# Differentiability of the diffusion coefficient for a family of intermittent
maps

Fanni M. Sélley  Leiden University Mathematical Institute, Niels Bohrweg 1
2333 CA Leiden<EMAIL_ADDRESS>
###### Abstract.

It is well known that the Liverani–Saussol–Vaienti map satisfies a central
limit theorem for Hölder observables in the parameter regime where the
correlations are summable. We show that when $C^{2}$ observables are
considered, the variance of the limiting normal distribution is a $C^{1}$
function of the parameter. We first show this for the first return map to the
base of the second branch by studying the Green-Kubo formula, then conclude
the result for the original map using Kac’s lemma and relying on linear
response.

###### Key words and phrases:

Intermittent maps, central limit theorem, diffusion coefficient, linear
response

_ORCID of the author_ : https://orcid.org/my-orcid?orcid=0000-0002-8694-6715

##  1\. Introduction

We consider a one-parameter family $\\{f_{\alpha}\\}_{\alpha\in\mathcal{A}}$
of Pomeau–Manneville type maps introduced by Liverani, Saussol and Vaienti
[18]. Provided that $\alpha\in[0,1)$, each map preserves a unique probability
measure $\nu_{\alpha}$ and exhibits polynomial decay of correlations for
Hölder observables [18, 20]. Restricting $\alpha$ further to $[0,1/2)$, the
central limit theorem also holds: ergodic sums of centered observables,
normalized by $\sqrt{n}$, converge in distribution to a Gaussian random
variable with mean zero and variance $\sigma^{2}(\alpha)$ [20, 9]. In this
paper we study the smoothness of the mapping
$\alpha\mapsto\sigma^{2}(\alpha)$.

The question is motivated by the deterministic diffusion on the real line
defined by [16]. Diffusion can be characterized by the linear growth rate of
the mean square displacement of an ensemble of moving particles, called the
_diffusion coefficient_. A short calculation shows that the said system on the
real line is conjugated to a Pomeau–Manneville type map of the interval, and
the corresponding quantity of the diffusion coefficient is
$\sigma^{2}(\alpha)$ associated to a particular observable. So knowing the
regularity of $\sigma^{2}(\alpha)$ gives information about the parameter
dependence of the diffusion itself.

Continuity properties of the diffusion coefficient associated to different
types of dynamics were studied extensively, for an overview see the book [13]
and references therein. In [7] a diffusion on the real line was defined where
trajectories for most time are localized to boxes
$B_{N}=[N,N+1)\times[N,N+1)$, but at certain instances they can move to either
$B_{N-1}$ or $B_{N+1}$. The analytic expression of the map is such that the
system on the real line is conjugated to an interval map, which aides
calculations a great deal. Focusing on this simple setting, the piecewise
linear case was studied in [14, 15, 11] and low regularity was reported both
from the analytic and geometric measure theory perspective. In [11] the
setting was generalized to a family of maps exhibiting exponential decay of
correlations with uniform parameters, and log-Lipschitz continuity of the
diffusion coefficient was proved with methods relying mainly on the transfer
operator approach. The paper [16], providing our motivation, proposed to
consider a family of intermittent maps and study the so-called generalized
diffusion coefficient in the parameter regime of anomalous diffusion. In the
regime of normal diffusion [3] proved the continuity of the diffusion
coefficient for a wide variety of ($f_{\alpha}$-independent) potentials. In
fact, for the zero potential (measure of maximal entropy) they proved that the
diffusion coefficient as a function of $f_{\alpha}$ varies in a continuously
differentiable way.

In this paper we consider the regime of normal diffusion, i.e.
$\alpha\in[0,1/2)$ (and the SRB measure $\nu_{\alpha}$). Then the variance
$\sigma^{2}(\alpha)$ can be given as the sum of correlations by the Green–Kubo
formula, so the smoothness of correlations gives a good guess for the
smoothness of $\sigma^{2}(\alpha)$. The first (self-correlation) term in the
formula is $\int\psi^{2}d\nu_{\alpha}$. The smooth differentiability of the
mapping $\alpha\mapsto\int\psi^{2}d\nu_{\alpha}$ is the question of linear
response, which is well understood for the family of maps in question [4, 17].
The higher order terms $\int\psi\cdot\psi\circ T_{\alpha}^{k}\>d\nu_{\alpha}$
have an additional $\alpha$-dependence in the integrand $\psi\cdot\psi\circ
T_{\alpha}^{k}$, however since $\alpha\mapsto T_{\alpha}^{k}$ is smooth, the
smoothness of $\alpha\mapsto\int\psi\cdot\psi\circ
T_{\alpha}^{k}\>d\nu_{\alpha}$ follows. Continuity of $\sigma^{2}(\alpha)$
follows from the summability of correlations, but in order to show continuous
differentiability, one also has to show that the $\alpha$-derivatives of the
correlations are summable. Making these steps precise we show that smooth
differentiability also holds for the higher order terms of the Green–Kubo
formula, furthermore, $\alpha\mapsto\sigma^{2}(\alpha)\in
C^{1}[\alpha_{-},\alpha_{+}]$ for any $0<\alpha_{-}<\alpha_{+}<1/2$.

The idea of the proof is to define the usual first return map, prove smooth
differentiability for the corresponding variance $\tilde{\sigma}^{2}(\alpha)$
and conclude the same regularity for $\sigma^{2}(\alpha)$ by using Kac’s
formula. The calculations make use of the linear response result and technical
estimates of [17], but the issues sketched above require work that goes beyond
being a mere corollary of linear response.

We note that another important quantity characterizing diffusion is the _drift
coefficient_ (the expectation of the normal random variable given by the
central limit theorem in case of a non-centered observable $\varphi$). This
translates to the integral $\int\varphi\>d\nu_{\alpha}$, hence the smooth
differentiability as a function of $\alpha$ is covered by the linear response
result of [4, 17].

The structure of this paper is as follows: in Section 2 we introduce our
setting, main results and give a sketch of the proof. Section 3 contains the
proof of our main theorem. Section 4 is devoted to concluding remarks on
possible further directions of research.
Acknowledgements. I would like to express my gratitude to Wael Bahsoun for
providing me perspective and guidance for this work. I would also like to
thank Alexey Korepanov, Julien Sedro and Dalia Terhesiu helpful discussions.
Notational remark. Throughout these notes, we will denote the Lebesgue measure
on $[0,1]$ by $m$. Furthermore $C>0$ will denote a generic constant whose
value might change from one line to the next.

##  2\. Setting and main result

We consider the Liverani–Saussol–Vaienti map $f_{\alpha}:[0,1]\to[0,1]$
defined as

$f_{\alpha}(x)=\begin{cases}x(1+2^{\alpha}x^{\alpha})&\quad\text{for}\quad
0\leq x\leq 1/2\\\ 2x-1&\quad\text{for}\quad 1/2<x\leq 1.\end{cases}$ (1)

It is well known that this map preserves a unique (Lebesgue) absolutely
continuous measure $\nu_{\alpha}$, with density function $\rho_{\alpha}$ [18].

The following theorem is also well known, see for example [20, Theorem 6] or
[9, Theorem 4.1].

######  Theorem 2.1.

Let $0\leq\alpha<1/2$ and $\psi:[0,1]\to\mathbb{R}$ a $C^{2}$ function. Denote
$S_{n}\psi=\sum_{i=0}^{n-1}\psi\circ f_{\alpha}^{i}$ and
$\hat{\psi}_{\alpha}=\psi-\int\psi\rho_{\alpha}dm$. Then there exists
$\sigma\geq 0$ such that $\frac{S_{n}\hat{\psi}_{\alpha}}{\sqrt{n}}$ converges
in distribution to a random variable $\mathcal{N}(0,\sigma^{2})$.

Actually, [20, Theorem 6] considers $\psi$ that is Hölder continuous, while
[9, Theorem 4.1] is stated for $C^{1}$ observables. We state this weaker
version of both theorems for $C^{2}$ observables, as this is the setting that
we will use.

Obviously $\sigma^{2}=\sigma^{2}(\alpha)$. The goal of these notes is to study
the smoothness of the mapping $\alpha\mapsto\sigma^{2}(\alpha)$. Our main
statement is the following:

######  Theorem 2.2.

Let $0<\alpha_{-}<\alpha_{+}<1/2$. In the setting of Theorem 2.1,

$\alpha\mapsto\sigma^{2}(\alpha)\in C^{1}([\alpha_{-},\alpha_{+}]).$

In the rest of this section we give the outline of the proof.

