(x,x_-,s)&=e^{d \beta_x(o,x_*)} e^{d\beta_{x_-}(o,x_*)} \dd m_o(x) \dd m_o(x_-)\dd s,
\end{align*}

where $m_o$ is the unique probability measure on $\partial(\mathbb{H}^{d+1})$ which is invariant under the stabilizer of $o$ in $G$.

These measures are all left $\Gamma$-invariant and induce measures on $\operatorname{T}^1(\Gamma\backslash \mathbb{H}^{d+1})$, which we will denote by $m^{\operatorname{BR}}$, $m^{\operatorname{BR_*}}$ and $m^{\operatorname{Haar}}$ respectively. Here we do not normalize the BMS measure to a probability measure, which is different from the previous part.

By  <cit.>, Theorem <ref> implies exponential decay of matrix coefficients.

There exists $\eta>0$ such that for any compactly supported functions $\phi, \psi\in C^1(\T^1(M))$, we have

\begin{equation*}
e^{(d-\delta)t}\int_{\T^1(M)} \phi\cdot\psi\circ\calG_t\ \dd m^{\operatorname{Haar}}=\frac{m^{\operatorname{BR_*}} (\phi) m^{\operatorname{BR}} (\psi)}{\bms(\T^1(M))}+O(\lVert \phi \rVert_{C^1} \lVert \psi\rVert_{C^1}e^{-\eta t})
\end{equation*}

for all $t>0$, where $O$ depends on the supports of $\phi,\psi$.

For $x,y\in\H^{d+1}$ and $T>0$, let

\[N(T,x,y)=\#\{\gamma\in\Gamma\,|\,d(x,\gamma y)\leq T \}, \]

where $d$ is the hyperbolic distance on $\H^{d+1}$.
In [34], it was shown that Theorem <ref> implies the following:

There exists $\eta>0$ such that for any $x,y\in\H^{d+1}$ and $T>0$, we have

\[N(T,x,y)=c_{x,y}e^{\delta T}+O(e^{(\delta-\eta)T}), \]

where $c_{x,y}>0$ is a constant depending on $x,y$.

For $x,y\in\H^{d+1}$ and $s\in\C$ with $\Re s>\delta$, let $P_s(x,y)$ be the Poincaré series defined by

\[P_s(x,y)=\sum_{\gamma\in\Gamma} e^{-sd(x,\gamma y)}. \]

We first prove that $P_s(x,y)$ is meromorphic on $\Re s>\delta-\eta$ with a unique pole $s=\delta$.
By Fubini's theorem

\[P_s(x,y)=\int_{0}^\infty \frac{1}{s}e^{-sT}N(T,x,y)\dd T=\int_{0}^\infty \frac{1}{s}e^{-(s-\delta)T}c_{x,y}\dd T+\int_{0}^\infty \frac{1}{s}e^{-sT}(N(T,x,y)-c_{x,y}e^{\delta T})\dd T.  \]

The first part is a meromorphic function of $s$ having a unique pole at $s=\delta$. The second part, it follows from Corollary <ref> that it is absolutely convergence if $\Re s>\delta-\eta$, hence it is analytic on $\Re s>\delta-\eta$. Then we use  <cit.> to deduce  that the resolvent $R_M(s)$ is also analytic on $\{s\in \mathbb{C}:\, \delta-\eta<\Re s<\delta\}$.

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Institut für Mathematik, Universität Zürich, 8057 Zürich

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Department of Mathematics, University of Chicago, Chicago, IL, 60637

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