:
1 2022 Symmetry and Geometry in Neural Representations

# See and Copy: Generation of complex compositional movements from modular and
geometric RNN representations

Sunny Duan†<EMAIL_ADDRESS>
Mikail Khona†<EMAIL_ADDRESS>
Adrian Bertagnoli†<EMAIL_ADDRESS>
Sarthak Chandra<EMAIL_ADDRESS>
Ila Fiete<EMAIL_ADDRESS>
[$\dagger$] Denotes co-first authors

###### Abstract

A hallmark of biological intelligence and control is combinatorial
generalization: animals are able to learn various things, then piece them
together in new combinations to produce appropriate outputs for new tasks.
Inspired by the ability of primates to readily imitate seen movement
sequences, we present a model of motor control using a realistic model of arm
dynamics, tasked with imitating a guide that makes arbitrary two-segment
drawings. We hypothesize that modular organization is one of the keys to such
flexible and generalizable control. We construct a modular control model
consisting of separate encoding and motor RNNs and a scheduler, which we train
end-to-end on the task. We show that the modular structure allows the model to
generalize not only to unseen two-segment trajectories, but to new drawings
consisting of many more segments than it was trained on, and also allows for
rapid adaptation to perturbations. Finally, our model recapitulates
experimental observations of the preparatory and execution-related processes
unfolding during motor control, providing a normative explanation for
functional segregation of preparatory and execution-related activity within
the motor cortex.

###### keywords:

Modularity, Motor Control, Neural Representational Geometry

##  1 Introduction

Animal behavior is believed to be atomic: composed of a discrete set of
“syllables” or motifs which together form a “grammar” Wiltschko et al. (2015);
Markowitz et al. (2018). Action sequences are then generated by combining
syllables and executing them sequentially. The compositional structure
inherent in this scheme allows animals to flexibly recombine motor primitives
to generate novel, ecologically relevant movement patterns. This system
affords animals a combinatorially large movement repertoire built of out
simpler components. However, the specialized structures in the motor system
required to implement this scheme efficiently remain unknown.

Existing experiments in the motor cortices of non-human primates performing
reaching movements have indicated that the process of movement generation is
comprised of three interdependent stages: a preparatory stage, a trigger
signal and action execution. During the preparatory stage, the parameters for
a motion are able to be decoded from the motor cortex Churchland et al. (2010)
indicating that the planned trajectory is present in the motor cortex despite
the lack of motion production. After the onset of the trigger signal, the
motor cortex exhibits dynamics which produce downstream muscle activations
leading to the desired motion.

In order to combine multiple motor primitives into complex sequences, there
must be additional structure to support these longer sequences. Recent work
studying the neural dynamics of rhesus macaques performing skilled, practiced
compound reaches has indicated that conjunctive movement consists of separate
independent chains of motor processes sharing the same underlying neural
substrate Zimnik and Churchland (2021). In order to skillfully and smoothly
execute a sequence of movements, the motor cortex needs to simultaneously
prepare for an upcoming movement while current motor commands are being
executed. This multi-tasking capability requires functional segregation
preparatory activity and execution within the underlying motor cortical
circuitry. Existing research suggests that this is implemented in biological
systems by separating preparatory activity and execution into orthogonal
subspaces.

We demonstrate that a task-optimized neural network is able to implement
arbitrary motor subroutines in a flexible and reusable manner. Our model
demonstrates remarkable generalization capabilities due to its structure.
Through learning, our model exhibits emergent self-organization of its latent
representation which facilitates robust production of movement patterns.
Furthermore, our model is able to continuously produce sequences of motion
interference, demonstrating properties of functional segregation exhibited in
biological neural circuits.

fig:fig1

Figure 1: a) Schematic of encoder-decoder training setup b) Example
trajectories used for training the model, consisting of procedurally generated
single and 2-segment trajectories of various length and angles. c) Examples of
complex, out-of-distribution sequences generated by the trained model. d)
Polar histogram of the number of neurons which fire preferentially in a given
direction. e) Distribution of preferred reach direction for different starting
locations within the drawing board, computed by finding the reach direction
which minimized error.

##  2 Methods

Our model architecture resembles the sequence-to-sequence models used in
natural language processing Sutskever et al. (2014). The input data (from the
observed “guide”) consists of procedurally generated movements consisting of
up to two straight segments, represented by a sequence of x,y coordinates. The
agent constructs an embedding by sequentially ingesting the x,y coordinates of
the guide sequence into an encoder implemented as a recurrent neural network
(RNN) consisting of continuous-time neurons. The encoder produces a single
(static) readout, or embedding, at the end of each segment. These embeddings
are fed into the motor RNN at pre-specified times by a scheduler, which
modulates the embedding by a ramping function such that motion onset is
triggered by the falling edge of the ramping signal Hennequin et al. (2014).
The beginning of the ramp signal was timed such that the embedding signal was
provided 7 steps before the end of the previous segment such that the go
signal aligns with the end of the previous segment. This temporal overlap
forced the network to simultaneously process both the future motor command
while completing the current segment.

The motor RNN takes in the ramp-modulated static encoder embedding and
produces a sequence of muscle activation commands which are implemented by a
realistic simulation of an over-actuated two-link arm moving in two
dimensional space Kalidindi et al. (2021), Lillicrap and Scott (2013). The
entire system is trained end-to-end on one or two segment motions using
stochastic gradient descent and supervised learning with additional
biologically relevant loss terms penalizing neural activity and simultaneous
activation of antagonistic muscle pairs.

##  3 Results

Although our model was trained on sequences of straight lines with at most two
segments (fig:fig1b), we find that our network is able to generalize to
sequences of arbitrarily many more segments, enabling the network to draw
complex figures which it has never been trained on before (fig:fig1c). We
posit that this ability arises from the implicit modular structure of our
network which is inspired by recordings from the primate motor cortical system
Zimnik and Churchland (2021). We find that bifurcation of movement production
into a preparatory phase and execution-related phase encouraged the network to
produce relaxation dynamics which permitted the network to reuse existing
circuitry for the production of conjunctive sequences. We also demonstrate
that the structural priors imposed on our model give rise to characteristic
behaviors of biological motor control including orthogonality of preparatory
and execution-related activity and rapid adaptation to changes in physics.

fig:fig2

Figure 2: a) PCA embedding of the encoder latent state which is fed into the
motor RNN over a dataset of single-segment reaches starting from the same
initial position, colored by reach angle and with size corresponding to the
reach distance. b) PCA embedding of the decoder latent state one step before
motion onset, similarly colored and sized.

###  3.1 Preferred movement directions

We analyzed the directions in which neurons preferentially fired. After
counting the number of neurons whose firing was the greatest in each
direction, we found a bimodal distribution with peaks located approximately
180 degrees from one another fig:fig1d. This phenomenon has been previously
reported by Lillicrap and Scott (2013), in which they attribute this effect as
the optimal neural activity for the biomechanical properties of the limb.
Accordingly, we estimated the optimal movement direction for a given arm
configuration by computing the direction which minimized the cumulative loss
for a single movement. As shown in fig:fig1e, this direction was tightly
distributed around 270 degrees. Notably, the axis on which the optimal
movement direction lied, was nearly perpendicular to the axis at which neurons
preferentially fire. This is due to the fact that at certain orientations,
larger neural activations are needed in order to perform the corresponding
reaches.

###  3.2 Structured latent representations of movement

We investigated how guide trajectories were represented in our motor RNN. We
instructed our model to hold for a fixed period during which the embedding of
the trajectory was fed into the decoder, emulating experiments on non-human
primates Churchland et al. (2010). In our experiments, we are able to decode
future movement from the motor RNN one time-step prior to motion onset with an
mean average error of 0.129 radians on a held-out dataset indicating that the
relevant movement parameters were present in the motor RNN state. We also
observed that low-dimensional PCA projections of neural activity during
reaches of varying angle and length exhibit rich topological structure.
Visualizing the projections onto the top 2 principle components show that the
manifold of latent representations in both the encoder and the motor RNN are
homeomorphic to a cylinder parameterized by the angle and length of the reach
(fig:fig2). This smooth mapping of movement parameters presumably allows the
network to robustly encode the desired trajectory and provides an
initialization of the motor RNN state which evolves autonomously.

fig:fig3

Figure 3: a) Visualization of the the neural activity during the preparatory
period (blue) and execution period (red) with principle components computed
only on the neural activation data during preparation (left) and execution
(right). b) Computed subspaces consisting of the first 4 principle components
of the neural activity during the preparatory period and execution period. We
quantify the orthogonality of the subspaces by taking the dot products of the
PCs. We observe that the top PC of the execution-period components are
completely orthogonal to all of the components of the preparation period
components.

###  3.3 Orthogonality of neural activity during preparation and execution

Additionally we sought to understand how our network was able to
simultaneously execute an existing motor command while reading in future
commands without interference. In line with experimental results, we observed
that our network separated preparatory activity and execution into orthogonal
subspaces Churchland et al. (2010). Visual inspection of the principle
components of preparatory activity and execution-related activity reveals this
relationship (fig:fig3a). Quantifying this relationship by computing the top 4
principle components which account for 97.7% and 96.7% of the preparatory and
execution-related activity respectively shows that these components are
largely orthogonal and the top principle component of the execution period is
orthogonal to all of the components of the activity in the preparatory period
(fig:fig3b). This indicates that orthogonality of these processes emerges as a
consequence of the need for the circuit to support both preparatory and
execution-related processes simultaneously.

fig:fig4

Figure 4: Comparison of neural activity for 2-segment trajectories and
1-segment trajectories consisting of the second half of the 2-segment
trajectories. The first 6 PCs account for 92% of the variance and the neural
activity of the second half of the 2-segment trajectories converges almost
identically to the corresponding 1-segment trajectory, implying reuse of the
neural circuit despite the additional separating work the network must do in
the two-segment trajectories to deal with the initial overlap of the two
segments.