Let $\tau_{\alpha}(x)=\min\\{n\geq 1:T_{\alpha}^{k}(x)\in(1/2,1]\\}$ be the
return time to the interval $(1/2,1]$. Define the induced map
$F_{\alpha}:(1/2,1]\to(1/2,1]$ as

$F_{\alpha}(x)=f_{\alpha}^{\tau_{\alpha}(x)}(x),$ (2)

and the induced observable

$\hat{\Psi}_{\alpha}(x)=\sum_{k=0}^{\tau_{\alpha}(x)-1}\hat{\psi}_{\alpha}(f_{\alpha}^{k}(x)).$

As a slight abuse of notation, we denote the ergodic sums of
$\hat{\Psi}_{\alpha}$ under $F_{\alpha}$ also by $S_{n}\hat{\Psi}_{\alpha}$,
that is,

$S_{n}\hat{\Psi}_{\alpha}=\sum^{n-1}_{i=0}\hat{\Psi}_{\alpha}\circ
F^{i}_{\alpha}.$

Since $F_{\alpha}$ is a mixing Gibbs–Markov map (see [1, Lemma 3.60]) with a
unique invariant measure $\mu_{\alpha}$ (see [2, Lemma 4.4.1]), and
$\hat{\Psi}_{\alpha}\in L^{2}([1/2,1])$, we can conclude that the induced map
also satisfies a Central Limit Theorem in the sense that
$\frac{S_{n}\hat{\Psi}_{\alpha}}{\sqrt{n}}$ converges in distribution to a
random variable $\mathcal{N}(0,\tilde{\sigma}^{2}(\alpha))$.

Using [9, Theorem 4.8] and Kac’s lemma, we show that

$\sigma^{2}(\alpha)=\frac{\tilde{\sigma}^{2}(\alpha)}{\int\tau_{\alpha}d\mu_{\alpha}},$
(3)

By [17] (see in particular the proof of [17, Theorem 1.1]), the map
$\alpha\mapsto\int\tau_{\alpha}d\mu_{\alpha}$ is continuously differentiable
(and trivially bounded from below), hence it is sufficient to study the
smoothness of $\alpha\mapsto\tilde{\sigma}^{2}(\alpha)$.

######  Remark 2.3.

By Kac’s lemma we also obtain that

$\int\hat{\Psi}_{\alpha}d\mu_{\alpha}=\frac{\int\hat{\psi}_{\alpha}d\nu_{\alpha}}{\int\tau_{\alpha}d\mu_{\alpha}}=0.$

We express $\tilde{\sigma}^{2}(\alpha)$ with the Green–Kubo formula:

$\tilde{\sigma}^{2}(\alpha)=\int\hat{\Psi}_{\alpha}^{2}\>d\mu_{\alpha}+2\sum_{k\geq
1}\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\>d\mu_{\alpha}.$ (4)

We first prove (in Proposition 3.5) that all correlations are individually
continuously differentiable functions of $\alpha$ in the sense that

$\begin{split}&\alpha\mapsto\int\hat{\Psi}_{\alpha}^{2}\>d\mu_{\alpha}\in
C^{1}([\alpha_{-},\alpha_{+}])\\\
&\alpha\mapsto\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\>d\mu_{\alpha}\in C^{1}([\alpha_{-},\alpha_{+}]),\>k\geq
1.\end{split}$ (5)

We then show (in Proposition 3.6) that the series

$\sum_{k\geq 1}\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\>d\mu_{\alpha}\quad\text{and}\quad\sum_{k\geq
1}\partial_{\alpha}\left(\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\>d\mu_{\alpha}\right)$ (6)

converge uniformly, from which we can conclude that

$\alpha\mapsto\tilde{\sigma}^{2}(\alpha)\in C^{1}([\alpha_{-},\alpha_{+}]),$

finishing the proof of Theorem 2.2.

In the next section we will make these steps precise.

##  3\. Proof

In the first part of this section we argue the central limit theorem for the
induced map and Equation (3) giving the connection between the variances
$\sigma^{2}$ and $\tilde{\sigma}^{2}$. These results are part of the general
folklore, but we give an argument for the sake of completeness as exact
references are hard to track down in the literature.

In the second part we study the Green–Kubo formula (4) in depth, proving (5)
and (6). The calculation makes use of the linear response result of [17], and
takes things a few steps further to obtain continuous differentiability of
$\tilde{\sigma}(\alpha)$.

###  3.1. Central limit theorem for the induced map.

Let $x_{0}=1$, set $x_{n+1}=f_{\alpha}^{-1}(x_{n})\cap[0,1/2]$ and
$y_{n+1}=f_{\alpha}^{-1}(x_{n})\cap(1/2,1]$. Let $I_{n}=(y_{n+1},y_{n})$ and
$J_{n}=(x_{n+1},x_{n})$. Define the partition $\pi_{I}$ as the collection of
intervals $\\{I_{n}\\}_{n\in\mathbb{N}}$, and similarly $\pi_{J}$ as
$\\{J_{n}\\}_{n\in\mathbb{N}}$. Note that for $z\in I_{n}$,
$\tau_{\alpha}(z)=n$ – meaning that $\pi_{I}$ partitions $(1/2,1]$ according
to their return time.

Let $s(x,y)$ be the separation time of $x,y\in(1/2,1]$ under $F_{\alpha}$:

$s(x,y)=\inf\\{n\in\mathbb{N}:\text{$F_{\alpha}^{n}(x)$ and
$F_{\alpha}^{n}(y)$ are in separate elements of $\pi_{I}$}\\}$

We extend this definition to $[0,1/2]$ as follows: let $x,y\in[0,1/2]$ and
denote by $x^{\prime},y^{\prime}$ their first returns to $(1/2,1]$. If
$f_{\alpha}^{i}(x)$ and $f_{\alpha}^{i}(y)$ stay in the same partition
elements of $\pi_{J}$ until their first return to $(1/2,1]$, set
$s(x,y)=s(x^{\prime},y^{\prime})+1$, otherwise $s(x,y)=0$.

Define $d_{\theta}(x,y)=\theta^{-s(x,y)}$ for some $\theta\in(1,2]$. Then
$((1/2,1],\mu_{\alpha},F_{\alpha})$ is also Gibbs–Markov for the metric
$d_{\theta}$. 111Indeed, expansion is evident since
$\theta^{-s(F_{\alpha}(x),F_{\alpha}(y))}=\theta^{-s(x,y)+1}=\theta\cdot\theta^{-s(x,y)}$
and we get the distortion estimate by using $|F_{\alpha}(x)-F_{\alpha}(y)|\leq
C\theta^{-s(F_{\alpha}(x),F_{\alpha}(y))}$.

We now study the induced observable. Let $g$ be a function from
$I_{n}\in\pi_{I}$ to $\mathbb{R}$. Define the Lipschitz semi-norm of $g$ as

$|g|_{Lip(I_{n})}=\inf\\{C>0:\forall x,y\in I_{n}\quad|g(x)-g(y)|\leq
C\theta^{-s(x,y)}\\}.$

######  Proposition 3.1.

$\hat{\Psi}_{\alpha}\in L^{2}$ and
$\sum_{r=1}^{\infty}\mu_{\alpha}(I_{r})|\hat{\Psi}_{\alpha}|_{Lip(I_{r})}<\infty$.

