text:
\section{Introduction}

Let $\alpha \in [0,1]$. Let us define the map $T_{\alpha} : [\alpha-1, \alpha] \rightarrow [\alpha-1, \alpha] $ as $T_\alpha(0) = 0$ and
\begin{equation} \label{definition}
T_{\alpha}(x) = \frac{1}{|x|} - a_{\alpha}(x)
\end{equation}
with $a_{\alpha}(x) := \left\lfloor \frac{1}{|x|} + 1 - \alpha \right\rfloor$.
These systems were introduced by Nakada \cite{Nakada81} and are known in the literature as \emph{$\alpha$-continued fractions}, or
 \emph{Japanese continued fractions}.
By taking $x_{n, \alpha} = T_{\alpha}^n(x)$, $a_{n, \alpha} = a_{\alpha}(x_{{n-1}, \alpha})$, $\epsilon_{n, \alpha} = \textup{Sign}(x_{n-1, \alpha})$,
the orbit under $T_\alpha$ generates the generalized continued fraction expansion
$$ x = a_{0, \alpha}  + \frac{\epsilon_{1, \alpha}}{a_{1,\alpha} + \frac{\epsilon_{2,\alpha}}{a_{2, \alpha} +_{\ddots}}} $$

The algorithm, analogously to the Gauss map in the classical case, provides rational approximations of real numbers.
It is known that for each $\alpha \in (0,1]$ there exists a unique invariant measure $\mu_{\alpha}(dx) = \rho_{\alpha}(x)dx$ absolutely
continuous w.r.t. Lebesgue measure, and this measure is ergodic (see \cite{LuzziMarmi}). In this paper we will focus on the metric entropy
of the $T_{\alpha}$'s, which is given by Rohlin's formula (see \cite{Rohlin})

\begin{equation} \label{Rohlin}
h(T_{\alpha}) = \int_{\alpha-1}^{\alpha} \log |T_\alpha'| d\mu_\alpha
\end{equation}

Nakada \cite{Nakada81} computed exact values of $h(T_\alpha)$ for $\alpha \geq \frac{1}{2}$, showing
that in this interval $h(T_\alpha)$ is continuous, and smooth except for the point $\alpha = \frac{\sqrt{5}-1}{2}$, where the left and right
 derivatives do not coincide.

In \cite{LuzziMarmi}, Luzzi and Marmi studied the behaviour of $h(T_\alpha)$ as a function of $\alpha$ for all parameters $\alpha \in (0,1)$.
 They gave numerical evidence that this function is continuous but not smooth. Nakada and Natsui
\cite{NakadaNatsui} then proved that this function is indeed not monotone, by giving explicit construction of infinitely many monotonicity
intervals. An extensive numerical study of these intervals has been carried out in \cite{CMPT}, and a complete characterization of
all monotonicity intervals is given in \cite{CT}. Even though the entropy is conjecturally smooth on any such interval, there are
points at which it is not even locally monotone, and the set of bifurcation parameters has a complicated
self-similar structure.

The first major result of this paper is the

\begin{theorem} \label{Holdermain}
The entropy function $\alpha \mapsto h(T_\alpha)$ is H\"older-continuous of any exponent $0 < s < \frac{1}{2}$.
\end{theorem}

The proof follows from spectral analysis of the transfer operator acting on the space of functions of bounded variation.
First (section \ref{LY}), we prove a uniform bound on the essential spectral radius (Lasota-Yorke inequality).
 Then (section \ref{Kellerdistsect}), we prove that a suitable distance between the transformations $T_\alpha$ is H\"older-continuous
in $\alpha$, and use a stability result of the spectral decomposition \cite{KellerLiverani} to prove H\"older-continuity of the invariant
 densities $\rho_\alpha$ in $L^1$-norm. Note that invariant densities are not continuous in $BV$-norm (see remark \ref{rmk}).

\medskip
The second part of the paper deals with central limit theorems.
In \cite{LuzziMarmi}, entropy is computed by approximating it with Birkhoff averages for the observable $\log |T_\alpha'|$, and
 numerical evidence is given (\cite{LuzziMarmi}, figure 3) that Birkhoff sums for different orbits distribute normally around the average.
We first show (section \ref{CLTBVsect}) that the methods of \cite{Broise} can be used to prove a central limit theorem
for observables of bounded variation.

Moreover, in section \ref{CLTunbsect} we expand the class of observables we use in order to encompass unbounded observables such as the logarithm.
Indeed, we will define a new family of Banach spaces $B_{K, \delta}$, consisting of possibly unbounded functions
whose total variation grows slowly on intervals which approach zero. Such functions will be called \emph{of mild growth},
 and we prove the central limit theorem to hold in these larger spaces:

\begin{theorem} \label{CLTunbounded}
Let $\alpha \in (0,1]$, $0 < \delta < \frac{1}{2}$, and $K$ sufficiently large. Then, for every non-constant real-valued $f \in B_{K, \delta}$ there exists $\sigma > 0$ s.t.
$$\lim_{n \rightarrow \infty} \mu_{\alpha}\left( \frac{S_n(f - \int_{I_\alpha} f d\mu_\alpha)}{\sqrt{n}} \leq v \right) = \frac{1}{\sigma \sqrt{2\pi}}\int_v^{+\infty} e^{-\frac{t^2}{2\sigma^2}} dt \qquad \forall v \in \mathbb{R}$$
\end{theorem}

As a corollary, Birkhoff sums for the observable $\log |T_\alpha'|$ distribute normally around the average value $h(T_\alpha)$.

Finally, in section \ref{last} we discuss the dependence of the standard deviation of Birkhoff averages on the parameter $\alpha$.
More precisely, given some observable $f$ of class $C^1$, for which we proved the central limit theorem to hold, we will
prove that the variance $\sigma^2_{\alpha, f}$ of the limit Gaussian distribution is continuous in $\alpha$.
The result is motivated by numerical data in (\cite{CMPT}, section 2.3).

\medskip
Many different authors have studied the spectral
properties of transfer operators of expanding maps. For instance,
a spectral decomposition for individual expanding maps
is proved in \cite{Broise},  \cite{Rychlik} and \cite{Viana}.
A brief historical account with references is given in \cite{KellerLiverani}.
In our case, however, it is essential to prove estimates on the spectral radius which are uniform in $\alpha$.
Since new branches of $T_\alpha$ appear as $\alpha$ moves, and $T_\alpha$ develops an indifferent fixed point
as $\alpha \to 0$, proving uniformity requires more work.
Unfortunately, although a uniform Lasota-Yorke inequality holds, the
proof provided by \cite{LuzziMarmi} contains a bug; we shall therefore
produce a new proof in prop. \ref{LY}.
Another proof of continuity (not H\"older) of entropy is given in the very recent paper \cite{KSS}
 via a study of natural extensions.

Let us finally remark that our functional-analytic methods only use a few properties of $T_\alpha$, hence they
can be applied to a wider class of one-parameter families of expanding interval maps.
For instance, they apply to the case of $(a,b)$-continued fraction transformations studied in \cite{KU}
for parameters on the critical line $b-a = 1$.

\section{Basic properties}

Let us start by setting up the framework needed for the rest of the paper, and establishing a basic spectral decomposition
for the transfer operator. The literature on thermodynamic formalism for interval maps is huge: the sources we mainly refer to are
\cite{Broise}, \cite{Rychlik} and \cite{Viana}, which already make use of functions of bounded variation.

The total variation of a function $f$ on a set $X \subseteq \mathbb{R}$ is
$$\var_X f := \sup \sum_{i = 1}^n |f(x_i)-f(x_{i+1})|$$
where the sup is taken over all finite increasing sequences ${x_1 \leq x_2 \leq \dots \leq x_n}$ of points of $X$.
Given an interval $I$, let us denote $BV(I)$ the Banach space of complex-valued bounded variation functions of the interval $I$,
modulo equality almost everywhere. The space is endowed with the norm
$$ \Vert f \Vert_{BV(I)} := \inf \left\{ \var_{I} g + \int_{I} |g(x)| dx  \ : \ g = f \ a.e.\right\}$$
Observe that every $f \in BV(I)$ has a (not necessarily unique) representative of minimal total variation, namely such that
$$f(x) \in [\lim_{y \to x^-} f(y), \lim_{y \to x^+} f(y) ] \qquad \forall x \in I$$
In the following, we will always choose representatives for our functions of minimal variation. Other basic properties of total variation are stated in the appendix.