###  3.4 Embedding reuse in multi-segment sequences

We compared movements consisting of two segments with single segment movements
to evaluate whether the network reused the same neural activity for both the
multi-segment and the single-segment movement. Our model was commanded to
perform randomly sampled 2-segment movements and then, independently, only the
second half of the 2-segment movements. Computing PCA on all of the neural
activity showed that there were 6 principle components accounting for 92% of
the variance. Visualizing the evolution of these components over time reveal
that the second half of the 2-segment movements quickly converged to the same
trajectories as the 1-segment movements even in cases when the activity had
not returned to zero yet (fig:fig4). This demonstrates that our network not
only reuses the same neural representation for conjunctive movements but also
is able to avoid interference by neural activity from the first segment.

###  3.5 Rapid adaptation to changes in arm physics

Many experimental studies have used perturbations of underlying physics of the
task in order to assess the role of the motor cortex (Cherian et al. (2013);
Vyas et al. (2018); Sun et al. (2022)). We modeled this by assessing our
models ability to recover from changes in biomechanical arm parameters. After
increasing the moment of inertia of our simulated arm by 30%, we observed
significant position errors. However, after less than 20 additional gradient
steps confined to the weights of only the motor RNN (keeping the encoder RNNs
fixed), our model was able to rapidly adapt to the change in inertia to
recover its original performance, assessed on a held out evaluation set of 40
movements. (fig:fig5a)

###  3.6 Feedback knockout

The dynamics of the motor network are driven by a combination of internal
recurrence, and time-varying feedback from the arm. We sought to characterize
how much of our dynamics were derived from autonomous temporal dynamics in the
motor network based on the static information contained in the preparatory
state set by the encoding network output. We knocked out feedback from the arm
and observed how its performance degraded while repeatedly drawing a diamond.
The knockout model still produced approximately the correct shape, but
gradually drifted away from the target location, indicating that the model
utilized feedback mostly as a correctional signal rather than a major source
of dynamical drive.

fig:fig5

Figure 5: a) Performance of our model while recovering from changing the arm
moment of inertia by 30%, training only the motor RNN portion of the model. b)
Comparison of our model tracing a diamond multiple times with and without
feedback.

##  4 Discussion

Our model takes inspiration from the versatility of animals in producing novel
movement outputs and of primates in their ability to “mirror” seen movements
to generate combinatorially flexible movements with a biophysical multi-
jointed model arm. Although it is difficult to make claims about how faithful
RNNs are as a model of biology, our model provides confirmation that observed
phenomena of motor cortical dynamics have functional utility in allowing
neural circuits to reuse previously seen motor primitives and recombine them
into complex sequences of movements. We found that the orthogonality of
preparatory and execution-related activity emerged in our end-to-end trained
model from the need to separate incoming planned movement instructions from
generating ongoing movements and is indicative that this phenomenon is needed
in order to prevent interference between subsequent movements. Additionally,
the separation of movement production into preparatory and execution phases
allowed our model to robustly generalize to significantly more complicated
movements. We find that both of these phenomena are critical to facilitating
modularity and generalization in RNNs and our experiments provide insight into
why these structures may exist in biology.

In our study, we focused on how the motor RNN is able to learn and composibly
reuse movements. However, a complete model of compositional control would
require the ability to decompose arbitrary drawings into modular sub-routines,
which, in our study was supplied by an external trigger signal. Additionally,
we focused on a limited set of movements consisting of combinations of
straight reaches whereas realistic drawing require a larger repertoire of
primitive movements. These are natural directions for future study.

The ability of an agent to decompose complex tasks into subroutines is a core
component in how intelligent agents are able to adapt to new environments and
form complex behaviors. This skill acquisition capability has been studied in
the context of reinforcement learning in the hopes of producing artificial
agents which are able to solve novel tasks Sharma et al. (2019) Xu et al.
(2020). Biological systems are able to efficiently acquire new skills and
recombine them in novel environments, solving tasks which are inaccessible to
current state of the art artificially intelligence systems. These systems have
been evolutionarily optimized to efficiently learn and perform tasks that are
critical for survival; thus, they provide clues for how to engineer
artificially intelligent systems to exhibit similarly remarkable capabilities.

## References

  * Cherian et al. (2013) Anil Cherian, Hugo L Fernandes, and Lee E Miller.  Primary motor cortical discharge during force field adaptation reflects muscle-like dynamics.  _Journal of neurophysiology_ , 110(3):768–783, 2013.
  * Churchland et al. (2010) Mark M Churchland, John P Cunningham, Matthew T Kaufman, Stephen I Ryu, and Krishna V Shenoy.  Cortical preparatory activity: representation of movement or first cog in a dynamical machine?  _Neuron_ , 68(3):387–400, 2010.
  * Hennequin et al. (2014) Guillaume Hennequin, Tim P Vogels, and Wulfram Gerstner.  Optimal control of transient dynamics in balanced networks supports generation of complex movements.  _Neuron_ , 82(6):1394–1406, June 2014.
  * Kalidindi et al. (2021) Hari Teja Kalidindi, Kevin P Cross, Timothy P Lillicrap, Mohsen Omrani, Egidio Falotico, Philip N Sabes, and Stephen H Scott.  Rotational dynamics in motor cortex are consistent with a feedback controller.  _eLife_ , 10:e67256, nov 2021.  ISSN 2050-084X.  10.7554/eLife.67256.  URL https://doi.org/10.7554/eLife.67256.
  * Lillicrap and Scott (2013) Timothy P Lillicrap and Stephen H Scott.  Preference distributions of primary motor cortex neurons reflect control solutions optimized for limb biomechanics.  _Neuron_ , 77(1):168–179, 2013.
  * Markowitz et al. (2018) Jeffrey E Markowitz, Winthrop F Gillis, Celia C Beron, Shay Q Neufeld, Keiramarie Robertson, Neha D Bhagat, Ralph E Peterson, Emalee Peterson, Minsuk Hyun, Scott W Linderman, et al.  The striatum organizes 3d behavior via moment-to-moment action selection.  _Cell_ , 174(1):44–58, 2018.
  * Sharma et al. (2019) Archit Sharma, Shixiang Gu, Sergey Levine, Vikash Kumar, and Karol Hausman.  Dynamics-aware unsupervised discovery of skills.  _arXiv preprint arXiv:1907.01657_ , 2019.
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##  Appendix A Model Architecture

There are two neural network models in our model: The encoder model and the
motor RNN. They both contain a continuous time recurrent neural network which
is a variant of the traditional recurrent neural network.

###  A.1 Continuous Time Recurrent Neural Network (CTRNN)

The CTRNN component of our network is a continuous time RNN with tanh non-
linearity, discretized using the forward eurler method as a discrete
approximation for the continuous time dynamics.

$\tau\frac{h[t+1]-h[t]}{\Delta t}=\sigma(x+Wh[t])$

where $h[t],h[t+1]$ represents the vector-valued state of the hidden units at
time step $t$, $W$ is the recurrent weight matrix, $x$ is the input from
upstream layers, $\sigma$ is the tanh non-linearity and $dt$ and $\tau$ are
constants with the values $dt=1.0$ and $\tau=100$. In our network the hidden
layer size is 1024 units.

###  A.2 Input/Output Layers

In addition to the CTRNN module, our encoder is furnished with a 2-layer
multi-layer perceptron (MLP) which takes in the x,y coordinates as input and
has sizes $2\times 1024$ and $1024\times 1024$ with an intermediate ReLU non-
linearity. The output of this network is fed in to the CTRNN as input.

The decoder also has an input MLP which takes in as input the concatenation of
the 1028 unit embedding produced by the encoder along with the x,y coordinates
of the current tip location, the two angles specifying the position of the arm
joints, as well as the current angular velocity of the two arm joints for a
full input size of 1034. This is fed into a 2-layer MLP with sizes $1034\times
1024$ and $1024\times 1024$ with a ReLU as the intermediate non-linearity. The
decoder is also furnished with a 2-layer MLP with sizes $1024\times 1024$ and
$1024\times 6$ which produces a 6 dimensional muscle activation vector which
is consumed by our realistic arm environment.


# Decay in norm of transfer operators for semiflows

Ian Melbourne Mathematics Institute, University of Warwick, Coventry, CV4 7AL,
UK Nicolò Paviato Mathematics Institute, University of Warwick, Coventry, CV4
7AL, UK Dalia Terhesiu Mathematisch Instituut, University of Leiden, Niels
Bohrweg 1, 2333 CA Leiden, Netherlands

(9 April 2021)

###### Abstract

We establish exponential decay in Hölder norm of transfer operators applied to
smooth observables of uniformly and nonuniformly expanding semiflows with
exponential decay of correlations.

##  1 Introduction

Exponential decay of correlations is well-understood for large classes of
uniformly and nonuniformly expanding maps, see for example [8, 11, 13, 14, 16,
20, 21, 22, 23, 26]. The typical method of proof is to establish a spectral
gap for the associated transfer operator $L$. Such a spectral gap yields a
decay rate $\|L^{n}v-\int v\|\leq C_{v}e^{-an}$ for $v$ lying in a suitable
function space, where $a$, $C_{v}$ are positive constants. Decay of
correlations is an immediate consequence of such decay for $L^{n}$.

Results on decay of correlations lead to numerous statistical limit theorems.
Although not needed for results such as the central limit theorem, strong norm
control on $L^{n}v$ is often useful for finer statistical properties. For
example, rates of convergence in the central limit theorem [12] and the
associated functional central limit theorem [4] rely heavily on control of
operator norms.

In this paper, we consider norm decay of transfer operators for uniformly and
nonuniformly expanding semiflows. Here, the standard method is to deduce decay
of the correlation function from analyticity of Laplace transforms, bypassing
spectral properties of $L_{t}$, see [10, 15, 19]. As far as we know, the only
result on spectral gaps for transfer operators of semiflows is due to Tsujii
[24]. However, this result is for suspension semiflows over the doubling map
with a $C^{3}$ roof function, where the smoothness of the roof function is
crucial and very restrictive. A similar result for contact Anosov flows is
proved in [25]. Both of the papers [24, 25] obtain spectral gaps for $L_{t}$
acting on a suitable anisotropic Banach space. Apart from these, there are
apparently no previous results on norm decay of transfer operators for
semiflows and flows.