###### Proof.

We first show the induced observable is in $L^{2}$. Recall that

$\hat{\Psi}_{\alpha}(x)=\sum_{k=0}^{\tau_{\alpha}(x)-1}\hat{\psi}_{\alpha}(f_{\alpha}^{k}(x)).$

and $\hat{\psi}_{\alpha}\in C^{2}$. Write

$\displaystyle\hat{\Psi}_{\alpha}(x)$
$\displaystyle=\hat{\Psi}_{\alpha}(x)-\tau_{\alpha}(x)\hat{\psi}_{\alpha}(0)+\tau_{\alpha}(x)\hat{\psi}_{\alpha}(0)$
$\displaystyle=\tau_{\alpha}(x)\hat{\psi}_{\alpha}(0)+\sum_{k=0}^{\tau_{\alpha}(x)-1}\hat{\psi}_{\alpha}(f_{\alpha}^{k}(x))-\hat{\psi}_{\alpha}(0),$

We have $x_{n+1}-x_{n}\sim 1/n^{1+1/\alpha}$ (for instance by [1, Equation (3.
149)]), thus

$\displaystyle\int\tau_{\alpha}^{2}\asymp\sum_{k\geq
1}k^{2}\cdot(x_{k+1}-x_{k})\asymp\sum_{k\geq 1}k^{2}\cdot
1/k^{1+1/\alpha}<\infty$

for all $\alpha\in(0,1/2)$. For the other term, we first exploit that $\psi$
is Lipschitz continuous with constant $C$. This implies that
$\hat{\psi}_{\alpha}$ is Lipschitz with constant $C$, hence

$\displaystyle\sum_{k=0}^{\tau_{\alpha}(x)-1}|\hat{\psi}_{\alpha}(f_{\alpha}^{k}(x))-\hat{\psi}_{\alpha}(0)|\leq
C\sum_{k=0}^{\tau_{\alpha}(x)-1}|f_{\alpha}^{k}(x)|.$

For $x\in(y_{n+1},y_{n})$, we have $f_{\alpha}^{i}(x)\in(x_{n+1-i},x_{n-i})$,
thus $|f_{\alpha}^{i}(x)|\leq c(n-i)^{-1/\alpha}$. Then

$|\hat{\Psi}_{\alpha}(x)-\tau_{\alpha}(x)\cdot\hat{\psi}_{\alpha}(0)|\leq
c+c\sum_{j=1}^{n-1}(1/j)^{1/\alpha}\leq cn^{1-1/\alpha}$

and

$\int(\hat{\Psi}_{\alpha}-\tau_{\alpha}\cdot\hat{\psi}_{\alpha}(0))^{2}\asymp\sum_{k\geq
1}k^{2-2/\alpha}\cdot 1/k^{1+1/\alpha}=\sum_{k\geq 1}k^{1-3/\alpha},$

which is summable for all $\alpha\in(0,1)$. Thus $\hat{\Psi}_{\alpha}\in
L^{2}$.

We now prove the second statement. Since the induced map on $(1/2,1]$ is
uniformly expanding with expansion factor $\lambda=2$, it is clear that
$|x-y|<C\theta^{-s(x,y)}$. Using that $\hat{\psi}_{\alpha}$ is Lipschitz
continuous, we get $|\hat{\psi}_{\alpha}(x)-\hat{\psi}_{\alpha}(y)|\leq
C|x-y|\leq C\theta^{-s(x,y)}$.

By the definition of the separation time, we can see that for $x,y\in I_{r}$,

$\sum_{k=0}^{r-1}|\hat{\psi}_{\alpha}(f_{\alpha}^{k}(x))-\hat{\psi}_{\alpha}(f_{\alpha}^{k}(y))|\leq\sum_{k=0}^{r-1}C\theta^{-(f_{\alpha}^{k}(x),f_{\alpha}^{k}(y))}\leq
Cr\theta^{-s(x,y)}$

and thus $|\hat{\Psi}_{\alpha}|_{Lip(I_{r})}\leq Cr$. Furthermore, since
$d\mu_{\alpha}=h_{\alpha}dm$, where $h_{\alpha}\in L^{\infty}$, we have
$|h_{\alpha}|\leq C_{\alpha}$ and

$|\mu_{\alpha}(I_{r})|\leq C|m(I_{r})|\leq C(1/r)^{1+1/\alpha},$

similarly to previous computations. This gives

$\sum_{r\geq 1}\mu_{\alpha}(I_{r})|\hat{\Psi}_{\alpha}|_{Lip(I_{r})}\leq
C\sum_{r\geq 1}r^{-1/\alpha}<\infty.$

∎

The central limit theorem for the induced map (as stated below) follows by [5,
Theorem 1.4].

######  Theorem 3.2.

$\frac{S_{n}\hat{\Psi}_{\alpha}}{\sqrt{n}}\to\mathcal{N}(0,\tilde{\sigma}^{2})\quad\text{in
distribution},$

and

$\tilde{\sigma}^{2}=\int\hat{\Psi}_{\alpha}^{2}\>d\mu_{\alpha}+2\sum_{k\geq
1}\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\>d\mu_{\alpha}.$

Next we prove the relation between the variances corresponding to the original
and the induced map.

######  Proposition 3.3.

$\sigma^{2}(\alpha)=\frac{\tilde{\sigma}^{2}(\alpha)}{\int\tau_{\alpha}d\mu_{\alpha}}$

###### Proof.

Let $Y=(1/2,1]$. Recall that

$\frac{S_{n}\hat{\Psi}_{\alpha}}{\sqrt{n}}\to\mathcal{N}(0,\tilde{\sigma}^{2}(\alpha)).$

We first show that

$\frac{S_{n}\tau_{\alpha}-n/\nu_{\alpha}(Y)}{\sqrt{n}}\text{ is tight and
}\sup_{0\leq k\leq\sqrt{n}}\frac{|S_{k}\hat{\Psi}_{\alpha}|}{\sqrt{n}}\text{
tends to 0 in probability.}$ (7)

Indeed, consider the observable $g$ equal to $1-1/\nu_{\alpha}(Y)$ on $Y$ and
1 elsewhere (this observable is not $C^{2}$, but $C^{2}$ on [0,1/2] and
[1/2,1] which is sufficient). Then the corresponding induced observable is
$\tau_{\alpha}-1/\nu_{\alpha}(Y)$, and by Theorem 3.2,
$\frac{S_{n}\tau_{\alpha}-n/\nu_{\alpha}(Y)}{\sqrt{n}}$ converges to a
Gaussian random variable, hence the sequence
$\frac{S_{n}\tau_{\alpha}-n/\nu_{\alpha}(Y)}{\sqrt{n}}$ is tight. To show that
$\sup_{0\leq k\leq\sqrt{n}}\frac{|S_{k}\hat{\Psi}_{\alpha}|}{\sqrt{n}}$ tends
to 0 in probability, it suffices to show that
$\frac{S_{k}\hat{\Psi}_{\alpha}}{k}$ tends almost surely to 0. This follows
from the Birkhoff ergodic theorem, since $\hat{\Psi}_{\alpha}$ is integrable.

Thus by [9, Theorem 4.8], we have

$\frac{S_{n}\hat{\psi}_{\alpha}}{\sqrt{\lfloor
n\nu_{\alpha}(Y)\rfloor}}\to\mathcal{N}(0,\tilde{\sigma}^{2}(\alpha))$

By recalling that
$\frac{S_{n}\hat{\psi}_{\alpha}}{\sqrt{n}}\to\mathcal{N}(0,\sigma^{2}(\alpha))$,
we obtain that $\tilde{\sigma}^{2}(\alpha)\nu_{\alpha}(Y)=\sigma^{2}(\alpha)$.
By Kac’s formula, $\nu_{\alpha}(Y)=\frac{1}{\int\tau_{\alpha}d\mu_{\alpha}}$
giving the result.

∎

###  3.2. Analysis of the Green–Kubo formula.

In this section we will prove that

$\alpha\mapsto\int\hat{\Psi}_{\alpha}^{2}h_{\alpha}\>dm+2\sum_{k\geq
1}\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ F_{\alpha}^{k}\cdot
h_{\alpha}\>dm\in C^{1}([\alpha_{-},\alpha_{+}])$

for any $0<\alpha_{-}<\alpha_{+}<1/2$. As a first step we prove that the
correlations are continuously differentiable.

We first recall a key lemma from [17] which we will use on many occasions. To
state this lemma we introduce some notation. Denote by $g_{\alpha}$ the
inverse of the left branch of $f_{\alpha}$, and for $z\in[0,1]$ let
$z_{r}=g_{\alpha}^{r}(z)$. Furthermore, denote
$F_{\alpha}|_{I_{n}}=F_{\alpha,n}$ and
$G_{\alpha,n}=(F_{\alpha,n}^{-1})^{\prime}$. Finally, define

$\operatorname{logg}x=\begin{cases}1&\quad x\leq e\\\ \log x&\quad
x>e\end{cases}$

where $\log$ is the logarithm with base $e$.