\subsection{Cylinders} \label{cyl}

For each $\alpha \in  (0,1)$, the dynamical system $T_\alpha$ defined in the introduction acts on the interval
$I_{\alpha} := [\alpha-1, \alpha]$.
Observe that there exists a partition of $I_\alpha$ in a countable number of intervals $I_j$
such that for every $j$ the restriction $T_{\alpha}\mid_{I_j}$ is a strictly monotone, $C^{\infty}$ function and it extends to a $C^{\infty}$
function on the closure of every $I_j$. The least fine of such partitions will be called $\mathcal{P}_1$, the \emph{partition associated } to
$T_{\alpha}$.
More specifically, $\mathcal{P}_1 = \{I_j^+\}_{j \geq j_{min}} \cup \{I_j^- \}_{j \geq 2}$ with $j_{min} = \lceil \frac{1}{\alpha} - \alpha \rceil$ where
$$I_j^+ = \left( \frac{1}{j+\alpha}, \frac{1}{j-1+\alpha} \right) \textup{ if } j\geq j_{min}+1 \quad I_{j_{min}}^+ = \left( \frac{1}{j_{min}+\alpha}, \alpha \right)$$
$$I_j^- = \left( -\frac{1}{j-1+\alpha}, -\frac{1}{j+\alpha}  \right) \textup{ if } j\geq 3 \quad I_{2}^- = \left(\alpha-1,  \frac{1}{2+\alpha} \right)$$
Moreover, for every $n > 1$, the set
$$\{ I_{j_1}^{\epsilon_1} \cap T_{\alpha}^{-1}(I_{j_2}^{\epsilon_2}) \cap \dots \cap T^{-(n-1)}(I_{j_n}^{\epsilon_{n}})\ |\ I_{j_1}^{\epsilon_1}, \dots, I_{j_n}^{\epsilon_n} \in \mathcal{P}_1 \}$$
where $\epsilon_i\in \{+,-\}$, is a partition of $I_{\alpha}$ in a countable number of intervals such that on each of these the restriction
of  $T_{\alpha}^n$ is monotone and $C^{\infty}$: such a partition will be denoted by $\mathcal{P}_n$ and its elements called \emph{cylinders}.
The cylinder $I_{j_1}^{\epsilon_1} \cap T_{\alpha}^{-1}(I_{j_2}^{\epsilon_2}) \cap \dots \cap T^{-(n-1)}(I_{j_n}^{\epsilon_{n}})$ will be denoted
either by $(I_{j_1}^{\epsilon_1}, \dots, I_{j_n}^{\epsilon_n})$ or by $((j_1, \epsilon_1), \dots, (j_n, \epsilon_n))$.
The cylinders $I_j \in \mathcal{P}_n$ such that $T_\alpha^n(I_j) = I_\alpha$ will be called \emph{full cylinders}.

Let us define the function
$$g_{n, \alpha}(x) := \sum_{j \in \mathcal{P}_n} \frac{1}{|(T^n_{\alpha})'(x)|}\chi_{I_j}(x)$$
The following estimates, proven in the appendix, will be used throughout the paper:

\begin{proposition} \label{classeC}
For every $\alpha \in (0, 1)$ and for every $n \geq 1$

\begin{enumerate}
 \item
$$ \Vert g_{n, \alpha} \Vert_\infty \leq \gamma_\alpha^n $$
where $\gamma_\alpha := \max\{ \alpha^2, (\alpha-1)^2 \}$.
\item
$$ \sup_{j \in \mathcal{P}_n} \sup_{x \in I_j} \left| g_{n, \alpha}'(x) \right| \leq \frac{2}{1-\gamma_\alpha } $$
\item The set $\{ T^n_{\alpha}(I_j)\ |\ I_j \in \mathcal{P}_n \}$ is finite; more precisely,
$$ \#\{ T^n_{\alpha}(I_j)\ |\ I_j \in \mathcal{P}_n \} \leq 2n + 1$$
\item
The total variation of $g_{1, \alpha}$ is universally bounded, i.e. there is a constant $C_0$ such that
$$ \var_{I_\alpha} g_{1, \alpha} \leq C_0 < +\infty \qquad \forall \alpha \in (0,1)$$

\end{enumerate}
\end{proposition}

\subsection{Spectral decomposition } \label{spectral}

The \emph{transfer operator} (also known as \emph{Ruelle-Perron-Frobenius operator}) $\Phi_\alpha : L^1(I_\alpha) \rightarrow L^1(I_\alpha)$ is
defined via the duality
$$\int_{I_\alpha} \Phi_\alpha(f) g dx = \int_{I_\alpha} f (g\circ T_\alpha) dx \quad \forall f\in L^\infty(I_\alpha)$$
Let us recall that the $n^{th}$ iterate of the transfer operator is given by
\begin{equation} \label{transfer}
\Phi_{\alpha}^n(f) = \sum_{j \in \mathcal{P}_n} \frac{f \circ  \sigma_j}{|(T^n_{\alpha})' \circ \sigma_j|}\chi_{T_{\alpha}^n(I_j)}
\end{equation}
where $\sigma_j : T^n_\alpha(I_j) \rightarrow I_j$ is the inverse of the restriction $T^n_\alpha\mid_{I_j} : I_j \rightarrow T^n_\alpha(I_j)$.

Even though $\Phi_\alpha$ is so far defined on $L^1$, it turns out that the transfer operator
preserves the subspace $BV(I_\alpha)$, and indeed it has good convergence properties in $BV$-norm.
More precisely, we can now prove the

\begin{theorem} \label{spectraldec}
Let $\Phi_{\alpha} : L^1(I_{\alpha}) \rightarrow L^1(I_{\alpha})$ be the transfer operator for the system $T_{\alpha}$, with
$\alpha \in (0,1)$. Then one can write
$$\Phi_{\alpha} = \Pi_\alpha + \Psi_\alpha $$
where $\Pi_\alpha$ and $\Psi_\alpha$ are commuting, linear bounded operators on $BV(I_\alpha)$. Moreover,
$\Psi_\alpha$ is a linear bounded operator on $BV(I_{\alpha})$ of spectral radius strictly less than $1$,
and $\Pi_\alpha$ is a projector onto the one-dimensional eigenspace relative to the eigenvalue $1$. It is given by
$$\Pi_\alpha(f) = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k = 1}^{n} \Phi_{\alpha}^k(f)$$
where the convergence is in $L^1$.
\end{theorem}

\begin{corollary}
For every $\alpha \in (0,1)$, $T_{\alpha}$ has exactly one invariant probability measure $\mu_\alpha$ which is absolutely continuous w.r.t.
Lebesgue measure.
Its density will be denoted by $\rho_\alpha$.
\end{corollary}

\begin{proof}
Let us fix $\alpha \in (0,1)$. By proposition \ref{classeC}, we can apply (\cite{Broise}, prop. 4.1), which
yields via Ionescu-Tulcea and Marinescu's theorem \cite{ITM} the following spectral decomposition
$$\Phi_{\alpha} = \sum_{i = 0}^p \lambda_i \Phi_i + \Psi_\alpha $$
where $|\lambda_i| = 1$, and the $\Phi_i$ are linear bounded operators on $BV(I_\alpha)$ with finite-dimensional image, and $\rho(\Psi_\alpha) < 1$.
Now, it is known (\cite{LuzziMarmi}, lemma 1) that $T_{\alpha}$ is \emph{exact} in Rohlin's sense (see \cite{Rohlin}); this implies that
the invariant measure we have found is ergodic and mixing, which in turn implies that the only eigenvalue of $\Phi_{\alpha}$ of modulus $1$
is $1$ itself and that its associated eigenspace is one-dimensional (see \cite{Viana}, chap. 3).
\end{proof}

The spectral decomposition also immediately implies the following exponential decay of correlations:

\begin{proposition}
For any $\alpha \in (0,1)$ there exist $C, \lambda$, $0 < \lambda < 1$ such that for every $n \in \mathbb{N}$ and for every $f_1, f_2 \in BV(I_\alpha)$
$$\left| \int_{I_\alpha} f_1(x) f_2(T^n_\alpha(x)) d\mu_\alpha -   \int_{I_\alpha} f_1(x) d\mu_\alpha \int_{I_\alpha} f_2(x) d\mu_\alpha \right| \leq C\lambda^n \Vert f_1 \Vert_{BV} \Vert f_2 \Vert_{L^1}$$
\end{proposition}

\begin{proof}
One can take any $\lambda$ s.t. $\rho(\Psi_\alpha) < \lambda <1$ and $C = 2 \Vert \rho_\alpha \Vert_{BV} \sup_{n \in \mathbb{N}} \frac{\Vert \Psi_\alpha^n \Vert_{BV}}{\lambda^n}$.
\end{proof}

\section{Continuity of entropy} \label{LasotaYorke}

The goal of this section is to prove theorem \ref{Holdermain}, namely the H\"older-continuity of the function
$\alpha \mapsto h(T_\alpha)$.