Recently, in [17], we showed that spectral gaps are impossible in Hölder
spaces with exponent greater than $\frac{1}{2}$ (and in any Banach space that
embeds in such a Hölder space). Nevertheless, our aim of controlling the
Hölder norm of $L_{t}v$ for a large class of semiflows and observables $v$
remains viable, and our main result is the first in this direction. We
consider uniformly and nonuniformly expanding semiflows satisfying a
Dolgopyat-type estimate [10]. Such an estimate plays a key role in proving
exponential decay of correlations for the semiflow. Theorem 2.2 below shows
how to use this estimate to prove exponential decay of $L_{t}v$ in a Hölder
norm for smooth mean zero observables satisfying a good support condition.
Apart from the Dolgopyat estimate, the main ingredient is an operator renewal
equation for semiflows [18] which enables consideration of the operator
Laplace transform $\int_{0}^{\infty}e^{-st}L_{t}\,dt$.

The remainder of the paper is organised as follows. In Section 2, we recall
the setup for nonuniformly expanding semiflows with exponential decay of
correlations and state our main result, Theorem 2.2, on decay in norm. In
Section 3, we prove Theorem 2.2.

#### Notation

We use “big O” and $\ll$ notation interchangeably, writing $a_{n}=O(b_{n})$ or
$a_{n}\ll b_{n}$ if there are constants $C>0$, $n_{0}\geq 1$ such that
$a_{n}\leq Cb_{n}$ for all $n\geq n_{0}$.

##  2 Setup and statement of the main result

In this section, we state our result on Hölder norm decay of transfer
operators for uniformly and nonuniformly expanding semiflows.

Let $(Y,d)$ be a bounded metric space with Borel probability measure $\mu$ and
an at most countable measurable partition $\\{Y_{j}\\}$. Let $F:Y\to Y$ be a
measure-preserving transformation such that $F$ restricts to a measure-
theoretic bijection from $Y_{j}$ onto $Y$ for each $j$. Let $g=d\mu/(d\mu\circ
F)$ be the inverse Jacobian of $F$.

Fix $\eta\in(0,1)$. Assume that there are constants $\lambda>1$ and $C>0$ such
that $d(Fy,Fy^{\prime})\geq\lambda d(y,y^{\prime})$ and $|\log g(y)-\log
g(y^{\prime})|\leq Cd(Fy,Fy^{\prime})^{\eta}$ for all $y,y^{\prime}\in Y_{j}$,
$j\geq 1$. In particular, $F$ is a Gibbs-Markov map as in [2] (see also [1,
3]) with ergodic (and mixing) invariant measure $\mu$.

Let $\varphi:Y\to[2,\infty)$ be a piecewise continuous roof function. We
assume that there is a constant $C>0$ such that

$|\varphi(y)-\varphi(y^{\prime})|\leq Cd(Fy,Fy^{\prime})^{\eta}$ (2.1)

for all $y,y^{\prime}\in Y_{j}$, $j\geq 1$. Also, we assume exponential tails,
namely that there exists $\delta_{0}>0$ such that

${\textstyle\sum_{j}}\mu(Y_{j})e^{\delta_{0}|1_{Y_{j}}\varphi|_{\infty}}<\infty.$
(2.2)

Define the suspension $Y^{\varphi}=\\{(y,u)\in
Y\times[0,\infty):u\in[0,\varphi(y)]\\}/\sim$ where
$(y,\varphi(y))\sim(Fy,0)$. The suspension semiflow $F_{t}:Y^{\varphi}\to
Y^{\varphi}$ is given by $F_{t}(y,u)=(y,u+t)$ computed modulo identifications.
We define the ergodic $F_{t}$-invariant probability measure
$\mu^{\varphi}=(\mu\times{\rm Lebesgue})/\bar{\varphi}$ where
$\bar{\varphi}=\int_{Y}\varphi\,d\mu$. 111We call such semiflows “nonuniformly
expanding” since they are the continuous time analogue of maps that are
nonuniformly expanding in the sense of Young [26]. “Uniformly expanding”
semiflows are those with $\varphi$ bounded; they have bounded distortion as
well as uniform expansion.

Let $L_{t}:L^{1}(Y^{\varphi})\to L^{1}(Y^{\varphi})$ denote the transfer
operator corresponding to $F_{t}$ (so
$\int_{Y^{\varphi}}L_{t}v\,w\,d\mu^{\varphi}=\int_{Y^{\varphi}}v\,w\circ
F_{t}\,d\mu^{\varphi}$ for all $v\in L^{1}(Y^{\varphi})$, $w\in
L^{\infty}(Y^{\varphi})$, $t>0$) and let $R_{0}:L^{1}(Y)\to L^{1}(Y)$ denote
the transfer operator for $F$. Recall (see for example [2]) that
$(R_{0}v)(y)={\textstyle\sum_{j}}g(y_{j})v(y_{j})$ where $y_{j}$ is the unique
preimage of $y$ under $F|Y_{j}$, and there is a constant $C>0$ such that

$|g(y)|\leq C\mu(Y_{j}),\qquad|g(y)-g(y^{\prime})|\leq
C\mu(Y_{j})d(Fy,Fy^{\prime})^{\eta},$ (2.3)

for all $y,y^{\prime}\in Y_{j}$, $j\geq 1$.

#### Function space on $Y^{\varphi}$

Let $Y^{\varphi}_{j}=\\{(y,u)\in Y^{\varphi}:y\in Y_{j}\\}$. Fix
$\eta\in(0,1]$, $\delta>0$. For $v:Y^{\varphi}\to{\mathbb{R}}$, define
$|v|_{\delta,\infty}=\sup_{(y,u)\in Y^{\varphi}}e^{-\delta u}|v(y,u)|$ and

$\|v\|_{\delta,\eta}=|v|_{\delta,\infty}+|v|_{\delta,\eta},\qquad|v|_{\delta,\eta}=\sup_{j\geq
1}\sup_{(y,u),(y^{\prime},u)\in Y_{j}^{\varphi},\,y\neq y^{\prime}}e^{-\delta
u}\frac{|v(y,u)-v(y^{\prime},u)|}{d(y,y^{\prime})^{\eta}}.$

Then ${\mathcal{F}}_{\delta,\eta}(Y^{\varphi})$ consists of observables
$v:Y^{\varphi}\to{\mathbb{R}}$ with $\|v\|_{\delta,\eta}<\infty$.

Next, define $\partial_{u}v$ to be the partial derivative of $v$ with respect
to $u$ at points $(y,u)\in Y^{\varphi}$ with $u\in(0,\varphi(y))$ and to be
the appropriate one-sided partial derivative when $u\in\\{0,\,\varphi(y)\\}$.
For $m\geq 0$, define ${\mathcal{F}}_{\delta,\eta,m}(Y^{\varphi})$ to consist
of observables $v:Y^{\varphi}\to{\mathbb{R}}$ such that
$\partial_{u}^{j}v\in{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})$ for
$j=0,1,\dots,m$, with norm
$\|v\|_{\delta,\eta,m}=\max_{j=0,\dots,m}\|\partial_{u}^{j}v\|_{\delta,\eta}$.

###### Definition 2.1

We say that a function $u:Y^{\varphi}\to{\mathbb{R}}$ has good support if
there exists $r>0$ such that $\operatorname{supp}v\subset\\{(y,u)\in
Y\times{\mathbb{R}}:u\in[r,\varphi(y)-r]\\}$.

For functions with good support, $\partial_{u}v$ coincides with the derivative
$\partial_{t}v=\lim_{h\to 0}(v\circ F_{h}-v)/h$ in the flow direction.

Let

${\mathcal{F}}_{\delta,\eta,m}^{0}(Y^{\varphi})=\\{v\in{\mathcal{F}}_{\delta,\eta,m}(Y^{\varphi}):{\textstyle\int}_{Y^{\varphi}}v\,d\mu^{\varphi}=0\\}.$

We write ${\mathcal{F}}_{\delta,\eta}(Y^{\varphi})$ and
${\mathcal{F}}_{\delta,\eta}^{0}(Y^{\varphi})$ when $m=0$.

#### Function space on $Y$

For $v:Y\to{\mathbb{R}}$, define

$\|v\|_{\eta}=|v|_{\infty}+|v|_{\eta},\qquad|v|_{\eta}=\sup_{j\geq
1}\sup_{y,y^{\prime}\in Y_{j},\,y\neq
y^{\prime}}|v(y)-v(y^{\prime})|/d(y,y^{\prime})^{\eta}.$

Let ${\mathcal{F}}_{\eta}(Y)$ consist of observables $v:Y\to{\mathbb{R}}$ with
$\|v\|_{\eta}<\infty$.

#### Dolgopyat estimate

Define the twisted transfer operators

$\widehat{R}_{0}(s):L^{1}(Y)\to
L^{1}(Y),\qquad\widehat{R}_{0}(s)v=R_{0}(e^{-s\varphi}v).$

We assume that there exists $\gamma\in(0,1)$, $\epsilon>0$, $m_{0}\geq 0$,
$A,D>0$ such that

$\displaystyle{\|\widehat{R}_{0}(s)^{n}\|}_{{\mathcal{F}}_{\eta}(Y)\mapsto{\mathcal{F}}_{\eta}(Y)}\leq|b|^{m_{0}}\gamma^{n}$
(2.4)

for all $s=a+ib\in{\mathbb{C}}$ with $|a|<\epsilon$, $|b|\geq D$ and all
$n\geq A\log|b|$. Such an assumption holds in the settings of [5, 6, 7, 10].

Now we can state our main result on norm decay for $L_{t}$.

###### Theorem 2.2

Under these assumptions, there exists $\epsilon>0$, $m\geq 1$, $C>0$ such that

$\|L_{t}v\|_{\delta,\eta,1}\leq Ce^{-\epsilon
t}\|v\|_{\delta,\eta,m}\quad\text{for all $t>0$}$

for all $v\in{\mathcal{F}}_{\delta,\eta,m}^{0}(Y^{\varphi})$ with good
support.