######  Lemma 3.4.

For $n\in\mathbb{N}$,

  1. $(K0)$

$|z^{\prime}_{n}|\leq 1$ [17, Equation 5.5]

  2. $(K1)$

$|z^{\prime}_{n}|\leq Cn^{-(\alpha+1)/\alpha}$ [17, Lemma 5.3]

  3. $(K2)$

$|z^{\prime\prime}_{n}/z^{\prime}_{n}|\leq
C\Rightarrow\|G^{\prime}_{\alpha,n}/G_{\alpha,n}\|_{\infty}\leq C$ [17, Lemma
5.4]

  4. $(K3)$

$|z^{\prime\prime\prime}_{n}/z^{\prime}_{n}|\leq
C\Rightarrow\|G^{\prime\prime}_{\alpha,n}/G_{\alpha,n}\|_{\infty}\leq C$ [17,
Lemma 5.5]

  5. $(K4)$

$|\partial_{\alpha}z_{n}|\leq Cn^{-1/\alpha}(\operatorname{logg}n)^{2}$ [17,
Lemma 5.6]

  6. $(K5)$

$|\partial_{\alpha}z^{\prime}_{n}/z^{\prime}_{n}|\leq
C(\operatorname{logg}n)^{3}\Rightarrow\|\partial_{\alpha}G_{\alpha,n}/G_{\alpha,n}\|_{\infty}\leq
C(\operatorname{logg}n)^{3}$ [17, Lemma 5.7]

  7. $(K6)$

$\|\partial_{\alpha}F^{-1}_{\alpha,n}\|_{\infty}\leq
Cn^{-1/\alpha}(\operatorname{logg}n)^{2}\leq C(\operatorname{logg}n)^{2}$ [17,
Lemma 5.6 and Equation 5.1]

Recall that the transfer operator $P$ of a nonsingular map $T:I\to I$ is
defined as the left adjoint of the composition (Koopman operator), that is,

$\int_{I}\varphi\cdot P\psi\text{ d}m=\int_{I}\varphi\circ T\cdot\psi\text{
d}m,\qquad\varphi\in L^{\infty},\psi\in L^{1}.$

We will denote the transfer operator of $T_{\alpha}$ by $P_{\alpha}$.

######  Proposition 3.5.

$\alpha\mapsto\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\cdot h_{\alpha}\>dm\in C^{1}([\alpha_{-},\alpha_{+}])$

for all $k\geq 0$.

###### Proof.

Write

$\displaystyle\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\cdot h_{\alpha}\>dm$ $\displaystyle=\int
P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\cdot\hat{\Psi}_{\alpha}\>dm$
$\displaystyle=\int
P_{\alpha}\left(P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\cdot\hat{\Psi}_{\alpha}\right)\>dm$

For $\underline{n}=(n_{1},\dots,n_{k})$ let
$F_{\alpha,\underline{n}}=F_{\alpha,n_{1}}\circ\dots\circ F_{\alpha,n_{k}}$
and $G_{\alpha,\underline{n}}=(F_{\alpha,\underline{n}}^{-1})^{\prime}$. With
this notation we can write

$\displaystyle\int
P_{\alpha}\left(P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\cdot\hat{\Psi}_{\alpha}\right)\>dm$
$\displaystyle=\int\sum_{n_{k}=1}^{\infty}\dots\sum_{n_{0}=1}^{\infty}(h_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ
F_{\alpha,n_{0}}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ
F_{\alpha,n_{0}}^{-1})(\hat{\Psi}_{\alpha}\circ F_{\alpha,n_{0}}^{-1})$
$\displaystyle\qquad\times G_{\alpha,\underline{n}}\circ
F_{\alpha,n_{0}}^{-1}\cdot G_{\alpha,n_{0}}\>dm$ (8)

and

$\displaystyle\partial_{\alpha}\int
P_{\alpha}\left(P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\cdot\hat{\Psi}_{\alpha}\right)\>dm$
$\displaystyle=\int\sum_{n_{k}=1}^{\infty}\dots\sum_{n_{0}=1}^{\infty}\partial_{\alpha}\\{(h_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ
F_{\alpha,n_{0}}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ
F_{\alpha,n_{0}}^{-1})(\hat{\Psi}_{\alpha}\circ F_{\alpha,n_{0}}^{-1})$
$\displaystyle\qquad\times G_{\alpha,\underline{n}}\circ
F_{\alpha,n_{0}}^{-1}\cdot G_{\alpha,n_{0}}\\}\>dm$ (9)

It is clear that the summands in (8) are jointly continuous functions of
$\alpha$ and the spatial variable $x$, and this also holds for their partial
derivatives with respect to $\alpha$ appearing in (9). Indeed, continuity of
terms involving $h_{\alpha}$ follows by [17, Theorem 2.1]), and for the terms
involving $\hat{\Psi}_{\alpha}$ we rely on [17, Theorem 1.1]).

In the rest of the proof we show that the series

$\displaystyle\sum_{n_{k}=1}^{\infty}\dots\sum_{n_{0}=1}^{\infty}(h_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ
F_{\alpha,n_{0}}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ
F_{\alpha,n_{0}}^{-1})(\hat{\Psi}_{\alpha}\circ F_{\alpha,n_{0}}^{-1})$
$\displaystyle\hskip 59.75095pt\times G_{\alpha,\underline{n}}\circ
F_{\alpha,n_{0}}^{-1}\cdot G_{\alpha,n_{0}}$ (10)
$\displaystyle\sum_{n_{k}=1}^{\infty}\dots\sum_{n_{0}=1}^{\infty}\partial_{\alpha}\\{(h_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ
F_{\alpha,n_{0}}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ
F_{\alpha,n_{0}}^{-1})(\hat{\Psi}_{\alpha}\circ F_{\alpha,n_{0}}^{-1})$
$\displaystyle\hskip 59.75095pt\times G_{\alpha,\underline{n}}\circ
F_{\alpha,n_{0}}^{-1}\cdot G_{\alpha,n_{0}}\\}$ (11)

converge uniformly.

We first show that (10) converges uniformly. This will imply the continuity of
$\alpha\mapsto\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\cdot h_{\alpha}\>dm$. According to [17, Theorem 2.1],
$h_{\alpha}\in C^{2}([1/2,1])$, hence $\|h_{\alpha}\|_{\infty}\leq K$ where
$K$ does not depend on $\alpha$.

First compute that

$\displaystyle(\hat{\Psi}_{\alpha}\circ F_{\alpha,n_{0}}^{-1})(z)$
$\displaystyle=\hat{\psi}_{\alpha}\left(\frac{z_{n_{0}}+1}{2}\right)+\sum_{j=0}^{n_{0}-1}\hat{\psi}_{\alpha}(f_{\alpha}^{j}(z_{n_{0}}))$
$\displaystyle=\hat{\psi}_{\alpha}\left(\frac{z_{n_{0}}+1}{2}\right)+\sum_{j=1}^{n_{0}}\hat{\psi}_{\alpha}(z_{j})$

giving $\|\hat{\Psi}_{\alpha}\circ F_{\alpha,n_{0}}^{-1}\|_{\infty}\leq
C\|\hat{\psi}_{\alpha}\|_{\infty}(n_{0}+1)\leq 2C\|\psi\|_{\infty}(n_{0}+1)$.
The same argument gives also $\|\hat{\Psi}_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ F_{\alpha,n_{0}}^{-1}\|_{\infty}\leq
C(n_{0}+1)$.

We have
$\|G_{\alpha,n_{0}}\|_{\infty}=\frac{1}{2}\sup_{z}|z_{n_{0}}^{\prime}|\leq
Cn_{0}^{-\left(1+\frac{1}{\alpha}\right)}$ by (K1) from Lemma 3.4.
Furthermore,

$G_{\alpha,\underline{n}}=(F_{\alpha,n_{k}}^{-1}\circ\dots\circ
F_{\alpha,n_{1}}^{-1})^{\prime}=\Pi_{i=1}^{k}(F_{\alpha,n_{i}}^{-1})^{\prime}(F_{\alpha,n_{i-1}}^{-1}\circ\dots\circ
F_{\alpha,n_{1}}^{-1})$

hence
$\|G_{\alpha,\underline{n}}\|_{\infty}\leq\|G_{\alpha,n_{1}}\|_{\infty}\dots\|G_{\alpha,n_{k}}\|_{\infty}\leq
C(n_{1}\dots n_{k})^{-\left(1+\frac{1}{\alpha}\right)}$ (note that $C=C(k)$,
but this causes no issue for the present argument as we view $k$ fixed.)

Putting all this together, we can bound the $k$-fold sum in (8) by

$C\sum_{n_{k}=1}^{\infty}\dots\sum_{n_{0}=1}^{\infty}(n_{0}+1)^{2}(n_{0}n_{1}\dots
n_{k})^{-\left(1+\frac{1}{\alpha}\right)},$

which is finite if $\alpha\in(0,1/2)$.