The first step is to prove an estimate of the essential spectral radius of the transfer operator
acting on the space of BV functions (Lasota-Yorke inequality).
If one can prove a bound which is uniform in $\alpha$, then the invariant densities $\rho_\alpha$
turn out to be continuous in the $L^1$-topology and their $BV$-norms are bounded.
This method has been undertaken in \cite{LuzziMarmi}, but unfortunately their estimates prove to be too
optimistic\footnote{The mistake in \cite{LuzziMarmi} consists in using, in eq. (12), the estimate (1) of lemma \ref{BVprop} of
the present paper on the sets $\tilde{I}_{\xi}^{(n)}$, which are not intervals if $n >1$.}:
the bulk of section \ref{LYsect} (prop. \ref{LY}) is another proof of this uniform Lasota-Yorke inequality.

The second step (section \ref{Kellerdistsect}) is to estimate the modulus of continuity of $h(T_\alpha)$:
we will prove H\"older-continuous dependence of the invariant densities $\rho_\alpha$ in the $L^1$-topology,
by using a stability result for the spectral projectors \cite{KellerLiverani}.
The theorem then follows from Rohlin's formula.

\subsection{Spectral radius estimate} \label{LYsect}

We are going to give a proof of the following uniform Lasota-Yorke inequality (in order to simplify notation,
from now on $\var_{I_\alpha} f$ will just be denoted $\var f$):

\begin{proposition} \label{LY}
Let $\underline{\alpha} \in (0,1)$. Then there exist a neighbourhood $U$ of $\underline{\alpha}$
and constants $0 < \lambda < 1, C > 0, D > 0$ such that for every $\alpha \in U$
$$\var \Phi^n_\alpha(f) \leq C \lambda^n \var f + D \Vert f \Vert_{L^1} \qquad \qquad \forall n \geq 1,
\ \forall f \in BV(I_\alpha)$$
\end{proposition}

Although several inequalities of this type are present in the literature, (i.e. in \cite{Rychlik}),
these are generally given for individual maps. However, for the goal of this paper it is absolutely essential that
coefficients $\lambda, C, D$ can be chosen uniformly in $\alpha$, hence one needs to take this dependence into account.
As $\alpha$ moves, even just in a neighbourhood of some fixed $\underline{\alpha}$,
 topological bifurcations are present (for instance if $\underline{\alpha}$ is a fixed point of some branch of $T_{\underline{\alpha}}$)
hence in the formula \eqref{transfer} new boundary terms appear, requiring a very careful control.

\begin{lemma} For each $\alpha \in (0,1)$, for each $f \in BV(I_\alpha)$
$$\var \Phi^n_{\alpha}(f) \leq  \var(f \cdot g_{n, \alpha})$$
\end{lemma}

\begin{proof}

\begin{small}

$$\var \Phi^n_{\alpha}(f) = \var \left( \sum_{j \in \mathcal{P}_n} \frac{f \circ  \sigma_j}{|(T^n_{\alpha})' \circ \sigma_j|}\chi_{T_{\alpha}^n(I_j)} \right) \leq
\sum_{j \in \mathcal{P}_n} \var \left( \frac{f \circ  \sigma_j}{|(T^n_{\alpha})' \circ \sigma_j|}\chi_{T_{\alpha}^n(I_j)} \right) =$$
$$ = \sum_{j \in \mathcal{P}_n} \var \left( \frac{f}{|(T_\alpha^n)'|}\chi_{I_j} \right) =
\var \left( f \sum_{j \in \mathcal{P}_n} \frac{1}{|(T_\alpha^n)'|} \chi_{I_j} \right) =
\var (fg_{n, \alpha})$$
\end{small}

\end{proof}

Observe that $g_{n, \alpha}$ has infinitely many jumps discontinuities (indeed it is zero on the boundary of any interval
of the partition $\mathcal{P}_n$), but all those jumps sum up to a finite total variation. We will, however, need to prove
the stronger statement that the total variation of $g_{n, \alpha}$ decays exponentially fast in $n$, and uniformly in $\alpha$.
The idea of the proof is to control the total variation of $g_{n, \alpha}$ by writing it as a sum of two functions,
$h_{n, \alpha}$ and $l_{n, \alpha}$ in such a way that the total variation of $l_{n, \alpha}$ is always very small,
and $h_{n, \alpha}$ has always a finite, controlled number, of jump discontinuities. The following lemma is the key lemma:

\begin{lemma} \label{technical}

For each $\epsilon > 0$, for each $n \geq 1$, for each $\alpha \in (0,1)$ there exist two non-negative functions
$h_{n, \alpha}$ and $l_{n, \alpha}$ such that
$$g_{n, \alpha} = h_{n,\alpha} + l_{n, \alpha}$$
and for each $\alpha$
\begin{enumerate}
\item $\var_{I_\alpha} l_{n, \alpha} \leq 3^n C_0^{n-1} \epsilon$, \ where $C_0$ is the constant in lemma \ref{classeC};
\item $h_{n, \alpha}$ is smooth with $|h_{n, \alpha}'| \leq 1$ outside a finite set $J_{n, \alpha}$, where $h_{n, \alpha}$ has jump
discontinuities. Moreover, for each $\alpha$ there exists a
neighbourhood $U = (\alpha-\eta, \alpha+\eta)$ of $\alpha$ and $r>0$ such that:
\begin{itemize}
 \item[a.]
 For each $\beta \in U$, \quad $J_{n, \beta} \subseteq B(J_{n, \alpha}, r)$
\item[b.]
For each $x \in J_{n, \alpha}$,\quad $\#|J_{n, \beta} \cap B(x, r)| \leq n+1$
\item[c.]
For each $y \in J_{n, \beta} \cap B(x, r)$, \quad  $|x-y| \leq |\alpha - \beta|$
\end{itemize}
\end{enumerate}

\end{lemma}

\begin{proof}
By induction on $n$. If $n = 1$, let us note that

$$g_{1, \alpha}(x) := \left\{
\begin{array}{ll} x^2 & \textup{if }x \textup{ belongs to some }I_j \\
		  0 & \textup{otherwise}

\end{array}
\right.$$
hence we can choose $L := [-\frac{1}{K}, \frac{1}{K}]$ an interval around $0$ such that, for all $\alpha$,
$\var_L g_{1, \alpha}\leq \epsilon$ and
define
$$l_{1, \alpha} := g_{1, \alpha} \chi_L \qquad h_{1, \alpha} := g_{1, \alpha} \chi_{I_\alpha \setminus L}$$
1. is clearly verified. To verify 2., note that given $x \in J_{1, \alpha}, x\neq \alpha, \alpha-1$,
for $\beta$ sufficiently close to $\alpha$, $J_{1, \beta}$ intersects a neighbourhood of $x$ in only one point.
The same happens if $x = \alpha, \alpha-1$ and $T_\alpha(x) \neq \alpha-1$.
On the other hand, if $x = \alpha$ and $T_\alpha(\alpha) = \alpha-1$, then $J_{1,\beta} \cap [\beta-\eta, \beta] = \{y, \beta\}$
contains at most two points, where $y = T_\beta^{-1}(\beta-1) \cap [\beta-\eta, \beta]$ and, since $T_\beta$ is expanding,
$|y - \alpha| \leq |\alpha - \beta|$. The case $x = \alpha-1$, $T_\alpha(\alpha-1) = \alpha-1$ is similar.