###### Remark 2.3

Since the norm applied to $v$ is stronger than the norm applied to $L_{t}v$,
Theorem 2.2 does not imply a spectral gap for $L_{t}$. We note that the norm
on ${\mathcal{F}}_{\delta,\eta,1}$ gives no Hölder control in the flow
direction when passing through points of the form $(y,\varphi(y))$. This lack
of control is a barrier to mollification arguments of the type usually used to
pass from smooth observables to Hölder observables. In fact, such arguments
are doomed to fail at the operator level by [17, Theorem 1.1] when
$\eta>\frac{1}{2}$ and hence seem unlikely for any $\eta$.

###### Remark 2.4

Usually, we can take $m_{0}\in(0,1)$ in (2.4) in which case $m=3$ suffices in
Theorem 2.2.

There are numerous simplifications when $\\{Y_{j}\\}$ is a finite partition.
In particular, conditions (2.1) and (2.2) are redundant and we can take
$\delta=0$.

##  3 Proof of Theorem 2.2

Our proof of norm decay is broken into three parts. In Subsection 3.1, we
recall a continuous-time operator renewal equation [18] which enables
estimates of Laplace transforms of transfer operators at the level of $Y$. In
Subsection 3.2, we show how to pass to estimates of Laplace transforms of
$L_{t}$. In Subsection 3.3, we invert the Laplace transform to obtain norm
decay of $L_{t}$.

###  3.1 Operator renewal equation

Let ${\widetilde{Y}}=Y\times[0,1]$ and define

$\widetilde{F}:{\widetilde{Y}}\to{\widetilde{Y}},\qquad\widetilde{F}(y,u)=(Fy,u),$

with transfer operator $\widetilde{R}:L^{1}({\widetilde{Y}})\to
L^{1}({\widetilde{Y}})$. Also, define

$\widetilde{\varphi}:{\widetilde{Y}}\to[2,\infty),\quad\widetilde{\varphi}(y,u)=\varphi(y).$

Define the twisted transfer operators

$\widehat{R}(s):L^{1}({\widetilde{Y}})\to
L^{1}({\widetilde{Y}}),\qquad\widehat{R}(s)v=\widetilde{R}(e^{-s\widetilde{\varphi}}v).$

Let ${\widetilde{Y}}_{j}=Y_{j}\times[0,1]$. For
$v:{\widetilde{Y}}\to{\mathbb{R}}$, define

$\|v\|_{\eta}=|v|_{\infty}+|v|_{\eta},\qquad|v|_{\eta}=\sup_{j\geq
1}\sup_{(y,u),(y^{\prime},u)\in{\widetilde{Y}}_{j},\,y\neq
y^{\prime}}|v(y,u)-v(y^{\prime},u)|/d(y,y^{\prime})^{\eta}.$

Let ${\mathcal{F}}_{\eta}({\widetilde{Y}})$ consist of observables
$v:{\widetilde{Y}}\to{\mathbb{R}}$ with $\|v\|_{\eta}<\infty$. Let

${\mathcal{F}}_{\eta}^{0}({\widetilde{Y}})=\\{v\in{\mathcal{F}}_{\eta}({\widetilde{Y}}):{\textstyle\int}_{\widetilde{Y}}v\,d\tilde{\mu}=0\\}$

where $\tilde{\mu}=\mu\times\operatorname{Leb}_{[0,1]}$.

###### Lemma 3.1

Write $s=a+ib\in{\mathbb{C}}$. There exists $\epsilon>0$, $m_{1}\geq 0$, $C>0$
such that

  * (a)

$s\mapsto(I-\widehat{R}(s))^{-1}:{\mathcal{F}}_{\eta}^{0}({\widetilde{Y}})\to{\mathcal{F}}_{\eta}({\widetilde{Y}})$
is analytic on $\\{|a|<\epsilon\\}$;

  * (b)

$s\mapsto(I-\widehat{R}(s))^{-1}:{\mathcal{F}}_{\eta}({\widetilde{Y}})\to{\mathcal{F}}_{\eta}({\widetilde{Y}})$
is analytic on $\\{|a|<\epsilon\\}$ except for a simple pole at $s=0$;

  * (c)

${\|(I-\widehat{R}(s))^{-1}\|}_{{\mathcal{F}}_{\eta}({\widetilde{Y}})\mapsto{\mathcal{F}}_{\eta}({\widetilde{Y}})}\leq
C|b|^{m_{1}}$ for $|a|\leq\epsilon$, $|b|\geq 1$.

###### Proof.

It suffices to verify these properties for $Z(s)=(I-\widehat{R}_{0}(s))^{-1}$
on $Y$. They immediately transfer to $(I-\widehat{R}(s))^{-1}$ on
${\widetilde{Y}}$ since $(\widehat{R}v)(y,u)=(\widehat{R}_{0}v^{u})(y)$ where
$v^{u}(y)=v(y,u)$.

The arguments for passing from (2.4) to the desired properties for $Z(s)$ are
standard. For completeness, we sketch these details now recalling arguments
from [5]. Define ${\mathcal{F}}_{\eta}(Y)$ with norm $\|\;\|_{\eta}$ by
restricting to $u=0$ (this coincides with the usual Hölder space on $Y$). Let
$A$, $D$, $\epsilon$ and $m_{0}$ be as in (2.4). Increase $A$ and $D$ so that
$D>1$ and $|b|^{m_{0}}\gamma^{[A\log|b|]}\leq\frac{1}{2}$ for $|b|\geq D$.
Suppose that $|a|\leq\epsilon$, $|b|\geq D$. Then
$\|\widehat{R}_{0}(s)^{[A\log|b|]}\|_{\eta}\leq|b|^{m_{0}}\gamma^{[A\log|b|]}\leq\frac{1}{2}$
and $\|(I-\widehat{R}_{0}(s)^{[A\log|b|]})^{-1}\|_{\eta}\leq 2.$

As in [5, Proposition 2.5], we can shrink $\epsilon$ so that
$s\to\widehat{R}_{0}(s)$ is continuous on ${\mathcal{F}}_{\eta}(Y)$ for
$|a|\leq\epsilon$. The simple eigenvalue $1$ for $\widehat{R}_{0}(0)=R_{0}$
extends to a continuous family of simple eigenvalues $\lambda(s)$ for
$|s|\leq\epsilon$. Hence we can choose $\epsilon$ so that
$\frac{1}{2}<\lambda(a)<2$ for $|a|\leq\epsilon$. By [5, Corollary 2.8],
$\|\widehat{R}_{0}(s)^{n}\|_{\eta}\ll|b|\lambda(a)^{n}\leq|b|2^{n}$ for all
$n\geq 1$, $|a|\leq\epsilon$, $|b|\geq D$. Hence

$\displaystyle\|Z(s)\|_{\eta}$
$\displaystyle\leq\big{(}1+\|\widehat{R}_{0}(s)\|_{\eta}+\dots+\|\widehat{R}_{0}(s)^{[A\log|b|]-1}\|_{\eta}\big{)}\|(I-\widehat{R}_{0}(s)^{[A\log|b|]})^{-1}\|_{\eta}$
$\displaystyle\ll(\log|b|)\,|b|\,2^{A\log|b|}\leq|b|^{m_{1}},$

with $m_{1}=1+A\log 2$. This proves analyticity on the region
$\\{|a|<\epsilon,|b|>D\\}$ with the desired estimates for property (c) on this
region.

For $|a|\leq\epsilon$, $|b|\leq D$, we recall arguments from the proof of [5,
Lemma 2.22] (where $\widehat{R}_{0}(s)$ is denoted $Q_{s}$). For $\epsilon$
sufficiently small, the part of spectrum of $\widehat{R}_{0}(s)$ that is close
to $1$ consists only of isolated eigenvalues. Also, the spectral radius of
$\widehat{R}_{0}(s)$ is at most $\lambda(a)$ and $\lambda(a)<1$ for
$a\in[0,\epsilon]$, so $s\mapsto Z(s)$ is analytic on $\\{0<a<\epsilon\\}$.

Suppose that $\widehat{R}_{0}(ib)v=v$ for some $v\in{\mathcal{F}}_{\eta}(Y)$,
$b\neq 0$. Choose $q\geq 1$ such that $q|b|>D$. Since $\widehat{R}_{0}(s)$ is
the $L^{2}$ adjoint of $v\mapsto e^{s\varphi}v\circ F$, we have
$e^{ib\varphi}v\circ F=v$. Hence $e^{iqb\varphi}v^{q}\circ F=v^{q}$ and so
$\widehat{R}_{0}(iqb)v^{q}=v^{q}$. But $\|Z(iqb)v^{q}\|_{\eta}<\infty$, so
$v=0$. Hence $1\not\in\operatorname{spec}\widehat{R}_{0}(ib)$ for all $b\neq
0$. It follows that for all $b\neq 0$ there exists an open set
$U_{b}\subset{\mathbb{C}}$ containing $ib$ such that
$1\not\in\operatorname{spec}\widehat{R}_{0}(s)$ for all $s\in U_{b}$, and so
$s\mapsto Z(s)$ is analytic on $U_{b}$.

Next, we recall that for $s$ near to zero, $\lambda(s)=1+cs+O(s^{2})$ where
$c<0$. Hence $s\mapsto Z(s)$ has a simple pole at zero. It follows that there
exists $\epsilon>0$ such that $s\mapsto Z(s)$ is analytic on
$\\{|a|<\epsilon,|b|<2D\\}$ except for a simple pole at $s=0$. Combining this
with the estimates on $\\{|a|<\epsilon,|b|\geq D\\}$ we have proved properties
(b) and (c) for $Z(s)$.