To show that (11) converges uniformly, we first study the $\partial_{\alpha}$
partial derivatives of the summands in (8), starting with

$\displaystyle\partial_{\alpha}(h_{\alpha}\circ
F_{\alpha,(\underline{n},n_{0})}^{-1})=(\partial_{\alpha}h_{\alpha})\circ
F_{\alpha,(\underline{n},n_{0})}^{-1}+h^{\prime}_{\alpha}\circ
F_{\alpha,(\underline{n},n_{0})}^{-1}\partial_{\alpha}F_{\alpha,(\underline{n},n_{0})}^{-1}$

By [17, Theorem 2.1] $\partial_{\alpha}h_{\alpha}\in C^{1}([1/2,1])$ and
$\|\partial_{\alpha}h_{\alpha}\|_{\infty}\leq K$, for a constant $K>0$
independent of $\alpha$. Since

$\displaystyle\partial_{\alpha}(F_{\alpha,n_{k}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})$
$\displaystyle=\partial_{\alpha}(F_{\alpha,n_{k}}^{-1})\circ(F_{\alpha,n_{k-1}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})$ $\displaystyle+(F_{\alpha,n_{k}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})^{\prime}\partial_{\alpha}(F_{\alpha,n_{k-1}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})$
$\displaystyle=\partial_{\alpha}(F_{\alpha,n_{k}}^{-1})\circ(F_{\alpha,n_{k-1}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})$
$\displaystyle+G_{\alpha,(n_{0},\dots,n_{k})}\partial_{\alpha}(F_{\alpha,n_{k-1}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})$

we obtain by induction that

$\|\partial_{\alpha}(F_{\alpha,n_{k}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})\|_{\infty}\leq
C\sum_{j=0}^{k}\|\partial_{\alpha}F_{\alpha,n_{j}}^{-1}\|_{\infty}$ (12)

(where we repeatedly used the fact that
$\|G_{\alpha,\underline{n}}\|_{\infty}\leq C(k+1)$ when $|\underline{n}|\leq
k+1$.)

By (K6) from Lemma 3.4 we have
$\|\partial_{\alpha}F_{\alpha,n_{j}}^{-1}\|_{\infty}\leq
C(\operatorname{logg}n_{j})^{2}$ and hence

$\displaystyle\|\partial_{\alpha}(h_{\alpha}\circ
F_{\alpha,(\underline{n},n_{0})}^{-1})\|_{\infty}\leq
C\sum_{j=0}^{k}(\operatorname{logg}n_{j})^{2}$ (13)

Next we compute that

$\displaystyle\partial_{\alpha}(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n_{0}}^{-1})(z)=\psi^{\prime}\left(\frac{z_{n_{0}}+1}{2}\right)\partial_{\alpha}z_{n_{0}}+\sum_{j=1}^{n_{0}}\psi^{\prime}(z_{j})\partial_{\alpha}z_{n_{j}}-(n_{0}+1)\partial_{\alpha}\hat{\psi}_{\alpha}$

Since $|\partial_{\alpha}z_{n_{j}}|\leq C$ by (K4) from Lemma 3.4 and
$|\partial_{\alpha}\hat{\psi}_{\alpha}|\leq C$ by [17, Theorem 1.1] we obtain
the bound $\|\partial_{\alpha}(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n_{0}}^{-1})\|_{\infty}\leq C(\|\psi\|_{C^{1}}+1)(n_{0}+1)$ and by a
similar argument

$\|\partial_{\alpha}(\hat{\Psi}_{\alpha}\circ
F_{\alpha,(\underline{n},n_{0})}^{-1})\|_{\infty}\leq
C\|\psi\|_{C^{1}}(n_{0}+1)$ (14)

Next,

$G_{\alpha,\underline{n}}\circ
F_{\alpha,n_{0}}^{-1}=\Pi_{i=1}^{k}G_{\alpha,n_{i}}\circ
F_{\alpha,n_{i-1}}^{-1}\circ\dots\circ F_{\alpha,n_{0}}^{-1}$

and

$\displaystyle\partial_{\alpha}(G_{\alpha,\underline{n}}\circ
F_{\alpha,n_{0}}^{-1})=\sum_{j=1}^{k}$
$\displaystyle\partial_{\alpha}[G_{\alpha,n_{j}}\circ
F_{\alpha,n_{j-1}}^{-1}\circ\dots\circ F_{\alpha,n_{0}}^{-1}]$
$\displaystyle\times\Pi_{i\neq j}G_{\alpha,n_{i}}\circ
F_{\alpha,n_{i-1}}^{-1}\circ\dots\circ F_{\alpha,n_{0}}^{-1}$

We have

$\displaystyle\partial_{\alpha}(G_{\alpha,n_{j}}\circ
F_{\alpha,n_{j-1}}^{-1}\circ\dots\circ F_{\alpha,n_{0}}^{-1})$
$\displaystyle=\partial_{\alpha}(G_{\alpha,n_{j}})\circ
F_{\alpha,n_{j-1}}^{-1}\circ\dots\circ F_{\alpha,n_{0}}^{-1}$
$\displaystyle+(G_{\alpha,n_{j}}\circ F_{\alpha,n_{j-1}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})^{\prime}\cdot\partial_{\alpha}(F_{\alpha,n_{j-1}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})$
$\displaystyle=\partial_{\alpha}(G_{\alpha,n_{j}})\circ
F_{\alpha,n_{j-1}}^{-1}\circ\dots\circ F_{\alpha,n_{0}}^{-1}$
$\displaystyle+G_{\alpha,n_{j}}^{\prime}\circ
F_{\alpha,n_{j-1}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1}\cdot(F_{\alpha,n_{j-1}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})^{\prime}\cdot\partial_{\alpha}(F_{\alpha,n_{j-1}}^{-1}\circ\dots\circ
F_{\alpha,n_{0}}^{-1})$

By (K2) from Lemma 3.4 we have
$\|G^{\prime}_{\alpha,n_{j}}/G_{\alpha,n_{j}}\|_{\infty}\leq C$ and by (K5) we
see that $\|\partial_{\alpha}G_{\alpha,n_{j}}/G_{\alpha,n_{j}}\|_{\infty}\leq
C$. Using furthermore (12) we obtain

$\|\partial_{\alpha}(G_{\alpha,n_{j}}\circ
F_{\alpha,n_{j-1}}^{-1}\circ\dots\circ F_{\alpha,n_{0}}^{-1})\|_{\infty}\leq
Cn_{j}^{-\left(1+\frac{1}{\alpha}\right)}\left((\operatorname{logg}n_{j})^{3}+\sum_{m=0}^{j-1}(\operatorname{logg}n_{m})^{2}\right)$

and hence

$\|\partial_{\alpha}(G_{\alpha,\underline{n}}\circ
F_{\alpha,n_{0}}^{-1})\|_{\infty}\leq
C\Pi_{j=1}^{k}n_{j}^{-\left(1+\frac{1}{\alpha}\right)}\sum_{j=1}^{k}\left((\operatorname{logg}n_{j})^{3}+\sum_{m=0}^{j-1}(\operatorname{logg}n_{m})^{2}\right)$
(15)

Using the previously computed bounds on the supremums and the bounds (13),
(14) and (15) on the partial derivative, we obtain for (11)

$\displaystyle\sum_{n_{k}=1}^{\infty}\dots$
$\displaystyle\sum_{n_{0}=1}^{\infty}\partial_{\alpha}\\{(h_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ
F_{\alpha,n_{0}}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,\underline{n}}^{-1}\circ
F_{\alpha,n_{0}}^{-1})(\hat{\Psi}_{\alpha}\circ F_{\alpha,n_{0}}^{-1})$
$\displaystyle\hskip 22.76228pt\times G_{\alpha,\underline{n}}\circ
F_{\alpha,n_{0}}^{-1}\cdot G_{\alpha,n_{0}}\\}$ $\displaystyle\leq
C\sum_{n_{k}=1}^{\infty}\dots$
$\displaystyle\sum_{n_{0}=1}^{\infty}(n_{0}+1)^{2}(n_{0}\dots
n_{k})^{-\left(1+\frac{1}{\alpha}\right)}\Pi_{i=0}^{k}(\operatorname{logg}n_{i})^{2}$
$\displaystyle+2(n_{0}+1)^{2}(n_{0}\dots
n_{k})^{-\left(1+\frac{1}{\alpha}\right)}$
$\displaystyle+(n_{0}+1)^{2}(n_{0}\dots
n_{k})^{-\left(1+\frac{1}{\alpha}\right)}\Pi_{i=1}^{k}(\operatorname{logg}n_{i})^{3}\Pi_{i=0}^{k-1}(\operatorname{logg}n_{i})^{2(k-i)}$
$\displaystyle<\infty$

for $\alpha\in(0,1/2)$ and thus $\alpha\mapsto\partial_{\alpha}\int
P_{\alpha}\left(P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\cdot\hat{\Psi}_{\alpha}\right)$
is continuous. ∎

The next proposition claims that both the correlations and their partial
derivatives with respect to $\alpha$ are summable. Then it follows from
Propositions 3.5 and 3.6 that $\alpha\mapsto\tilde{\sigma}(\alpha)\in
C^{1}([\alpha_{-},\alpha_{+}])$ which concludes the proof of Theorem 2.2.