In order to prove the inductive step, let us remark that $g_{n+1, \alpha} = g_{n, \alpha} \circ T_\alpha \cdot g_{1, \alpha}$. Hence,
we can define

$$\begin{array}{lll}
    h_{n+1, \alpha} & := & h_{n, \alpha} \circ T_\alpha \cdot h_{1, \alpha} \\
    l_{n+1, \alpha} & := & l_{n, \alpha} \circ T_\alpha \cdot g_{1, \alpha} + h_{n, \alpha} \circ T_\alpha \cdot l_{1, \alpha}
\end{array}$$
and check all properties are satisfied.
First of all, we can prove by induction that
\begin{equation} \label{variterate}
\var_{I_\alpha} h_{n, \alpha} \leq 2^{n-1} C_0^n \qquad \forall \alpha \in (0,1), \forall n \geq 1
\end{equation}
Indeed,
\begin{small}
$$\var_{I_\alpha} h_{1, \alpha} \leq \var_{I_\alpha} g_{1, \alpha} \leq C_0$$
$$\var_{I_\alpha} h_{n+1, \alpha} = \sum_{k \in \mathcal{P}_1} \var_{\overline{I_k}} (h_{n, \alpha} \circ T_\alpha \cdot h_{1, \alpha}) \leq
\sum_{k \in \mathcal{P}_1} \var_{\overline{I}_k} (h_{n, \alpha} \circ T_\alpha) \sup_{\overline{I}_k} h_{1, \alpha} + \sup_{\overline{I}_k} (h_{n, \alpha} \circ T_\alpha) \var_{\overline{I}_k} h_{1, \alpha} \leq$$
\end{small}
and since $T_\alpha \mid_{I_k}$ is a homeomorphism
\begin{small}
$$ \leq \var_{I_\alpha} h_{n, \alpha} \sum_{k \in \mathcal{P}_1} \sup_{\overline{I}_k} h_{1, \alpha} + \sup_{I_\alpha} h_{n, \alpha} \sum_{k \in \mathcal{P}_1} \var_{\overline{I}_k} h_{1, \alpha}
\leq 2 \var_{I_\alpha} h_{n, \alpha} \var_{I_\alpha} h_{1, \alpha}\leq 2 \cdot 2^{n-1}C_0^{n} \cdot C_0$$
\end{small}
where in the penultimate inequality we used the fact that $\sup_I f \leq \var_I f$ if $f(x) = 0$ for some $x \in I$.

Let us now check 1.: similarly as before,
$$\var_{I_\alpha} l_{n+1, \alpha} =
\var_{I_\alpha}  (l_{n, \alpha} \circ T_\alpha \cdot g_{1, \alpha} + h_{n, \alpha} \circ T_\alpha \cdot l_{1, \alpha}) \leq
 2 \var_{I_\alpha} l_{n, \alpha} \var_{I_\alpha} g_{1, \alpha} + 2 \var_{I_\alpha} h_{n, \alpha} \var_{I_\alpha} l_{1, \alpha} \leq$$
and by inductive hypothesis and \eqref{variterate}
$$\leq 2 \cdot 3^n C_0^{n-1} \epsilon \cdot C_0 + 2 \cdot 2^{n-1} C_0^n \cdot \epsilon \leq 3^{n+1} C_0^n \epsilon$$

Since $h_{1, \alpha}$ is nonzero only on finitely many branches of $T_\alpha$, then $h_{n+1, \alpha}$ has only finitely many jump
discontinuities. Now, if $x$ is a jump discontinuity for $h_{n, \alpha} \circ T_\alpha$ and not for $h_{1, \alpha}$,
then $T_\beta$ is an expanding local homeomorphism at $x$ for all $\beta$ in a neighbourhood of $\alpha$, hence a., b. and c. follow.
Let now $x \neq \alpha, \alpha-1$ be on the boundary of some cylinder, i.e. $T_\alpha(x) = \alpha-1$.
Then by inductive hypothesis c., if $\beta > \alpha$ is sufficiently close to $\alpha$ and $\eta$ is sufficiently small, then
$$J_{n, \beta} \cap [\beta-1, \beta -1 + \eta] = \{\beta-1\}$$
hence $$J_{n+1, \beta} \cap B(x, r) = T_{\beta}^{-1}(J_{n, \beta}\cap [\beta-\eta, \beta]) \cap B(x, r)$$
and b. follows. c. follows from the fact that $T_\beta$ is expanding. If $\beta < \alpha$, similarly the claims follow because
$$J_{n+1, \beta} \cap B(x, r) = T_{\beta}^{-1}(J_{n, \beta}\cap [\beta-1, \beta-1+\eta]) \cap B(x, r)$$
If $x = \alpha$, then for $\beta$ sufficiently close to $\alpha$,
$$J_{n+1, \beta} \cap B(x, r) \subseteq (T_\beta^{-1}(J_{n, \beta}) \cup \{\beta\}) \cap B(x, r)$$
has cardinality at most $n+2$, and c. follows because $T_\alpha$ is expanding.
The case $x = \alpha-1$ is analogous.

\end{proof}

\begin{lemma} \label{goodpartition}
Let $\alpha \in (0,1)$, $n \geq 1$ and $\epsilon > 0$. Then there exist $\eta > 0$, $C > 0$ and a finite partition
 of $[\alpha-1-\eta, \alpha +\eta]$ in closed intervals $L_1, \dots L_r$ such that for each $\beta \in (\alpha - \eta, \alpha + \eta)$
and each $i \in {1, \dots, r}$ the following holds:
\begin{itemize}
 \item $ 0 < C \leq m(L_{i, \beta}) \leq \epsilon$
\item $\var_{L_{i, \beta}} g_{n, \beta} \leq 2(n+1) \Vert g_{n, \beta} \Vert_\infty + 2\epsilon$
\end{itemize}
where $L_{i, \beta} := L_i \cap [\beta-1, \beta]$.
\end{lemma}

\begin{proof}
Given $\alpha, n, \epsilon$, choose $L_1, \dots, L_r$ in such a way that $m(L_i) \leq \epsilon$, each element of $J_{n, \alpha}$
lies in the interior of some $L_i$ and no two such elements lie in the same $L_i$. Moreover,
set $\epsilon_1 := \epsilon/(3^nC_0^{n-1})$ and, for each $\beta$ sufficiently close to $\alpha$, choose a decomposition
$g_{n, \beta} = h_{n, \beta} + l_{n, \beta}$ as in lemma \ref{technical} relative to $\epsilon_1$.

$$\var_{L_{i, \beta}} h_{n, \beta} \leq \int_{L_{i, \beta} \setminus J_{n, \beta}} h'_{n, \beta}(x) dx + \sum_{x \in L_{i, \beta} \cap J_{n, \beta}} \lim_{y \to x^-} h_{n, \beta}(y) + \lim_{y \to x^+} h_{n, \beta}(y) \leq $$
$$\leq m(L_{i, \beta}) + 2 \#\{L_{i, \beta} \cap J_{n, \beta}\} \Vert h_{n, \beta} \Vert_\infty \leq \epsilon + 2(n+1) \Vert h_{n, \beta} \Vert_\infty$$
hence $\var_{L_{i, \beta}} g_{n, \beta} \leq \var_{L_{i, \beta}} h_{n, \beta} + l_{n, \beta} \leq 2 \epsilon + 2(n+1)\Vert g_{n, \beta} \Vert_\infty$.
\end{proof}