Finally, the spectral projection $\pi$ corresponding to the eigenvalue
$\lambda(0)=1$ for $\widehat{R}_{0}(0)=R$ is given by $\pi v=\int_{Y}v\,d\mu$.
Hence the pole disappears on restriction to observables of mean zero, proving
property (a) for $Z(s)$. ∎

Next define

$T_{t}v=1_{\widetilde{Y}}L_{t}(1_{\widetilde{Y}}v),\qquad
U_{t}v=1_{\widetilde{Y}}L_{t}(1_{\\{\widetilde{\varphi}>t\\}}v)$

and

$\qquad\widehat{T}(s)=\int_{0}^{\infty}e^{-st}T_{t}\,dt,\qquad\widehat{U}(s)=\int_{0}^{\infty}e^{-st}U_{t}\,dt,$

By [18, Theorem 3.3], we have the operator renewal equation

$\widehat{T}=\widehat{U}(I-\widehat{R})^{-1}.$

###### Proposition 3.2

There exists $\epsilon>0$, $C>0$ such that
$s\mapsto\widehat{U}(s):{\mathcal{F}}_{\eta}({\widetilde{Y}})\to{\mathcal{F}}_{\eta}({\widetilde{Y}})$
is analytic on $\\{|a|<\epsilon\\}$ and
${\|\widehat{U}(s)\|}_{{\mathcal{F}}_{\eta}({\widetilde{Y}})\mapsto{\mathcal{F}}_{\eta}({\widetilde{Y}})}\leq
C|s|$ for $|a|\leq\epsilon$.

###### Proof.

By [18, Proposition 3.4],

$(U_{t}v)(y,u)=\begin{cases}v(y,u-t)1_{[t,1]}(u)&0\leq t\leq 1\\\
(\widetilde{R}v_{t})(y,u)&t>1\end{cases}$

where $v_{t}(y,u)=1_{\\{t<\varphi(y)<t+1-u\\}}v(y,u-t+\varphi(y))$. Hence
$\widehat{U}(s)=\widehat{U}_{1}(s)+\widehat{U}_{2}(s)$ where

$(\widehat{U}_{1}(s)v)(y,u)=\int_{0}^{u}e^{-st}v(y,u-t)\,dt,\qquad\widehat{U}_{2}(s)v=\int_{1}^{\infty}e^{-st}\widetilde{R}v_{t}\,dt.$

It is clear that $\|\widehat{U}_{1}(s)v\|_{\eta}\leq
e^{\epsilon}\|v\|_{\eta}$. We focus attention on the second term

$(\widehat{U}_{2}(s)v)(y,u)={\textstyle\sum_{j}}g(y_{j})\int_{1}^{\infty}e^{-st}v_{t}(y_{j},u)\,dt={\textstyle\sum_{j}}g(y_{j})\widehat{V}(s)(y_{j},u),$

where $\widehat{V}(s)(y,u)=\int_{u}^{1}e^{s(t-u-\varphi)}v(y,t)\,dt$. Clearly,
$|1_{Y_{j}}\widehat{V}(s)|_{\infty}\leq
e^{\epsilon|1_{Y_{j}}\varphi|_{\infty}}|v|_{\infty}$. Also,

$\widehat{V}(s)(y,u)-\widehat{V}(s)(y^{\prime},u)=I+J,$

where

$\displaystyle I$
$\displaystyle=\int_{u}^{1}(e^{s(t-u-\varphi(y))}-e^{s(t-u-\varphi(y^{\prime}))})v(y,t)\,dt,$
$\displaystyle J$
$\displaystyle=\int_{u}^{1}e^{s(t-u-\varphi(y^{\prime}))}(v(y,t)-v(y^{\prime},t))\,dt.$

For $y,y^{\prime}\in Y_{j}$,

$|I|\leq|v|_{\infty}\int_{u}^{1}e^{\epsilon(|1_{Y_{j}}\varphi|_{\infty}+u-t)}|s||\varphi(y)-\varphi(y^{\prime})|\,dt\ll|s||v|_{\infty}\,e^{\epsilon|1_{Y_{j}}\varphi|_{\infty}}d(Fy,Fy^{\prime})^{\eta}$

by (2.1), and

$|J|\leq\int_{u}^{1}e^{\epsilon(|1_{Y_{j}}\varphi|_{\infty}+u-t)}|v(y,t)-v(y^{\prime},t)|\,dt\leq
e^{\epsilon|1_{Y_{j}}\varphi|_{\infty}}|v|_{\eta}\,d(y,y^{\prime})^{\eta}.$

Hence
$|\widehat{V}(s)(y,u)-\widehat{V}(s)(y^{\prime},u)|_{\eta}\ll|s|e^{\epsilon|1_{Y_{j}}\varphi|_{\infty}}\|v\|_{\eta}\,d(Fy,Fy^{\prime})^{\eta}$.

It follows from the estimates for $1_{Y_{j}}\widehat{V}(s)$ together with
(2.3) that
$\|\widehat{U}_{2}(s)v\|_{\eta}\ll{\textstyle\sum_{j}}|s|\mu(Y_{j})e^{\epsilon|1_{Y_{j}}\varphi|_{\infty}}\|v\|_{\eta}$.
By (2.2), $\|\widehat{U}_{2}(s)v\|_{\eta}\ll|s|\|v\|_{\eta}$ for $\epsilon$
sufficiently small. We conclude that
$\|\widehat{U}(s)v\|_{\eta}\ll|s|\|v\|_{\eta}$. ∎

###  3.2 From $\widehat{T}$ on ${\widetilde{Y}}$ to $\widehat{L}$ on
$Y^{\varphi}$

Lemma 3.1 and Proposition 3.2 yield analyticity and estimates for
$\widehat{T}=\widehat{U}(I-\widehat{R})^{-1}$ on ${\widetilde{Y}}$. In this
subsection, we show how these properties are inherited by
$\widehat{L}(s)=\int_{0}^{\infty}e^{-st}L_{t}\,dt$ on $Y^{\varphi}$.

###### Remark 3.3

The approach in this subsection is similar to that in [9, Section 5] but there
are some important differences. The rationale behind the two step
decomposition in Propositions 3.4 and 3.5 below is that the discreteness of
the decomposition in Proposition 3.4 simplifies many formulas significantly.
In particular, the previously problematic term $E_{t}$ in [9] becomes
elementary (and vanishes for large $t$ when $\varphi$ is bounded). The
decomposition in Proposition 3.5 remains continuous to simplify the estimates
in Proposition 3.8.

Since the setting in [9] is different (infinite ergodic theory, reinducing) we
keep the exposition here self-contained even where the estimates coincide with
those in [9].

Define

$\displaystyle A_{n}:L^{1}({\widetilde{Y}})\to L^{1}(Y^{\varphi}),$
$\displaystyle(A_{n}v)(y,u)=1_{\\{n\leq u<n+1\\}}(L_{n}v)(y,u),\;n\geq 0,$
$\displaystyle E_{t}:L^{1}(Y^{\varphi})\to L^{1}(Y^{\varphi}),$
$\displaystyle(E_{t}v)(y,u)=1_{\\{[t]+1\leq
u\leq\varphi(y)\\}}(L_{t}v)(y,u),\;t>0.$

###### Proposition 3.4

$\displaystyle L_{t}=\sum_{j=0}^{[t]}A_{j}1_{\widetilde{Y}}L_{t-j}+E_{t}$ for
$t>0$.

###### Proof.

For $y\in Y$, $u\in(0,\varphi(y))$,

$\displaystyle(L_{t}v)(y,u)$ $\displaystyle=\sum_{j=0}^{[t]}1_{\\{j\leq
u<j+1\\}}(L_{t}v)(y,u)+1_{\\{[t]+1\leq u\leq\varphi(y)\\}}(L_{t}v)(y,u)$
$\displaystyle=\sum_{j=0}^{[t]}(A_{j}L_{t-j}v)(y,u)+(E_{t}v)(y,u).$

Now use that $A_{n}=A_{n}1_{\widetilde{Y}}$. ∎

Next, define

$\displaystyle B_{t}:L^{1}(Y^{\varphi})\to L^{1}({\widetilde{Y}}),$
$\displaystyle B_{t}v=1_{\widetilde{Y}}L_{t}(1_{\Delta_{t}}v),$ $\displaystyle
G_{t}:L^{1}(Y^{\varphi})\to L^{1}({\widetilde{Y}}),$ $\displaystyle
G_{t}v=B_{t}(\omega(t)v),$ $\displaystyle H_{t}:L^{1}(Y^{\varphi})\to
L^{1}({\widetilde{Y}}),$ $\displaystyle
H_{t}v=1_{\widetilde{Y}}L_{t}(1_{\Delta^{\prime}_{t}}v),$

for $t>0$, where

$\displaystyle\Delta_{t}$ $\displaystyle=\\{(y,u)\in
Y^{\varphi}:\varphi(y)-t\leq u<\varphi(y)-t+1\\}$
$\displaystyle\Delta^{\prime}_{t}$ $\displaystyle=\\{(y,u)\in
Y^{\varphi}:u<\varphi(y)-t\\},\qquad\omega(t)(y,u)=\varphi(y)-u-t+1.$

###### Proposition 3.5

$\displaystyle
1_{\widetilde{Y}}L_{t}=\int_{0}^{t}T_{t-\tau}B_{\tau}\,d\tau+G_{t}+H_{t}$ for
$t>0$.