######  Proposition 3.6.

The series

$\sum_{k\geq 1}\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\cdot h_{\alpha}\>dm\quad\text{and}\quad\sum_{k\geq
1}\partial_{\alpha}\left(\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\cdot h_{\alpha}\>dm\right)$

converge uniformly.

###### Proof.

We first show that

$\sum_{k\geq 1}\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\cdot h_{\alpha}\>dm=\sum_{k\geq 1}\int
P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\cdot\hat{\Psi}_{\alpha}\>dm$

converges uniformly. Our first observation is the following:

######  Lemma 3.7.

$x\mapsto P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})(x)\in C^{2}$

###### Proof.

Since $P_{\alpha}:C^{2}\to C^{2}$, it is enough to prove the lemma for $k=1$.

$P_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})=\sum_{n=1}^{\infty}(h_{\alpha}\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ F_{\alpha,n}^{-1})G_{\alpha,n}$

where $h_{\alpha}\circ F_{\alpha,n}^{-1}$, $\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}$ and $G_{\alpha,n}$ are continuously differentiable in $x$,
and we have seen previously that $\|h_{\alpha}\|_{\infty}\leq K$,
$\|\hat{\Psi}_{\alpha}\circ F_{\alpha,n}^{-1}\|_{\infty}\leq C(n+1)$ and
$\|G_{\alpha,n}\|_{\infty}\leq Cn^{-\left(1+\frac{1}{\alpha}\right)}$. Thus

$\displaystyle\sum_{n=1}^{\infty}|(h_{\alpha}\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})G_{\alpha,n}|\leq
C\sum_{n=1}^{\infty}(n+1)n^{-\left(1+\frac{1}{\alpha}\right)}<\infty$ (16)

implying that the sum converges uniformly and $x\mapsto
P_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})(x)$ is continuous.

We now study

$\sum_{n=1}^{\infty}\partial_{x}[(h_{\alpha}\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ F_{\alpha,n}^{-1})G_{\alpha,n}]$

According to [17, Theorem 2.1] we in fact have $\|h_{\alpha}\|_{C^{2}}\leq K$
and

$\displaystyle(\hat{\Psi}_{\alpha}\circ F_{\alpha,n}^{-1})^{\prime}(z)$
$\displaystyle=\left(\hat{\psi}_{\alpha}\left(\frac{z_{n}+1}{2}\right)+\sum_{j=0}^{n-1}\hat{\psi}_{\alpha}(f_{\alpha}^{j}(z_{n}))\right)^{\prime}$
$\displaystyle=\psi^{\prime}\left(\frac{z_{n}+1}{2}\right)z_{n}^{\prime}+\sum_{j=1}^{n}\psi^{\prime}(z_{j})z_{j}^{\prime}$

thus $\|(\hat{\Psi}_{\alpha}\circ F_{\alpha,n}^{-1})^{\prime}\|_{\infty}\leq
C\|\psi\|_{C^{1}}(n+1)$ (using that $z_{j}^{\prime}\leq 1$ by (K0) of Lemma
3.4).

Finally, $\|G_{\alpha_{n}}^{\prime}/G_{\alpha,n}\|_{\infty}\leq C$ by (K2)
from Lemma 3.4. Thus

$\displaystyle\sum_{n=1}^{\infty}|\partial_{x}[(h_{\alpha}\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})G_{\alpha,n}]|\leq
C\sum_{n=1}^{\infty}n^{-\left(1+\frac{1}{\alpha}\right)}(n+1)<\infty$

implying that
$x\mapsto(P_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha}))^{\prime}(x)$ is
continuous. We do similar calculations to prove that
$x\mapsto(P_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha}))^{\prime\prime}(x)$ is
continuous. We compute

$\sum_{n=1}^{\infty}\partial^{2}_{x}[(h_{\alpha}\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ F_{\alpha,n}^{-1})G_{\alpha,n}]$

and for fixed $n$ we get nine terms that are continuous individually. We can
easily compute that $\|(h_{\alpha}\circ
F_{\alpha,n}^{-1})^{\prime\prime}\|_{\infty}\leq C$,
$\|(\hat{\Psi}_{\alpha}\circ F_{\alpha,n}^{-1})^{\prime\prime}\|_{\infty}\leq
C\|\psi\|_{C^{2}}(n+1)$ (using that $|z_{r}^{\prime\prime}|\leq C$ by (K2)
from Lemma 3.4), and
$\|G_{\alpha,n}^{\prime\prime}/G_{\alpha,n}\|_{\infty}\leq C$ by (K3) from
Lemma 3.4. By using bounds computed previously, we obtain that

$\displaystyle\sum_{n=1}^{\infty}|\partial^{2}_{x}[(h_{\alpha}\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})G_{\alpha,n}]|\leq
C\sum_{n=1}^{\infty}(n+1)n^{-\left(1+\frac{1}{\alpha}\right)}<\infty,$

implying that the sum converges uniformly. ∎

By [17, Corollary 4.8]

$\|P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\|_{C^{i}}\leq
C(1-\theta)^{k}\|P_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})\|_{C^{i}}\qquad
i=1,2$ (17)

for some $\theta\in(0,1)$ and $C>0$. We obtain
$\|P_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})\|_{C^{1}}\leq C$ by the
computation of Lemma 3.7 and $\|\hat{\Psi}_{\alpha}\|_{1}\leq C$ by a
computation very similar to that in Proposition 3.1. Hence

$\displaystyle\sum_{k\geq 2}\left|\int
P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\cdot\hat{\Psi}_{\alpha}\>dm\right|$
$\displaystyle\leq
C\|P_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})\|_{C^{1}}\|\hat{\Psi}_{\alpha}\|_{L^{1}}\sum_{k\geq
2}(1-\theta)^{k}$ $\displaystyle\leq C\sum_{k\geq 2}(1-\theta)^{k},$

which proves the first statement of the proposition.

We now study

$\displaystyle\sum_{k\geq
1}\partial_{\alpha}\left(\int\hat{\Psi}_{\alpha}\cdot\hat{\Psi}_{\alpha}\circ
F_{\alpha}^{k}\cdot h_{\alpha}\>dm\right)=\sum_{k\geq
1}\int\partial_{\alpha}\left(P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\hat{\Psi}_{\alpha}\>dm\right).$

Write

$\displaystyle\int\partial_{\alpha}\left(P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\hat{\Psi}_{\alpha}\>dm\right)$
$\displaystyle=\int
P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\partial_{\alpha}\hat{\Psi}_{\alpha}\>dm+\int\partial_{\alpha}[P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})]\hat{\Psi}_{\alpha}\>dm$
$\displaystyle=(I)+(II)$

First,

$(I)=\int
P_{\alpha}(P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\partial_{\alpha}\hat{\Psi}_{\alpha})\>dm=:\int
P_{\alpha}(p_{\alpha}\partial_{\alpha}\hat{\Psi}_{\alpha})\>dm,$

where by Equation (17) we have $\|p_{\alpha}\|_{C^{1}}\leq C(1-\theta)^{k}$.
Write

$\int
P_{\alpha}(p_{\alpha}\partial_{\alpha}\hat{\Psi}_{\alpha})\>dm=\int\sum_{n\geq
1}p_{\alpha}\circ
F_{\alpha,n}^{-1}\cdot\partial_{\alpha}\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}\cdot G_{\alpha,n}\>dm$