\noindent \textit{Proof of proposition \ref{LY}. }
Consider the partition $L_1, \dots, L_r$ given by lemma \ref{goodpartition}. Then
$$\var (f \cdot g_{n, \alpha} ) = \sum_{i = 1}^r \var_{L_i} (f g_{n, \alpha}) \leq \sum_{i=1}^r \var_{L_i} f \sup_{L_i} g_{n, \alpha} +
 \var_{L_i} g_{n, \alpha} \sup_{L_i} f  \leq $$
$$ \leq \sum_{i=1}^r \Vert g_{n, \alpha}   \Vert_\infty  \var_{L_i} f + \var_{L_i} g_{n, \alpha} \left( \frac{1}{m(L_{i, \alpha})} \int_{L_i} f(x) dx + \var_{L_i} f \right) \leq $$
$$\leq [(2n+3) \Vert g_{n, \alpha} \Vert_\infty + 2 \epsilon] \var_{I_\alpha} f + \frac{(2n+2) \Vert g_{n,\alpha} \Vert_\infty + 2 \epsilon}{C} \int_{I_\alpha} f(x) dx $$
Now, since $\Vert g_{n, \alpha} \Vert_\infty \leq \gamma_\alpha^n$ decays exponentially,
 we can choose $n$ large enough so that $\lambda:= (2n+4) \gamma_\alpha^n < 1$, and we can also choose $2 \epsilon \leq \gamma_\alpha^n$,
hence we get that for some constant $D >0$, for each $\alpha \in (\underline{\alpha} - \eta, \underline{\alpha} + \eta)$,
\begin{equation} \label{finalLY}
\var \Phi_\alpha^n(f) \leq \lambda \var f + D \Vert f \Vert_1
\end{equation}
and by iteration and euclidean division (see e.g. \cite{Rychlik}, lemma 7 and prop. 1) the claim is proven.
\qed

\subsection{Stability of spectral decomposition} \label{Kellerdistsect}

The next step to prove H\"older-continuity is proving the continuous dependence of invariant densities $\rho_\alpha$ in $L^1$-norm.
In order to guarantee the stability of spectral projectors of the transfer operator, we will use the following theorem of Keller and Liverani \cite{KellerLiverani}:

\begin{theorem} \label{KellerLiv}
 Let $P_{\epsilon}$ be a family of bounded linear operators on a Banach space $(B, \Vert \cdot \Vert)$ which is also equipped with a second
norm $\vert \cdot \vert$ such that $\vert \cdot \vert \leq \Vert \cdot \Vert$. Let us assume that the following conditions hold:
\begin{enumerate}
 \item $\exists C_1, M > 0$ s.t. for all $\epsilon \geq 0$
$$|P_{\epsilon}^n| \leq C_1 M^n \qquad \forall n \in \mathbb{N}$$
\item $\exists C_2, C_3 > 0$ and $\lambda \in (0,1)$, $\lambda < M$, such that for all $\epsilon \geq 0$
$$\Vert P_\epsilon^n f \Vert \leq C_2 \lambda^n \Vert f \Vert + C_3 M^n \vert f \vert \quad \forall n \in \mathbb{N} \quad \forall f \in B$$
\item if $z \in \sigma(P_{\epsilon}), |z| > \lambda$, then $z$ is not in the residual spectrum of $P_\epsilon$
\item There is a monotone continuous function $\tau : [0, \infty) \rightarrow [0, \infty)$ such that $\tau(\epsilon) > 0$ if $\epsilon > 0$ and
$$ \vert \! \vert \! \vert P_0 - P_\epsilon \vert \! \vert \! \vert \leq \tau(\epsilon) \rightarrow 0 \quad as \ \epsilon \rightarrow 0$$
where the norm $| \! | \! | \cdot |\!|\!|$ is defined as
$$| \! | \! | Q | \! | \! | := \sup_{\Vert f \Vert \leq 1} |Qf|$$
\end{enumerate}

Let us now fix $\delta > 0$ and $r \in (\lambda, M)$ and define
$$V_{\delta, r} := \{ z \in \mathbb{C} : |z| \leq r \textrm{ or dist }(z, \sigma(P_0)) \leq \delta \}$$
and $\eta := \frac{\log(\lambda/r)}{\log(\lambda/M)}$. Then there exist $H, K > 0$ such that if $\tau(\epsilon) \leq H$ then $\sigma(P_{\epsilon}) \subseteq V_{\delta, r}$ and
$$\vert \! \vert \! \vert (z-P_{\epsilon})^{-1} - (z-P_0)^{-1} \vert \! \vert \! \vert \leq K \tau(\epsilon)^{\eta} \qquad \forall z \notin V_{\delta, r}$$
\end{theorem}
\noindent In our context, the norm $|\cdot|$ will be the $L^1$ norm and $\Vert \cdot \Vert$ will be the $BV$ norm.
Our goal is to apply this result to the family
$\{\Phi_\alpha\}_{\alpha \in U}$ where $U$ is a suitable neighbourhood of a given $\underline{\alpha} \in (0,1)$.

Hypothesis (1) is trivial since transfer operators have unit $L^1$-norm, and (2) is precisely
proposition \ref{LY}.
In the context of one-dimensional piecewise expanding maps, (3) is an immediate corollary of (2):

\begin{lemma} \label{henn}
For every $\underline{\alpha} \in (0,1)$ there exists $\epsilon > 0$ such that for
$|\alpha - \underline{\alpha}| < \epsilon$, $$\rho_{ess}(\Phi_{\alpha}) \leq \lambda$$
where $\lambda$ is the same as in proposition \ref{LY} and therefore condition (3) holds.
\end{lemma}

\begin{proof}
By a result of Hennion \cite{Hennion}, the uniform Lasota-Yorke inequality plus the fact that the injection
$BV(I) \rightarrow L^1(I)$ is compact implies the estimate on the essential spectral radius; therefore the elements
of the spectrum of modulus bigger than $\lambda$ are eigenvalues with finite multiplicity and cannot belong to the residual spectrum.
\end{proof}

To prove condition (4) it is necessary to estimate the distance between the $\Phi_{\alpha}$ as $\alpha$ varies in a neighbourhood of a fixed $\underline{\alpha}$; by a result of Keller \cite{Keller} the distance between the
transfer operators is related to the following distance between the transformations:

\begin{definition}
Let $T_1, T_2 : I \rightarrow I$ two maps of the interval $I$. We define the \emph{Keller distance} between $T_1$ and $T_2$ as
$$d(T_1, T_2) := \inf \{ \kappa > 0 \ | \ \exists A \subset I \textup{ measurable with }m(A) > 1 - \kappa, $$
$$  \exists \sigma : I \rightarrow I \ \textup{diffeo s.t. } T_1\mid_A = T_2\circ \sigma\mid_A, \ \sup_{x \in I}|\sigma(x) -x| < \kappa,\ \sup_{x \in I}\left|\frac{1}{\sigma'(x)}-1\right| < \kappa \}$$
\end{definition}

\begin{lemma}[\cite{Keller}, lemma 13] \label{Kellerlemma}
If $P_1$ and $P_2$ are the transfer operators associated to the interval maps $T_1$ and $T_2$, then $\vert \! \vert \! \vert P_1 - P_2 \vert \! \vert \! \vert \leq 12 d(T_1, T_2)$
where $d$ is the Keller distance.
\end{lemma}

We verify now that this convergence result applies to our case of $\alpha$-continued fractions. In order to do so, it is necessary to translate the maps in such a way that they are all defined on the same interval, which will be $[0,1]$ in our case. We therefore consider the maps $\tilde{T}_{\alpha} : [0,1] \rightarrow [0,1]$
$$ \tilde{T}_{\alpha}(x) = T_{\alpha}(x+\alpha-1) + 1-\alpha $$
The relative invariant densities will be
$$\tilde{\rho}_\alpha(x) = \rho_\alpha(x+ \alpha-1)$$

\begin{lemma} \label{Kellerdist}
Fix $\underline{\alpha} \in (0,1)$. Then there exists a neighbourhood $U$ of $\underline{\alpha}$ and
a positive constant $C$ such that, for $\alpha, \beta \in U$, we have
$$ d(\tilde{T}_{\alpha}, \tilde{T}_{\beta}) \leq C |\alpha -\beta|^{1/2}$$
\end{lemma}