###### Proof.

Let $y\in Y$, $u\in[0,\varphi(y)]$. Then

$\displaystyle\int_{0}^{t}1_{\Delta_{\tau}}(y,u)\,d\tau$
$\displaystyle=\int_{0}^{t}1_{\\{\varphi(y)-u\leq\tau\leq\varphi(y)-u+1\\}}\,d\tau$
$\displaystyle=1_{\\{t\geq\varphi(y)-u+1\\}}+1_{\\{\varphi(y)-u\leq
t<\varphi(y)-u+1\\}}(t-\varphi(y)+u)$
$\displaystyle=1-1_{\\{t<\varphi(y)-u+1\\}}+1_{\\{\varphi(y)-u\leq
t<\varphi(y)-u+1\\}}(t-\varphi(y)+u)$
$\displaystyle=1-1_{\Delta_{t}^{\prime}}(y,u)+1_{\Delta_{t}}(y,u)(t-\varphi(y)+u-1).$

Hence
$\int_{0}^{t}1_{\Delta_{\tau}}\,d\tau=1-1_{\Delta_{t}}\omega(t)-1_{\Delta^{\prime}_{t}}.$
It follows that

$\displaystyle\int_{0}^{t}T_{t-\tau}B_{\tau}$ $\displaystyle
v\,d\tau=1_{\widetilde{Y}}\int_{0}^{t}L_{t-\tau}1_{\widetilde{Y}}B_{\tau}v\,d\tau=1_{\widetilde{Y}}\int_{0}^{t}L_{t-\tau}B_{\tau}v\,d\tau$
$\displaystyle=1_{\widetilde{Y}}\int_{0}^{t}L_{t-\tau}L_{\tau}(1_{\Delta_{\tau}}v)\,d\tau=1_{\widetilde{Y}}L_{t}\Big{(}\int_{0}^{t}1_{\Delta_{\tau}}v\,d\tau\Big{)}=1_{\widetilde{Y}}L_{t}v-G_{t}v-H_{t}v$

as required. ∎

We have already defined the Laplace transforms $\widehat{L}(s)$ and
$\widehat{T}(s)$ for $s=a+ib$ with $a>0$. Similarly, define

$\displaystyle\widehat{B}(s)$
$\displaystyle=\int_{0}^{\infty}e^{-st}B_{t}\,dt,\qquad$
$\displaystyle\widehat{E}(s)$
$\displaystyle=\int_{0}^{\infty}e^{-st}E_{t}\,dt,$
$\displaystyle\widehat{G}(s)$
$\displaystyle=\int_{0}^{\infty}e^{-st}G_{t}\,dt,\qquad$
$\displaystyle\widehat{H}(s)$
$\displaystyle=\int_{0}^{\infty}e^{-st}H_{t}\,dt.$

Also, we define the discrete transform
$\displaystyle\widehat{A}(s)=\sum_{n=0}^{\infty}e^{-sn}A_{n}.$

###### Corollary 3.6

$\widehat{L}(s)=\widehat{A}(s)\widehat{T}(s)\widehat{B}(s)+\widehat{A}(s)\widehat{G}(s)+\widehat{A}(s)\widehat{H}(s)+\widehat{E}(s)$
for $a>0$.

###### Proof.

By Proposition 3.4,

$\displaystyle\widehat{L}(s)-\widehat{E}(s)$
$\displaystyle=\int_{0}^{\infty}e^{-st}\sum_{j=0}^{[t]}A_{j}1_{\widetilde{Y}}L_{t-j}\,dt=\sum_{j=0}^{\infty}e^{-sj}A_{j}1_{\widetilde{Y}}\int_{j}^{\infty}e^{-s(t-j)}L_{t-j}\,dt$
$\displaystyle=\widehat{A}(s)1_{\widetilde{Y}}\int_{0}^{\infty}e^{-st}L_{t}\,dt=\widehat{A}(s)1_{\widetilde{Y}}\widehat{L}(s).$

Hence $\widehat{L}=\widehat{A}1_{\widetilde{Y}}\widehat{L}+\widehat{E}$. In
addition, by Proposition 3.5, $\displaystyle
1_{\widetilde{Y}}\widehat{L}=\widehat{T}\widehat{B}+\widehat{G}+\widehat{H}$.
∎

###### Proposition 3.7

Let $\delta>\epsilon>0$. Then there is a constant $C>0$ such that

  * (a)

${\|\widehat{A}(s)\|}_{{\mathcal{F}}_{\eta}({\widetilde{Y}})\to{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})}\leq
1$,

  * (b)

${\|\widehat{E}(s)\|}_{{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})\to{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})}\leq
C$,

  * (c)

${\|\widehat{H}(s)\|}_{{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})\to{\mathcal{F}}_{\eta}({\widetilde{Y}})}\leq
e^{\delta}$,

for $|a|\leq\epsilon$.

###### Proof.

(a) Let $v\in{\mathcal{F}}_{\eta}({\widetilde{Y}})$. Let
$(y,u),\,(y^{\prime},u)\in Y_{j}^{\varphi}$, $j\geq 1$. Since
$(A_{n}v)(y,u)=1_{\\{n\leq u<n+1\\}}v(y,u-n)$,

$(\widehat{A}(s)v)(y,u)=\sum_{n=0}^{\infty}e^{-sn}1_{\\{n\leq
u<n+1\\}}v(y,u-n)=e^{-s[u]}v(y,u-[u]).$

Hence

$|(\widehat{A}(s)v)(y,u)|\leq e^{\epsilon
u}|v|_{\infty},\quad|(\widehat{A}(s)v)(y,u)-(\widehat{A}(s)v)(y^{\prime},u)|\leq
e^{\epsilon u}|v|_{\eta}\,d(y,y^{\prime})^{\eta}.$

That is, $|\widehat{A}(s)v|_{\epsilon,\infty}\leq|v|_{\infty}$,
$|\widehat{A}(s)v|_{\epsilon,\eta}\leq|v|_{\eta}$. Hence
$\|\widehat{A}(s)v\|_{\delta,\eta}\leq\|\widehat{A}(s)v\|_{\epsilon,\eta}\leq\|v\|_{\eta}$.
(b) We take $C=1/(\delta-\epsilon)$. Let
$v\in{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})$. Let $(y,u),\,(y^{\prime},u)\in
Y_{j}^{\varphi}$, $j\geq 1$. Note that $(E_{t}v)(y,u)=1_{\\{[t]+1\leq
u\\}}v(y,u-t)$, so

$(\widehat{E}(s)v)(y,u)=\int_{0}^{\infty}e^{-st}1_{\\{[t]+1\leq
u\\}}v(y,u-t)\,dt.$

Hence

$|(\widehat{E}(s)v)(y,u)|\leq\int_{0}^{\infty}e^{\epsilon
t}|v|_{\delta,\infty}\,e^{\delta(u-t)}\,dt=C|v|_{\delta,\infty}\,e^{\delta
u},$

and

$|(\widehat{E}(s)v)(y,u)-(\widehat{E}(s)v)(y^{\prime},u)|\leq\int_{0}^{\infty}e^{\epsilon
t}|v|_{\delta,\eta}\,d(y,y^{\prime})^{\eta}e^{\delta(u-t)}\,dt=Ce^{\delta
u}|v|_{\delta,\eta}\,d(y,y^{\prime})^{\eta}.$

That is, $|\widehat{E}(s)v|_{\delta,\infty}\leq|v|_{\delta,\infty}$ and
$|\widehat{E}(s)v|_{\delta,\eta}\leq|v|_{\delta,\eta}$.
(c) Let $v\in{\mathcal{F}}_{\epsilon,\eta}(Y^{\varphi})$. Let
$(y,u),\,(y^{\prime},u)\in{\widetilde{Y}}_{j}$, $j\geq 1$. Then
$(H_{t}v)(y,u)=1_{\\{t<u\\}}v(y,u-t)$ and
$(\widehat{H}(s)v)(y,u)=\int_{0}^{u}e^{-st}v(y,u-t)\,dt$. Hence,

$\displaystyle|\widehat{H}(s)v|_{\infty}\leq
e^{\delta}|v|_{\delta,\infty}\quad\text{and}\quad|(\widehat{H}(s)v)(y,u)-(\widehat{H}(s)v)(y^{\prime},u)|\leq
e^{\delta}|v|_{\delta,\eta}\,d(y,y^{\prime})^{\eta}.$

The result follows. ∎

###### Proposition 3.8

There exists $\delta>\epsilon>0$, $C>0$ such that

${\|\widehat{B}(s)\|}_{{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})\to{\mathcal{F}}_{\eta}({\widetilde{Y}})}\leq
C|s|\quad\text{and}\quad{\|\widehat{G}(s)\|}_{{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})\to{\mathcal{F}}_{\eta}({\widetilde{Y}})}\leq
C|s|\quad\text{for $|a|\leq\epsilon$.}$

###### Proof.

Let $v\in L^{1}(Y^{\varphi})$, $w\in L^{\infty}({\widetilde{Y}})$. Using that
$F_{t}(y,u)=(Fy,u+t-\varphi(y))$ for $(y,u)\in\Delta_{t}$,

$\displaystyle\int_{\widetilde{Y}}B_{t}v\,w\,d\tilde{\mu}$
$\displaystyle=\bar{\varphi}\int_{Y^{\varphi}}L_{t}(1_{\Delta_{t}}v)\,w\,d\mu^{\varphi}=\bar{\varphi}\int_{Y^{\varphi}}1_{\Delta_{t}}v\,w\circ
F_{t}\,d\mu^{\varphi}$ $\displaystyle=\int_{Y}\int_{0}^{\varphi(y)}1_{\\{0\leq
u+t-\varphi(y)<1\\}}v(y,u)w(Fy,u+t-\varphi)\,du\,d\mu$
$\displaystyle=\int_{Y}\int_{t-\varphi(y)}^{t}1_{\\{0\leq
u<1\\}}v(y,u+\varphi(y)-t)w(Fy,u)\,du\,d\mu$
$\displaystyle=\int_{\widetilde{Y}}v_{t}\,w\circ\widetilde{F}\,d\tilde{\mu}=\int_{\widetilde{Y}}\widetilde{R}v_{t}\,w\,d\tilde{\mu}$

where $v_{t}(y,u)=1_{\\{0<u+\varphi(y)-t<\varphi(y)\\}}v(y,u+\varphi(y)-t)$.

Hence $B_{t}v=\widetilde{R}v_{t}$ and it follows immediately that
$G_{t}v=\widetilde{R}(\omega(t)v)_{t}$. But

$(\omega(t)v)_{t}(y,u)=1_{\\{0<u+\varphi(y)-t<\varphi(y)\\}}(\omega(t)v)(y,u+\varphi(y)-t)=(1-u)v_{t}(y,u),$

so $(G_{t}v)(y,u)=(1-u)(B_{t}v)(y,u)$.