Now

$\displaystyle\partial_{\alpha}\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}=\partial_{\alpha}(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})-\hat{\Psi}_{\alpha}^{\prime}\circ
F_{\alpha,n}^{-1}\cdot\partial_{\alpha}F_{\alpha,n}^{-1}$
$\displaystyle=\partial_{\alpha}(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})-(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})^{\prime}(G_{\alpha,n})^{-1}\cdot\partial_{\alpha}F_{\alpha,n}^{-1}$

thus

$\|(\partial_{\alpha}\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})G_{\alpha,n}\|_{\infty}\leq
C(n+1)n^{-\frac{1}{\alpha}}(\operatorname{logg}n)^{2}$

by (14) and (K6) from Lemma 3.4, thus

$\sum_{n\geq 1}|p_{\alpha}\circ
F_{\alpha,n}^{-1}\cdot\partial_{\alpha}\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}\cdot G_{\alpha,n}|\leq C(1-\theta)^{k}\sum_{n\geq
1}C(n+1)(\operatorname{logg}n)^{2}n^{-\frac{1}{\alpha}}<\infty$

for $\alpha<1/2$, and

$(I)\leq C_{1}(1-\theta)^{k}.$ (18)

Next,

$\displaystyle(II)$ $\displaystyle=\int
P_{\alpha}^{k}(\partial_{\alpha}[\hat{\Psi}_{\alpha}h_{\alpha}])\hat{\Psi}_{\alpha}\>dm+\int\partial_{\alpha}[P_{\alpha}^{k}](\hat{\Psi}_{\alpha}h_{\alpha})\hat{\Psi}_{\alpha}\>dm$
$\displaystyle=(IIa)+(IIb)$

With $(IIa)$ we can do the same argument as with $\int
P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\hat{\Psi}_{\alpha}\>dm$,
provided that $x\mapsto
P_{\alpha}^{k}(\partial_{\alpha}\hat{\Psi}_{\alpha}h_{\alpha}+\hat{\Psi}_{\alpha}\partial_{\alpha}h_{\alpha})(x)\in
C^{1}$.

######  Lemma 3.8.

$x\mapsto
P_{\alpha}^{k}(\partial_{\alpha}\hat{\Psi}_{\alpha}h_{\alpha}+\hat{\Psi}_{\alpha}\partial_{\alpha}h_{\alpha})(x)\in
C^{1}$

###### Proof.

It is enough again to prove the statement for $k=1$. We write

$\displaystyle P_{\alpha}(\partial_{\alpha}[\hat{\Psi}_{\alpha}h_{\alpha}])$
$\displaystyle=\sum_{n=1}^{\infty}(\partial_{\alpha}h_{\alpha}\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})G_{\alpha,n}+(h_{\alpha}\circ
F_{\alpha,n}^{-1})(\partial_{\alpha}\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})G_{\alpha,n}$

All terms are continuous in $x$, and the sum uniformly converges since

$\displaystyle\sum_{n=1}^{\infty}|(\partial_{\alpha}h_{\alpha}\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})G_{\alpha,n}+(h_{\alpha}\circ
F_{\alpha,n}^{-1})(\partial_{\alpha}\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})G_{\alpha,n}|$
$\displaystyle\leq\sum_{n=1}^{\infty}(n+1)(\operatorname{logg}n)^{2}n^{-\frac{1}{\alpha}}<\infty,$

As for the continuity of
$x\mapsto(P^{k}_{\alpha}(\partial_{\alpha}\hat{\Psi}_{\alpha}h_{\alpha}+\hat{\Psi}_{\alpha}\partial_{\alpha}h_{\alpha}))^{\prime}(x)$,
we write

$\displaystyle(P_{\alpha}^{k}(\partial_{\alpha}[\hat{\Psi}_{\alpha}h_{\alpha}]))^{\prime}$
$\displaystyle=\sum_{n=1}^{\infty}[((\partial_{\alpha}h_{\alpha})^{\prime}\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})+(h_{\alpha}^{\prime}\circ
F_{\alpha,n}^{-1})(\partial_{\alpha}\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})]G_{\alpha,n}^{2}$
$\displaystyle+[(\partial_{\alpha}h_{\alpha}\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})^{\prime}+(h_{\alpha}\circ
F_{\alpha,n}^{-1})(\partial_{\alpha}\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})^{\prime}]G_{\alpha,n}$
$\displaystyle+[((\partial_{\alpha}h_{\alpha})\circ
F_{\alpha,n}^{-1})(\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})+(h_{\alpha}\circ
F_{\alpha,n}^{-1})(\partial_{\alpha}\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})]G^{\prime}_{\alpha,n}$ (19)

We compute that

$\displaystyle(\partial_{\alpha}[\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}])^{\prime}$
$\displaystyle=\psi^{\prime}\left(\frac{z_{n}+1}{2}\right)z_{n}^{\prime}\partial_{\alpha}z_{n}+\psi\left(\frac{z_{n}+1}{2}\right)\partial_{\alpha}z_{n}^{\prime}$
$\displaystyle+\sum_{i=1}^{n-1}\psi^{\prime}\left(z_{i}\right)z_{i}^{\prime}\partial_{\alpha}z_{i}+\psi\left(z_{i}\right)\partial_{\alpha}z_{i}^{\prime},$

and since $|z_{i}^{\prime}|,|\partial_{\alpha}z_{i}^{\prime}|\leq C$ and
$|\partial_{\alpha}z_{i}^{\prime}|\leq C(\operatorname{logg}i)^{3}$ we have
$\|(\partial_{\alpha}[\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}])^{\prime}\|_{\infty}\leq C(n+1)(\operatorname{logg}n)^{3}$
and thus

$\displaystyle\|(\partial_{\alpha}\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1})^{\prime}G_{\alpha,n}\|_{\infty}\leq\|(\partial_{\alpha}[\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}])^{\prime}\|_{\infty}\|G_{\alpha,n}\|_{\infty}$
$\displaystyle+\|(\hat{\Psi}_{\alpha}^{\prime}\circ
F_{\alpha,n}^{-1})^{\prime}G_{\alpha,n}\|_{\infty}\|\partial_{\alpha}F_{\alpha,n}^{-1}\|_{\infty}+\|(\hat{\Psi}_{\alpha}^{\prime}\circ
F_{\alpha,n}^{-1})G_{\alpha,n}\|_{\infty}\|\partial_{\alpha}(F_{\alpha,n}^{-1})^{\prime}\|_{\infty}$
$\displaystyle\leq
C(n+1)\left[(\operatorname{logg}n)^{3}n^{-\left(1+\frac{1}{\alpha}\right)}+(\operatorname{logg}n)^{2}n^{-\frac{1}{\alpha}}+(\operatorname{logg}n)^{5}n^{-\left(1+\frac{1}{\alpha}\right)}\right]$

thus

$\eqref{eq:dersum}\leq
C\sum_{n=1}^{\infty}n^{-\left(1+\frac{1}{\alpha}\right)}(n+1)(\operatorname{logg}n)^{5}n^{-\frac{1}{\alpha}}<\infty.$

∎

Returning to $(IIa)$, we write

$(IIa)=\int
P_{\alpha}(P_{\alpha}^{k}(\partial_{\alpha}[\hat{\Psi}_{\alpha}h_{\alpha}])\hat{\Psi}_{\alpha})\>dm=:\int
P_{\alpha}(q_{\alpha}\hat{\Psi}_{\alpha})\>dm,$

and we have

$\displaystyle\left|\int P_{\alpha}(q_{\alpha}\hat{\Psi}_{\alpha})dm\right|$
$\displaystyle\leq\sum_{n\geq 1}|q_{\alpha}\circ
F_{\alpha,n}^{-1}\cdot\hat{\Psi}_{\alpha}\circ F_{\alpha,n}^{-1}\cdot
G_{\alpha,n}|$ $\displaystyle\leq C(1-\theta)^{k}\sum_{n\geq
1}C(n+1)n^{-\left(1+\frac{1}{\alpha}\right)}$

giving

$(IIa)\leq C_{2}(1-\theta)^{k}.$ (20)

Finally, we study $(IIb)$. Write

$\partial_{\alpha}[P_{\alpha}^{k}]=\lim_{\alpha\to\alpha^{\prime}}\frac{1}{\alpha-\alpha^{\prime}}\sum_{i=1}^{k}P_{\alpha}^{k-i}(P_{\alpha}-P_{\alpha^{\prime}})P_{\alpha^{\prime}}^{i-1}=:\sum_{i=1}^{k}P_{\alpha}^{k-i}Q_{\alpha}P_{\alpha}^{i-1},$

using similar notation to [17]. Hence

$\displaystyle\int\partial_{\alpha}[P_{\alpha}^{k}](\hat{\Psi}_{\alpha}h_{\alpha})\hat{\Psi}_{\alpha}\>dm=\int\sum_{i=1}^{k}P_{\alpha}^{k-i}Q_{\alpha}P_{\alpha}^{i-1}(\hat{\Psi}_{\alpha}h_{\alpha})\hat{\Psi}_{\alpha}\>dm$