\begin{proof}
Having fixed $\alpha, \beta$, let us define
$$y(x) := \frac{x+\alpha-1}{1+(\beta-\alpha)|x+\alpha-1|} + 1 - \beta$$
It is immediate to verify that $\tilde{T}_{\alpha}(x) = \tilde{T}_{\beta}(y(x)) \  \forall x \in [0,1]$ and $y'(x) = \frac{1}{(1+(\beta-\alpha)|x+\alpha-1|)^2}$
We also have
$$\sup_{x \in [0,1]} |y(x) - x| = |y(1) - 1|$$
because when $|\alpha -\beta|$ is small we have that, for $\alpha > \beta$,  $y(x) - x$ has positive derivative and $y(0) > 0$, while, for $\alpha < \beta$, $y(x) -x$ has negative derivative and $y(0) < 0$.
Thus, for $|\alpha - \beta|$ sufficiently small,
$$ \sup_{x \in [0,1]} |y(x) - x| = |\alpha - \beta|\left| \frac{1+ \alpha \beta}{1+ (\beta-\alpha)\alpha}\right| \leq 2 |\alpha - \beta|$$
$$ \sup_{x \in [0,1]} \left| \frac{1}{y'(x)}-1 \right| = |\beta-\alpha| \sup_{x \in [0,1]} \left|2 |x+\alpha-1| + (\beta-\alpha)|x+\alpha-1|^2\right| \leq 3 |\alpha - \beta|$$
In order to compute the Keller distance we need to find a diffeomorphism $\sigma$ of the interval such that $\tilde{T}_{\alpha} = \tilde{T}_{\beta} \circ \sigma$ on a set of large measure; the $y$ defined so far is not a diffeo, so it is necessary to modify it a bit at the endpoints and we will do it by introducing two little linear bridges. Let $\delta$ be such that $\delta^2 = \sup_{x \in [0,1]}|y(x) - x | \leq 2 |\alpha -\beta|$;  we can define
$$ \sigma(x) = \left\{ \begin{array}{ll} \frac{y(\delta)}{\delta} x &  \textrm{for }x \leq \delta \\ y(x) & \textrm{for }\delta \leq x \leq 1-\delta \\
\frac{1-y(1-\delta)}{\delta}(x-1+\delta) + y(1-\delta) & \textrm{for }x \geq 1-\delta \end{array} \right. $$
For the sup norm we have
$$\sup_{x \in [0,1]} |\sigma(x) - x| \leq \max \left\{ |y(\delta) - \delta|, \sup_{x \in [\delta, 1-\delta]}  |y(x)-x|,  |y(1-\delta) -1+\delta| \right\} \leq $$
$$\leq \sup_{x \in[0,1]} |y(x) - x| \leq 2 |\beta - \alpha|$$
Since $|y(\delta)| \geq \delta - |y(\delta) - \delta| \geq \delta - \delta^2$, one gets
$ \sup_{x \in[0,\delta]}\left| \frac{1}{\sigma'(x)}-1 \right| \leq \frac{\delta}{1-\delta} $ and
$$\sup_{x \in [0,1]} \left| \frac{1}{\sigma'(x)}-1\right| \leq \max \left\{ 3|\alpha-\beta|, \frac{\delta}{1-\delta} \right\} \leq C |\alpha - \beta|^{1/2} $$

Now, $\sigma$ is a homeomorphism of $[0,1]$ with well defined, non-zero derivative except for the points $x = \delta, 1-\delta$.
Hence one can construct smooth approximations $\sigma_n$ of $\sigma$ which coincide with it except on
$[\delta-\frac{1}{2^n}, \delta+\frac{1}{2^n}] \cup [1-\delta -\frac{1}{2^n}, 1 - \delta+\frac{1}{2^n}]$ and such that previous estimates still
hold. These $\sigma_n$ will be diffeomorphisms of the interval s.t.
$\tilde{T}_\alpha(x) = \tilde{T}_{\beta}(\sigma_n(x))$ for $x \in [\delta+ \frac{1}{2^n}, 1-\delta -\frac{1}{2^n}]$.
Since $\sup m([\delta+ \frac{1}{2^n}, 1-\delta -\frac{1}{2^n}]) = 1 - 2\delta \geq 1 - 2 |\alpha - \beta|^{1/2}$, then
the claim is proven.
\end{proof}

\subsection{H\"older-continuity of entropy} \label{Holdercont}

By using the perturbation theory developed so far, we complete the proof that the function $\alpha \mapsto h(T_\alpha)$ is
locally H\"older-continuous. Note that the uniform Lasota-Yorke inequality proven in section \ref{LYsect} would already imply
 continuity by the methods in \cite{LuzziMarmi}, while here we get a quantitative bound on the continuity module.

\begin{proposition}
Let $\delta > 0$, and $0 < s < \frac{1}{2}$. Then there exists a constant $C > 0$ such that
$$| h(T_{\alpha})-h(T_{\beta}) | \leq C |\alpha - \beta|^s \qquad \forall \alpha, \beta \in [\delta, 1]$$
\end{proposition}

\begin{proof}
Let us fix $\eta \in (0,1)$, and choose $r$ such that $\eta = \frac{\log(\lambda/r)}{\log(\lambda)}$.
By thm \ref{KellerLiv} applied to the family $\Phi_\alpha$, (using proposition \ref{LY}, lemma \ref{henn} and lemma \ref{Kellerdist} as hypotheses),
for each $\alpha \in (0,1)$ there exist $\epsilon, C_1 > 0$ such that
$$\vert \! \vert \! \vert \Pi_{\alpha} - \Pi_{\beta} \vert \! \vert \! \vert \leq C_1 |\alpha-\beta|^{\eta/2} \qquad \forall \beta \in (\alpha-\epsilon, \alpha + \epsilon)$$

Now, in theorem \ref{KellerLiv} the bounds ($H, K$) depend only on the constants $C_1$, $C_2$, $C_3$, $\lambda$, $M$,
and in proposition \ref{LY} and lemma \ref{Kellerdist} these constants are locally uniformly bounded in $\alpha$, hence
the following stronger statement is true: for each $\underline{\alpha} \in (0,1)$ there is $C_1 >0$ and some neighbourhood $U$ of $\underline{\alpha}$ such that
$$\vert \! \vert \! \vert \Pi_{\alpha} - \Pi_{\beta} \vert \! \vert \! \vert \leq C_1 |\alpha-\beta|^{\eta/2} \qquad \forall \alpha, \beta \in U$$
Since $\tilde{\rho_{\alpha}} = \Pi_{\alpha}(1)$, the previous equation implies
$$\Vert \tilde{\rho}_{\alpha} - \tilde{\rho}_{\beta} \Vert_{L^1} = O(\left|\alpha-\beta\right|^{\frac{\eta}{2}})$$
By prop. \ref{LY}, $\Vert \tilde{\rho}_{\alpha} \Vert_{BV}$ is locally bounded, hence so is $\Vert \tilde{\rho}_\alpha \Vert_{\infty}$ and for any $p > 1$
$$\Vert \tilde{\rho}_{\alpha} - \tilde{\rho}_{\beta} \Vert_{L^p} = O(\left|\alpha-\beta\right|^{\frac{\eta}{2p}})$$
By Rohlin's formula, $h(T_\alpha) = -2 \int_0^1 \log|y+\alpha-1| \tilde{\rho}_\alpha(y)dy$, thus
$$|h(T_{\alpha}) - h(T_{\beta})| \leq 2\int_{0}^1 \left|\log |y+\alpha-1| \tilde{\rho}_{\alpha}(y) - \log |y + \beta -1| \tilde{\rho}_{\beta}(y)\right|dy \leq $$
by separating the product and applying H\"older's inequality, for any $p > 1$
\begin{small}
$$ \leq 2 \Vert \tilde{\rho}_\alpha \Vert_{\infty} \Vert \log |y+\alpha-1| - \log|y+\beta-1|\Vert_{L^1} + \Vert 2 \log(y+\beta -1)
 \Vert_{L^{p/p-1}} \Vert \tilde{\rho}_{\alpha} - \tilde{\rho}_{\beta} \Vert_{L^p} $$
\end{small}
Now, basic calculus shows $\Vert \log |y+\alpha-1| - \log|y+\beta-1|\Vert_{L^1} = O(-|\alpha- \beta| \log |\alpha-\beta|)$
and $\Vert 2 \log(y + \alpha - 1) \Vert_{L^{p/p-1}}$ is bounded independently of $\alpha$.
Since this is true $\forall \eta < 1$ and $\forall p > 1$, the claim follows.
\end{proof}

\begin{remark} \label{rmk}
One has to be careful with the norm he uses to get the convergence, because while $L^1$-convergence of the densities is assured by uniform Lasota-Yorke, the invariant densities in general DO NOT converge to each other in $BV$-norm. For example we have for $\alpha \geq \frac{\sqrt{5}-1}{2}$
$$ \rho_\alpha(x) = \frac{1}{\log(1+\alpha)} \left(\chi_{[0, \frac{1-\alpha^2}{\alpha}]}(x)\frac{1}{x+2} + \chi_{(\frac{1-\alpha^2}{\alpha}, 1]}(x) \frac{1}{x+1} \right) $$
so
\begin{small}
$$\var_{[0,1]}(\tilde{\rho}_\alpha - \tilde{\rho}_{\underline{\alpha}}) \geq \left| \lim_{x \rightarrow \left(\frac{1-\alpha^2}{\alpha}\right)^{-}} (\rho_\alpha - \rho_{\underline{\alpha}}) - \lim_{x \rightarrow \left(\frac{1-\alpha^2}{\alpha}\right)^{+}} (\rho_\alpha - \rho_{\underline{\alpha}}) \right|$$
\end{small}
which does not converge to $0$ as $\alpha \rightarrow \underline{\alpha}$.