Next, $\widehat{B}(s)v=\widetilde{R}\widehat{V}(s)$ where

$\displaystyle\widehat{V}(s)(y,u)=\int_{0}^{\infty}e^{-st}v_{t}(y,u)\,dt$
$\displaystyle=\int_{u}^{u+\varphi(y)}e^{-st}v(y,u+\varphi(y)-t)\,dt$
$\displaystyle=\int_{0}^{\varphi(y)}e^{-s(\varphi(y)+u-t)}v(y,t)\,dt.$

It is immediate that

$\displaystyle(\widehat{G}(s)v)(y,u)=(1-u)(\widehat{B}(s)v)(y,u).$ (3.1)

Suppose that $\delta>\epsilon>0$ are fixed. Let
$v\in{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})$. Let
$(y,u),\,(y^{\prime},u)\in{\widetilde{Y}}_{j}$, $j\geq 1$. Then

$|\widehat{V}(s)(y,u)|\leq\int_{0}^{\varphi(y)}e^{-a\,(\varphi(y)+u-t)}|v|_{\delta,\infty}\,e^{\delta
t}\,dt\ll e^{\delta\varphi(y)}|v|_{\delta,\infty}$

and so $|1_{Y_{j}}\widehat{V}(s)|_{\infty}\ll
e^{\delta|1_{Y_{j}}\varphi|_{\infty}}|v|_{\delta,\infty}$.

Next, suppose without loss that $\varphi(y^{\prime})\leq\varphi(y)$. Then

$\widehat{V}(s)(y,u)-\widehat{V}(s)(y^{\prime},u)=J_{1}+J_{2}+J_{3}$

where

$\displaystyle J_{1}$
$\displaystyle=\int_{0}^{\varphi(y)}(e^{-s(\varphi(y)+u-t)}-e^{-s(\varphi(y^{\prime})+u-t)})v(y,t)\,dt,$
$\displaystyle J_{2}$
$\displaystyle=\int_{0}^{\varphi(y)}e^{-s(\varphi(y^{\prime})+u-t)}(v(y,t)-v(y^{\prime},t))\,dt,$
$\displaystyle J_{3}$
$\displaystyle=\int_{\varphi(y^{\prime})}^{\varphi(y)}e^{-s(\varphi(y^{\prime})+u-t)}v(y^{\prime},t)\,dt.$

For notational convenience we suppose that $a\in(-\epsilon,0)$ since the range
$a\geq 0$ is simpler. Using (2.1),

$\displaystyle|J_{1}|$
$\displaystyle\leq\int_{0}^{\varphi(y)}e^{\epsilon(|1_{Y_{j}}\varphi|_{\infty}+1-t)}|s||\varphi(y)-\varphi(y^{\prime})||v|_{\delta,\infty}\,e^{\delta
t}\,dt$
$\displaystyle\ll|s|\varphi(y)e^{\delta|1_{Y_{j}}\varphi|_{\infty}}\,d(Fy,Fy^{\prime})^{\eta}|v|_{\delta,\infty}\ll|s|e^{2\delta|1_{Y_{j}}\varphi|_{\infty}}\,d(Fy,Fy^{\prime})^{\eta}|v|_{\delta,\infty},$
$\displaystyle|J_{2}|$
$\displaystyle\leq\int_{0}^{\varphi(y)}e^{\epsilon(|1_{Y_{j}}\varphi|_{\infty}+1-t)}|v|_{\delta,\eta}\,e^{\delta
t}d(y,y^{\prime})^{\eta}\,dt\ll
e^{\delta|1_{Y_{j}}\varphi|_{\infty}}\,d(y,y^{\prime})^{\eta}|v|_{\delta,\eta},$
$\displaystyle|J_{3}|$
$\displaystyle\leq\int_{\varphi(y^{\prime})}^{\varphi(y)}e^{\epsilon(|1_{Y_{j}}\varphi|_{\infty}+1-t)}|v|_{\delta,\infty}\,e^{\delta
t}\,dt\ll
e^{2\delta|1_{Y_{j}}\varphi|_{\infty}}|v|_{\delta,\infty}\,d(Fy,Fy^{\prime})^{\eta}.$

Hence

$|\widehat{V}(s)(y,u)-\widehat{V}(s)(y,u)|\ll|s|e^{2\delta|1_{Y_{j}}\varphi|_{\infty}}\|v\|_{\delta,\eta}\,d(Fy,Fy^{\prime})^{\eta}.$

Now, for $(y,u)\in{\widetilde{Y}}$,

$(\widehat{B}(s)v)(y,u)=(\widetilde{R}\widehat{V}(s))(y,u)={\textstyle\sum_{j}}g(y_{j})\widehat{V}(s)(y_{j},u),$

where $y_{j}$ is the unique preimage of $y$ under $F|Y_{j}$. It follows from
the estimates for $\widehat{V}(s)$ together with (2.3) that

$\|\widehat{B}(s)v\|_{\eta}\ll|s|{\textstyle\sum_{j}}\mu(Y_{j})e^{2\delta|1_{Y_{j}}\varphi|_{\infty}}\|v\|_{\delta,\eta}.$

Shrinking $\delta$, the desired estimate for $\widehat{B}$ follows from (2.2).
Finally, the estimate for $\widehat{G}$ follows from (3.1). ∎

###### Proposition 3.9

$\int_{\widetilde{Y}}\widehat{B}(0)v\,d\tilde{\mu}=\bar{\varphi}\int_{Y^{\varphi}}v\,d\mu^{\varphi}$
for $v\in L^{1}(Y^{\varphi})$.

###### Proof.

By the definition of $\widehat{B}$,

$\displaystyle\int_{\widetilde{Y}}$
$\displaystyle\widehat{B}(0)v\,d\tilde{\mu}=\int_{\widetilde{Y}}\int_{0}^{\infty}L_{t}(1_{\Delta_{t}}v)\,dt\,d\tilde{\mu}=\bar{\varphi}\int_{0}^{\infty}\int_{Y^{\varphi}}L_{t}(1_{\Delta_{t}}v)\,d\mu^{\varphi}\,dt$
$\displaystyle=\bar{\varphi}\int_{0}^{\infty}\int_{Y^{\varphi}}1_{\Delta_{t}}v\,d\mu^{\varphi}\,dt=\bar{\varphi}\int_{Y^{\varphi}}\int_{0}^{\infty}1_{\\{\varphi-u<t<\varphi-u+1\\}}v\,dt\,d\mu^{\varphi}=\bar{\varphi}\int_{Y^{\varphi}}v\,d\mu^{\varphi},$

as required. ∎

###### Lemma 3.10

Write $s=a+ib\in{\mathbb{C}}$. There exists $\epsilon>0$, $\delta>0$,
$m_{2}\geq 0$, $C>0$ such that

  * (a)

$s\mapsto\widehat{L}(s):{\mathcal{F}}_{\delta,\eta}^{0}(Y^{\varphi})\to{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})$
is analytic on $\\{|a|<\epsilon\\}$;

  * (b)

$s\mapsto\widehat{L}(s):{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})\to{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})$
is analytic on $\\{|a|<\epsilon\\}$ except for a simple pole at $s=0$;

  * (c)

$\|\widehat{L}(s)v\|_{\delta,\eta}\leq C|b|^{m_{2}}\|v\|_{\delta,\eta}$ for
$|a|\leq\epsilon$, $|b|\geq 1$,
$v\in{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})$.

###### Proof.

Recall that

$\widehat{L}=\widehat{A}\widehat{T}\widehat{B}+\widehat{A}\widehat{G}+\widehat{A}\widehat{H}+\widehat{E},\qquad\widehat{T}=\widehat{U}(I-\widehat{R})^{-1}$

where $\widehat{U}$, $\widehat{A}$, $\widehat{B}$, $\widehat{G}$,
$\widehat{H}$ and $\widehat{E}$ are analytic by Propositions 3.2, 3.7 and 3.8.
Hence part (b) follows immediately from Lemma 3.1(b). Also, part (c) follows
using Lemma 3.1(c).

By Proposition 3.9,
$\widehat{B}(0)({\mathcal{F}}_{\delta,\eta}^{0}(Y^{\varphi}))\subset{\mathcal{F}}_{\eta}^{0}({\widetilde{Y}})$.
Hence the simple pole at $s=0$ for $(I-\widehat{R})^{-1}\widehat{B}$
disappears on restriction to ${\mathcal{F}}_{\delta,\eta}^{0}(Y^{\varphi})$ by
Lemma 3.1(a). This proves part (a). ∎

###  3.3 Moving the contour of integration

###### Proposition 3.11

Let $m\geq 1$. Let $v\in{\mathcal{F}}_{\delta,\eta,m}(Y^{\varphi})$ with good
support. Then
$\widehat{L}(s)v=\sum_{j=0}^{m-1}(-1)^{j}s^{-(j+1)}\partial_{t}^{j}v+(-1)^{m}s^{-m}\widehat{L}(s)\partial_{t}^{m}v$
for $a>0$.

###### Proof.

Recall that $\operatorname{supp}v\subset\\{(y,u)\in
Y^{\varphi}:u\in[r,\varphi(y)-r]\\}$ for some $r>0$. For $h\in[0,r]$, we can
define $(\Psi_{h}v)(y,u)=v(y,u-h)$ and then $(\Psi_{h}v)\circ F_{h}=v$.

Let $w\in L^{\infty}(Y^{\varphi})$ and write
$\rho_{v,w}(t)=\int_{Y^{\varphi}}v\,w_{t}\,d\mu^{\varphi}$ where $w_{t}=w\circ
F_{t}$. Then for $h\in[0,r]$,

$\rho_{v,w}(t+h)=\int_{Y^{\varphi}}v\,w_{t}\circ
F_{h}\,d\mu^{\varphi}=\int_{Y^{\varphi}}(\Psi_{h}v)\circ F_{h}\,w_{t}\circ
F_{h}\,d\mu^{\varphi}=\int_{Y^{\varphi}}\Psi_{h}v\,w_{t}\,d\mu^{\varphi}.$

Hence
$h^{-1}(\rho_{v,w}(t+h)-\rho_{v,w}(t))=\int_{Y^{\varphi}}h^{-1}(\Psi_{h}v-v)\,w_{t}\,d\mu^{\varphi}$
so

$\rho^{\prime}_{v,w}(t)=-\int_{Y^{\varphi}}\partial_{t}v\,w_{t}\,d\mu^{\varphi}=-\int_{Y^{\varphi}}\partial_{t}v\,w\circ
F_{t}\,d\mu^{\varphi}=-\rho_{\partial_{t}v,w}(t).$

Inductively, $\rho^{(j)}_{v,w}(t)=(-1)^{j}\rho_{\partial_{t}^{j}v,w}(t)$.