We first look at the $i\geq 2$ terms of the sum. Then

$\displaystyle\sum_{i=2}^{k}|P_{\alpha}^{k-i}Q_{\alpha}P_{\alpha}^{i-1}(\hat{\Psi}_{\alpha}h_{\alpha})\hat{\Psi}_{\alpha}|$
$\displaystyle\leq
Ck(1-\theta)^{k-i}\|Q_{\alpha}P_{\alpha}^{i-1}(\hat{\Psi}_{\alpha}h_{\alpha})\|_{C^{1}}|\hat{\Psi}_{\alpha}|$
$\displaystyle\leq
Ck(1-\theta)^{k-i}\|P_{\alpha}^{i-1}(\hat{\Psi}_{\alpha}h_{\alpha})\|_{C^{2}}|\hat{\Psi}_{\alpha}|$
$\displaystyle\leq
Ck(1-\theta)^{k-2}\|P_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})\|_{C^{2}}|\hat{\Psi}_{\alpha}|$

using [17, Lemma 4.1] in the second step, and (17) in the first and third
step. We obtain $\|P_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})\|_{C^{2}}\leq C$
by the computation of Lemma 3.7 and $\|\hat{\Psi}_{\alpha}\|_{1}\leq C$ by a
computation very similar to that in Proposition 3.1. Thus

$\left|\int\sum_{i=2}^{k}P_{\alpha}^{k-i}Q_{\alpha}P_{\alpha}^{i-1}(\hat{\Psi}_{\alpha}h_{\alpha})\hat{\Psi}_{\alpha}\>dm\right|\leq
Ck(1-\theta)^{k-2}.$

As for the $i=1$ term, we show that
$Q_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})\in C^{1}$. Indeed,

$\displaystyle Q_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})=\sum_{n\geq 1}$
$\displaystyle\partial_{\alpha}[\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}\cdot h_{\alpha}\circ F_{\alpha,n}^{-1}\cdot G_{\alpha,n}]$
$\displaystyle=\sum_{n\geq 1}$
$\displaystyle\partial_{\alpha}[\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}]\cdot h_{\alpha}\circ F_{\alpha,n}^{-1}\cdot G_{\alpha,n}$
$\displaystyle+\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}\cdot\partial_{\alpha}[h_{\alpha}\circ
F_{\alpha,n}^{-1}]\cdot G_{\alpha,n}$ $\displaystyle+\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}\cdot h_{\alpha}\circ
F_{\alpha,n}^{-1}\cdot\partial_{\alpha}G_{\alpha,n}$

Each term is continuous in $x$, and by previously computed bounds we can upper
bound the sum by

$C\sum_{n\geq
1}(n+1)(1+(\operatorname{logg}n)^{2}+(\operatorname{logg}n)^{3})n^{-\left(1+\frac{1}{\alpha}\right)}<\infty,$

hence it converges uniformly. By similar argument we also get that
$(Q_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha}))^{\prime}$ is continuous: we
compute

$Q_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})=\sum_{n\geq
1}\partial_{x}\partial_{\alpha}[\hat{\Psi}_{\alpha}\circ
F_{\alpha,n}^{-1}\cdot h_{\alpha}\circ F_{\alpha,n}^{-1}\cdot G_{\alpha,n}],$

and we see that for fixed $n$ each of the nine terms are continuous in $x$. By
using previously computed bounds, we again upper bound the sum by

$C\sum_{n\geq
1}(n+1)(\operatorname{logg}n)^{3}n^{-\left(1+\frac{1}{\alpha}\right)}<\infty,$

proving that it converges uniformly.

Then by (17) we have

$\|P_{\alpha}^{k-1}Q_{\alpha}(\hat{\Psi}_{\alpha}h_{\alpha})\|_{C^{1}}\leq
C(1-\theta)^{k-1},$

thus

$(IIb)\leq\begin{cases}C_{3}&\text{ if }k=1\\\ C_{3}k(1-\theta)^{k-2}&\text{
if }k\geq 2\end{cases}$ (21)

We can conclude the proof of the proposition by combining (18), (20) and (21)
to obtain

$\sum_{k\geq
1}\int\partial_{\alpha}\left(P_{\alpha}^{k}(\hat{\Psi}_{\alpha}h_{\alpha})\hat{\Psi}_{\alpha}\>dm\right)\leq
C+C\sum_{k\geq 2}k(1-\theta)^{k-2}<\infty.$

∎

##  4\. Concluding remarks

A number of further questions would be interesting to study in the future. The
most straightforward one would be considering an observable for which
$\psi(0)=0$ and $\alpha\in(1/2,1)$. In this case the central limit theorem
holds [8] and according to a 2002 announcement by Hu, correlations are
summable – making the Green–Kubo formula well-defined. However, a proof of
this statement is not possible to track down in the literature. Provided that
this in fact holds, we can expect that $\alpha\mapsto\sigma^{2}(\alpha)\in
C^{1}[\alpha_{-},\alpha_{+}]$ for any $1/2<\alpha_{-}<\alpha_{+}<1$. However,
the calculations in these notes made use of $\alpha<1/2$ in several places so
the generalization is not completely straightforward.

For a general observable $\psi$, we have $(S_{n}\psi)/n^{\alpha}$ converging
in distribution to a random variable with a stable law of index $1/\alpha$
[8]. In this case the first task would be to give a proper definition for the
diffusion coefficient. Venturing to the regime of $\alpha\geq 1$, $f_{\alpha}$
preserves a $\sigma$-finite measure, and the corresponding anomalous diffusion
calls for the definition of a generalized diffusion coefficient, possibly
along the lines of [16]. It would be an intriguing task to check rigorously
the discontinuities and fractal properties of the diffusion coefficient
uncovered by numerics in [16].

Returning to the setting of the current paper, another interesting question
would be to study further regularity of the drift- and diffusion coefficient
as a function of $\alpha$. $C^{2}$-smoothness of the drift coefficient is
sometimes called _quadratic response_ in the literature and has essentially
only been studied in the uniformly expanding setting [10]. The first task
would be to clear quadratic response for the LSV map, then one could move on
to study higher order regularity of the diffusion coefficient.

$0$$1$$H$ Figure 1. LSV-type map with a non-full branch

Another possible direction of generalization would be to consider LSV-type
maps with a non-full branch, such as the one displayed on Figure 1. In this
setting one could study the regularity of the induced map’s diffusion
coefficient by the methods developed in [11] to obtain a log-Lipschitz modulus
of continuity, namely

$|\tilde{\sigma}^{2}(\alpha)-\tilde{\sigma}^{2}(\alpha^{\prime})|\leq
C|\alpha-\alpha^{\prime}|(1+|\log|\alpha-\alpha^{\prime}||)^{2},\quad\alpha,\alpha^{\prime}\in[\alpha_{-},\alpha_{+}]$

which would be inherited by $\sigma^{2}(\alpha)$, provided that
$\alpha\mapsto\int\tau_{\alpha}d\mu_{\alpha}$ has the same type of (or better)
regularity. For this, the regularity of $\alpha\mapsto\tau_{\alpha}$ and
$\alpha\mapsto h_{\alpha}$ has to be studied. According to [12],
$|h_{\alpha}-h_{\alpha^{\prime}}|_{L^{1}}\leq
C|\alpha-\alpha^{\prime}||\log|\alpha-\alpha^{\prime}||$ (the infinite number
of branches does not cause a difficulty in the proof), but the regularity of
$\alpha\mapsto\tau_{\alpha}$ remains to be studied.

It should be possible to study the diffusion coefficient as a function of both
the parameter $\alpha$ and the height $H$ of the second branch, and obtain

$|\tilde{\sigma}^{2}(\alpha,H)-\tilde{\sigma}^{2}(\alpha^{\prime},H^{\prime})|\leq
C(|\alpha-\alpha^{\prime}|+|H-H^{\prime}|)(1+|\log(|\alpha-\alpha^{\prime}|+|H-H^{\prime}|)|)^{2}$

for all
$(\alpha,H),(\alpha^{\prime},H^{\prime})\in[\alpha_{-},\alpha_{+}]\times[H_{-},H_{+}]$
where $0<\alpha_{-}<\alpha_{+}<1/2$ and $1/2<H_{-}<H_{+}<1$ (so that the
second branch is expanding.)

## Statements and declarations

### Funding and competing interests

The author has no relevant financial or non-financial interests to disclose.

### Data availability statement

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