\end{remark}

\section{Central limit theorems} \label{CLTsect}

The goal of this section is to prove a central limit theorem (CLT) for the systems $T_{\alpha}$.
Given an observable $f : I_\alpha \rightarrow \mathbb{R}$, we denote by $S_nf$ the \emph{Birkhoff sum}
$$S_n f = \sum_{j = 0}^{n-1} f \circ T_{\alpha}^j$$
The function $x \mapsto \frac{S_n f(x)}{n}$ is called \emph{Birkhoff average} and it can be seen as a random variable
on the space $I_\alpha = [\alpha-1, \alpha]$ endowed with the measure $\mu_{\alpha}$.
By ergodicity, this random  variable converges a.e. to a constant.
Our goal is to prove that the difference from such limit value converges in law to a Gaussian distribution.

Heuristically, this means the sequence of observables $\{f \circ T_\alpha^n\}$
(which can be seen as  identically distributed random variables on $I_\alpha$) behave as if they were independent, i.e. the system has
little memory of its past. A convergence property of this type is also useful to confirm numerical data, since it implies the variance of
Birkhoff averages up to the $n^{th}$ iterate decays as $\frac{1}{\sqrt{n}}$, hence one can get a good approximation of the limit
value by computing Birkhoff averages up to a suitable finite time $n$ (see \cite{LuzziMarmi}).

First (subsection \ref{CLTBVsect}), we will prove CLT for observables of bounded variation.
A particularly important observable is $\log|T_\alpha'|$, because by Rohlin's formula its expectation is the metric entropy.
Such observable, however, is not of bounded variation: in subsection \ref{CLTunbsect}, we will enlarge the class of observables we work with in order to encompass certain
unbounded functions, including $\log|T_\alpha'|$. In order to do so, we need to define \emph{ad hoc} Banach spaces.

\subsection{CLT for functions of bounded variation} \label{CLTBVsect}

\begin{theorem} \label{CLT}
Let $\alpha \in (0,1]$ and $f$ be a real-valued nonconstant element of $BV(I_{\alpha})$.  There exists $\sigma > 0$ such that the random variable
$\frac{S_n (f -\int f d\mu_{\alpha})}{\sigma \sqrt{n}}$ converges in law to a Gaussian $\mathcal{N}(0,1)$, i.e. for every $v \in \mathbb{R}$ we have
$$ \lim_{n \rightarrow \infty} \mu_{\alpha}\left(\frac{S_n f - n \int_I f d\mu_{\alpha}}{\sigma \sqrt{n}} \leq v \right) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^v e^{-x^2/2}dx$$
\end{theorem}

The proof of theorem \ref{CLT} follows a method developed by A. Broise \cite{Broise}.

\medskip
\noindent \textbf{Perturbations of $\Phi_{\alpha}$}

Given $f \in BV(I_\alpha)$ with real values and given $\theta \in \mathbb{C}$, let us define the operator $\Phi_f(\theta): BV(I_\alpha) \rightarrow BV(I_\alpha)$ with
$$\Phi_f(\theta)(g) = \Phi(\textup{exp}(\theta f)g)$$
For $f$ fixed, this family of operators has the property that $\Phi_f(0) = \Phi$ and the function $\theta \rightarrow \Phi_f(\theta)$ is analytic; the interest in this kind of perturbations resides in the identity
$$\Phi_f^n(\theta)(g) = \Phi^n(\textup{exp}(\theta S_n f)g)  \qquad \textup{with }S_nf = \sum_{k = 0}^{n-1} f \circ T_{\alpha}^k$$
Since in our case all eigenvalues of modulus $1$ are simple, the spectral decomposition transfers to the perturbed operator:

$$ \Phi^n_f(\theta)(g) = \lambda^n_0(\theta)\Phi_0(\theta)(g) + \Psi^n_f(\theta)(g) $$
where the functions $\theta \mapsto \Phi_0(\theta)$, $\theta \mapsto \lambda_0(\theta)$ and $\theta \mapsto \Psi_f(\theta)$ are analytic in a neighbourhood of $\theta = 0$. Moreover,
$\rho(\Psi_f(\theta)) \leq \frac{2 + \rho(\Psi)}{3} \leq |\lambda_0(\theta)|$.

\vskip 0.5 cm

\noindent Let us now consider the variance of $S_n f$:

\begin{proposition}[\cite{Broise}, thm. 6.1] \label{variance}
Let $\alpha \in (0,1]$ and $f$ be a real-valued element of $BV(I_{\alpha})$.
Then the sequence
$$M_n = \int_{I_\alpha}\left(\frac{S_n f- n \int f d\mu_{\alpha}}{\sqrt{n}}\right)^2 d\mu_{\alpha}$$ converges to a real nonnegative value, which will be denoted by $\sigma^2$. Moreover, $\sigma^2 = 0$ if and only if there exists
$u \in L^2(\mu_{\alpha})$ such that $u\rho_{\alpha} \in BV(I_{\alpha})$ and
\begin{equation} \label{cohom}
f - \int_{I_\alpha} f d\mu_{\alpha} = u - u \circ T_{\alpha}
\end{equation}
\end{proposition}

Now, if $\sigma > 0$, the method of (\cite{Broise}, chap. 6) yields the central limit theorem. The main steps in the argument are:

\begin{enumerate}
\item
$\lambda_0'(0) = \int_{I_{\alpha}} f d\mu_{\alpha}$
\item
If $\int_{I_\alpha} f d\mu_{\alpha} = 0$, then $\lambda_0''(0) = \sigma^2$
\item
If $\int_{I_\alpha} f d\mu_{\alpha} = 0$, then $\lim_{n \rightarrow \infty} \int_{I_\alpha} \Phi_f^n(\frac{it}{\sigma \sqrt{n}})(\rho_\alpha)dm = \textup{exp}(-\frac{t^2}{2})$
\end{enumerate}
CLT then follows by L\'evy's continuity theorem, LHS in previous equation being the characteristic function of the random variable $\frac{S_n(f  - \int_{I_\alpha} f d\mu_{\alpha})}{\sigma \sqrt{n}}$.

In order to prove the CLT for a given observable we are now left with checking that equation (\ref{cohom}) has no solutions. The following proposition completes the proof of theorem \ref{CLT}.