Now
$\int_{Y^{\varphi}}\widehat{L}(s)v\,w\,d\mu^{\varphi}=\int_{0}^{\infty}e^{-st}\int_{Y^{\varphi}}L_{t}v\,w\,d\mu^{\varphi}\,dt=\int_{0}^{\infty}e^{-st}\rho_{v,w}(t)\,dt$,
so repeatedly integrating by parts,

$\displaystyle\int_{Y^{\varphi}}\widehat{L}(s)v\,w\,d\mu^{\varphi}$
$\displaystyle=\sum_{j=0}^{m-1}s^{-(j+1)}\rho^{(j)}_{v,w}(0)+s^{-m}\int_{0}^{\infty}e^{-st}\rho^{(m)}_{v,w}(t)\,dt$
$\displaystyle=\sum_{j=0}^{m-1}(-1)^{j}s^{-(j+1)}\rho_{\partial_{t}^{j}v,w}(0)+(-1)^{m}s^{-m}\int_{0}^{\infty}e^{-st}\rho_{\partial_{t}^{m}v,w}(t)\,dt$
$\displaystyle=\int_{Y^{\varphi}}\sum_{j=0}^{m-1}(-1)^{j}s^{-(j+1)}\partial_{t}^{j}v\,w\,d\mu^{\varphi}+(-1)^{m}s^{-m}\int_{0}^{\infty}e^{-st}\rho_{\partial_{t}^{m}v,w}(t)\,dt.$

Finally,
$\int_{0}^{\infty}e^{-st}\rho_{\partial_{t}^{m}v,w}(t)\,dt=\int_{Y^{\varphi}}\widehat{L}(s)\partial_{t}^{m}v\,w\,d\mu^{\varphi}$
and the result follows since $w\in L^{\infty}(Y^{\varphi})$ is arbitrary. ∎

We can now estimate $\|L_{t}v\|_{\delta,\eta}$.

###### Corollary 3.12

Under the assumptions of Theorem 2.2, there exists $\epsilon>0$, $m_{3}\geq
1$, $C>0$ such that

$\|L_{t}v\|_{\delta,\eta}\leq Ce^{-\epsilon
t}\|v\|_{\delta,\eta,m_{3}}\quad\text{for all $t>0$}$

for all $v\in{\mathcal{F}}_{\delta,\eta,m_{3}}^{0}(Y^{\varphi})$ with good
support.

###### Proof.

Let $m_{3}=m_{2}+2$. By Lemma 3.10(a),
$\widehat{L}(s):{\mathcal{F}}_{\delta,\eta,m_{3}}^{0}(Y^{\varphi})\to{\mathcal{F}}_{\delta,\eta}(Y^{\varphi})$
is analytic for $|a|\leq\epsilon$. The alternative expression in Proposition
3.11 is also analytic on this region (the apparent singularity at $s=0$ is
removable by the equality with the analytic function $\widehat{L}$). Hence we
can move the contour of integration to $s=-\epsilon+ib$ when computing the
inverse Laplace transform, to obtain

$\displaystyle L_{t}v$
$\displaystyle=\int_{-\infty}^{\infty}e^{st}\Big{(}\sum_{j=0}^{m_{3}-1}(-1)^{j}s^{-(j+1)}\partial_{t}^{j}v+(-1)^{m_{3}}s^{-{m_{3}}}\widehat{L}(s)\partial_{t}^{m_{3}}v\Big{)}\,db$
$\displaystyle=e^{-\epsilon
t}\sum_{j=0}^{m_{3}-1}(-1)^{j}\partial_{t}^{j}v\int_{-\infty}^{\infty}e^{ibt}s^{-(j+1)}\,db+(-1)^{m_{3}}e^{-\epsilon
t}\int_{-\infty}^{\infty}e^{ibt}s^{-{m_{3}}}\widehat{L}(s)\partial_{t}^{m_{3}}v\,db.$

The final term is estimated using Lemma 3.10(b,c):

$\Big{\|}\int_{-\infty}^{\infty}e^{ibt}s^{-{m_{3}}}\widehat{L}(s)\partial_{t}^{m_{3}}v\,db\Big{\|}_{\delta,\eta}\ll\int_{-\infty}^{\infty}(1+|b|)^{-(m_{2}+2)}(1+|b|)^{m_{2}}\|\partial_{t}^{m_{3}}v\|_{\delta,\eta}\,db\ll\|v\|_{\delta,\eta,{m_{3}}}.$

Clearly, the integrals $\int_{-\infty}^{\infty}e^{ibt}s^{-(j+1)}\,db$ converge
absolutely for $j\geq 1$, while the integral for $j=0$ converges as an
improper Riemann integral. Hence altogether we obtain that
$\|L_{t}v\|_{\delta,\eta}\ll e^{-\epsilon t}\|v\|_{\delta,\eta,m_{3}}$. ∎

For the proof of Theorem 2.2, it remains to estimate
$\|\partial_{u}L_{t}v\|_{\delta,\eta}$. Recall that the transfer operator
$R_{0}$ for $F$ has weight function $g$. We have the pointwise formula
$(R_{0}^{k}v)(y)=\sum_{F^{k}y^{\prime}=y}g_{k}(y^{\prime})v(y^{\prime})$ where
$g_{k}=g\,\dots\,g\circ F^{k-1}$. Let
$\varphi_{k}=\sum_{j=0}^{k-1}\varphi\circ F^{j}$.

###### Proposition 3.13

Let $v\in L^{1}(Y^{\varphi})$. Then for all $t>0$, $(y,u)\in Y^{\varphi}$,

$(L_{t}v)(y,u)=\sum_{k=0}^{[t/2]}\sum_{F^{k}y^{\prime}=y}g_{k}(y^{\prime})1_{\\{0\leq
u-t+\varphi_{k}(y^{\prime})<\varphi(y^{\prime})\\}}v(y^{\prime},u-t+\varphi_{k}(y^{\prime})).$

###### Proof.

The lap number $N_{t}(y,u)\in[0,t/2]\cap{\mathbb{N}}$ is the unique integer
$k\geq 0$ such that $u+t-\varphi_{k}(y)\in[0,\varphi(F^{k}y))$. In particular,
$F_{t}(y,u)=(F^{N_{t}(y,u)}y,u+t-\varphi_{N_{t}(y,u)}(y))$. For $w\in
L^{\infty}(Y^{\varphi})$,

$\displaystyle\int_{Y^{\varphi}}$ $\displaystyle
L_{t}(1_{\\{N_{t}=k\\}}v)\,w\,d\mu^{\varphi}=\int_{Y^{\varphi}}1_{\\{N_{t}=k\\}}v\,w\circ
F_{t}\,d\mu^{\varphi}$
$\displaystyle=\bar{\varphi}^{-1}\int_{Y}\int_{0}^{\varphi(y)}1_{\\{0\leq
u+t-\varphi_{k}(y)<\varphi(F^{k}y)\\}}v(y,u)\,w(F^{k}y,u+t-\varphi_{k}(y))\,du\,d\mu$
$\displaystyle=\bar{\varphi}^{-1}\int_{Y}\int_{0}^{\varphi(F^{k}y)}1_{\\{0\leq
u-t+\varphi_{k}(y)<\varphi(y)\\}}v(y,u-t+\varphi_{k}(y))\,w(F^{k}y,u)\,du\,d\mu.$

Writing $v_{t,k}^{u}(y)=1_{\\{0\leq
u-t+\varphi_{k}(y)<\varphi(y)\\}}v(y,u-t+\varphi_{k}(y))$ and
$w^{u}(y)=w(y,u)$,

$\displaystyle\int_{Y^{\varphi}}$ $\displaystyle
L_{t}(1_{\\{N_{t}=k\\}}v)\,w\,d\mu^{\varphi}=\bar{\varphi}^{-1}\int_{0}^{\infty}\int_{Y}1_{\\{u<\varphi\circ
F^{k}\\}}v_{t,k}^{u}\,w^{u}\circ F^{k}\,d\mu\,du$
$\displaystyle=\bar{\varphi}^{-1}\int_{0}^{\infty}\int_{Y}1_{\\{u<\varphi\\}}R_{0}^{k}v_{t,k}^{u}\,w^{u}\,d\mu\,du=\int_{Y^{\varphi}}(R_{0}^{k}v_{t,k}^{u})(y)\,w(y,u)\,d\mu^{\varphi}.$

Hence,

$\displaystyle(L_{t}v)(y,u)=\sum_{k=0}^{[t/2]}(L_{t}(1_{\\{N_{t}=k\\}}v)(y,u)=\sum_{k=0}^{[t/2]}(R_{0}^{k}v_{t,k}^{u})(y).$

The result follows from the pointwise formula for $R_{0}^{k}$. ∎

Proof of Theorem 2.2 Let $m=m_{3}+1$. By Corollary 3.12,
$\|L_{t}v\|_{\delta,\eta}\ll e^{-\epsilon t}\|v\|_{\delta,\eta,m}$.

Recall that $\partial_{u}$ denotes the ordinary derivative with respect to $u$
at $0<u<\varphi(y)$ and denotes the appropriate one-sided derivative at $u=0$
and $u=\varphi(y)$. Since $v$ has good support, the indicator functions in the
right-hand side of the formula in Proposition 3.13 are constant on the support
of $v$. It follows that $\partial_{u}L_{t}v=L_{t}(\partial_{u}v)$. By
Corollary 3.12,

$\|\partial_{u}L_{t}v\|_{\delta,\eta}=\|L_{t}(\partial_{u}v)\|_{\delta,\eta}\ll
e^{-\epsilon t}\|\partial_{u}v\|_{\delta,\eta,m_{3}}\leq e^{-\epsilon
t}\|v\|_{\delta,\eta,m}.$

Hence, $\|L_{t}v\|_{\delta,\eta,1}\ll e^{-\epsilon t}\|v\|_{\delta,\eta,m}$ as
required. ∎

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