\begin{proposition}
For every real-valued nonconstant $f \in BV(I_\alpha)$, equation (\ref{cohom}) has no solutions.
\end{proposition}

\begin{proof}
By proposition \ref{classeC}, $T_\alpha$ satisfies the hypotheses of a theorem of Zweim\"uller \cite{Zweimuller}, which asserts that there
exists $C_\alpha > 0$ such that  $\rho_\alpha \geq C_\alpha$ on $\{\rho_\alpha \neq 0 \}$. Hence, the function
 $\frac{1}{\rho_{\alpha}} \chi_{\{\rho_{\alpha} \neq 0\}}$ belongs to $BV(I_\alpha)$, so if it exists $u$ such that
$f\rho_\alpha -(\int_{I_\alpha} f d\mu_\alpha) \rho_\alpha = u\rho_\alpha - u \circ T_\alpha \cdot \rho_\alpha$ in $BV(I_\alpha)$, then
 we can multiply by $\frac{1}{\rho_\alpha} \chi_{\{\rho_\alpha \neq 0\}}$ and get $f - \int_{I_\alpha} f d\mu_\alpha = u - u \circ T_\alpha$ in
 $BV(I_\alpha)$, with $u$ in $BV(I_\alpha)$ because $u\rho_\alpha \in BV(I_\alpha)$; by knowing that $f \in BV(I_\alpha)$ we get
$u \circ T_\alpha \in BV(I_\alpha)$. For each cylinder $I_j \in \mathcal{P}_1$, since $T_\alpha\mid_{I_j}: I_j \rightarrow I_\alpha$ is a homeomorphism,
$$\var_{I_j} (u \circ T_\alpha) = \var_{T_\alpha(I_j)} u$$
hence
$$ \var_{I_\alpha} (u \circ T_\alpha) \geq \sum_{\substack{I_j  \in \mathcal{P}_1 \\ I_j \textup{ full}}} \var_{I_j} (u \circ T_\alpha)
\geq \sum_{\substack{I_j \in \mathcal{P}_1 \\ I_j \textup{ full}}} \var_{T_\alpha(I_j)} u = \sum_{\substack{I_j \in \mathcal{P}_1 \\ I_j \textup{ full}}} \var_{(\alpha - 1, \alpha)} u$$
and, since the set of $j$ s.t. $I_j$ is full is infinite, $u \circ T_\alpha$ has a representative
with bounded variation only if $\var_{(\alpha-1, \alpha)} u = 0$, i.e. $u$ is constant a.e.
\end{proof}

\subsection{CLT for unbounded observables} \label{CLTunbsect}

In order to prove a central limit theorem for the entropy $h(T_\alpha)$
one has to consider the observable $x \mapsto \log|T_\alpha'(x)| = - 2 \log |x|$, which is not of bounded variation on intervals containing $0$.
 Therefore, one has to enlarge the space of functions to work with so that it contains such observable, and use some norm which still allows to
bound the essential spectral radius of the transfer operator. Such technique will be developed in this section.

The strategy is to use the Ionescu-Tulcea and Marinescu theorem to get a spectral decomposition of the transfer operator,
as we did in section \ref{spectral}. This theorem requires a pair of Banach spaces contained in each other such that the operator
preserves both. Traditionally, this is achieved by considering the pair $BV(I) \subset L^1(I)$. In our case, we will replace the space
of functions of bounded variation with newly-defined, larger spaces $B_{K, \delta} \subseteq L^1$, which allow for functions with a
mild singularity in $0$.

\subsubsection{A new family of Banach spaces}

Fix $\alpha \in (0, 1]$.
Given a positive integer $K$ and some $0 < \delta < 1$, let us define the $K, \delta$\emph{-norm} of a function
$f : I_\alpha \rightarrow \mathbb{C}$ as
$$\Vert f \Vert_{K, \delta} := \sup_{k \geq K} \left( k^{-\delta} \var_{L_k} f + \int_{L_k} |f(x)| dx \right)$$
where the $L_k$ are a sequence of increasing subintervals of $I_\alpha$, namely

$$L_k^+ := \overline{\bigcup_{j \geq k} I^+_j} = \left[ \frac{1}{k+\alpha}, \alpha \right] \quad L_k^- := \overline{\bigcup_{j \geq k} I^-_j} = \left[\alpha-1,
-\frac{1}{k+\alpha}\right]$$
and $L_k := L_k^+ \cup L_k^-$, with $\var_{L_k} f := \var_{L_k^+} f + \var_{L_k^-} f$.
Let us now define the space $B_{K, \delta}$ of \emph{functions of mild growth} as
$$B_{K, \delta} := \{ f \in L^1 \ : \ f \textup{ has a version }g \textup{ with }\Vert g \Vert_{K, \delta} < \infty \}$$
Let us now establish some basic properties of these spaces. First of all, they are Banach spaces:

\begin{proposition} For every $K \in \mathbb{N}$, $0 < \delta < 1$, the space
$B_{K, \delta}$ endowed with the norm
$$\Vert f \Vert_{K, \delta} := \inf \{ \Vert g \Vert_{K, \delta}, \ g = f \textup{ a.e.} \}$$
 is a Banach space.
\end{proposition}

\begin{proof}
This is obviously a normed vector space. Let us prove completeness. If $\{ f_n \}$ is a Cauchy sequence, then there exists for every $k$ a function $\overline{f}_k$ such that $f_n\mid_{L_k} \rightarrow \overline{f}_k$ in $BV(L_k)$-norm for $n \rightarrow \infty$. Also, by restricting $f_n\mid_{L_{k+1}} \rightarrow \overline{f}_{k+1}$ to $L_k$ one can conclude $\overline{f}_{k+1}\mid_{L_k} = \overline{f}_k$, hence one can define $f : [\alpha-1, \alpha]\setminus\{0\} \rightarrow \mathbb{C}$ s.t. $f\mid_{L_k} = \overline{f}_k$. Now, $\forall \epsilon > 0$ $\exists N$ $\forall m,n\geq N$ $\forall k \geq K$
$$\Vert f_m - f_n \Vert_{L^1(L_k)} + k^{-\delta} \var_{L_k}(f_m - f_n) \leq \epsilon$$
and by taking the limit for $n \rightarrow \infty$ one has $\Vert f_m - f \Vert_{K, \delta} \leq \epsilon$.
\end{proof}
Note that $\Vert f \Vert_{L^1(I_\alpha)} \leq \Vert f \Vert_{K, \delta}$, and $B_{K, \delta}$ is a $BV$-module, i.e.
$$f \in B_{K, \delta}, g \in BV \Rightarrow fg \in B_{K, \delta}$$
Another useful property of these spaces is the following:

\begin{lemma} \label{powergrowth}
For $K > \max \left\{\frac{1}{\alpha} ,\frac{1}{1-\alpha} \right\}
$ $(K \geq 1$ if $\alpha = 1)$, $0 < \delta < 1$, $\exists A >0$ s.t. $\forall f\in B_{K. \delta}$
$$|f(x)| \leq  \frac{A}{|x|^\delta} \Vert f \Vert_{K, \delta} \qquad \forall x \in [\alpha-1, \alpha]\setminus\{ 0\}$$
\end{lemma}

\begin{proof}
For $f \in B_{K, \delta}$, $x \in L_k^+\setminus L_{k-1}^+$
$$|f(x)| \leq |f(x)-f(\alpha)| + |f(\alpha)| \leq \var_{L_k^+} f + \sup_{L^+_{K}} |f| \leq $$
and since $x \leq \frac{1}{k-1+\alpha}$
$$ \leq k ^\delta \Vert f \Vert_{K, \delta} + \var_{L^+_{K}} f + \frac{ \Vert f \Vert_{L^1(I_\alpha)}}{m(L^+_{K})} \leq \left( \left( \frac{1}{|x|}+1 \right)^{\delta} + K^\delta + \frac{1}{m(L^+_{K})} \right)\Vert f \Vert_{K, \delta}$$
Similarly for $x < 0$.
\end{proof}

Moreover, just as in the case of $BV$, the inclusion $B_{K, \delta} \rightarrow L^1$ is compact.

\begin{proposition} \label{compactlog}
For every $K$ sufficiently large, $\delta >0$, the unit ball
$$ \mathcal{B} = \{ f \in B_{K, \delta} ,  \Vert f \Vert_{K, \delta} \leq 1 \}$$
is compact in the $L^1$-topology.
\end{proposition}

\begin{proof}
This fact is well-known when you consider $BV$ instead of $B_{K, \delta}$. Now, given $\{f_n\} \subseteq \mathcal{B}$, for any $k$ the sequence of restrictions $f_n\mid_{L_k}$ sits inside a closed ball in $BV(L_k)$ hence it has a subsequence which converges in $L^1(L_k)$ to some $F_k \in BV(L_k)$. By refining the subsequence as $k \rightarrow \infty$, one finds a subsequence $f_{n_l} \in \mathcal{B}$ such that for every $k$, $f_{n_l} \mid_{L_k} \rightarrow F_k$ in $L^1(L_k)$ and a.e. for $l \rightarrow \infty$. By uniqueness of the limit there exists $F$ such that $F \mid_{L_k} = F_k$. By lower semicontinuity of total variation, $k^{-\delta} \var_{L_k} F + \Vert F \Vert_{L^1(L_k)} \leq 1$, so $F \in \mathcal{B}$. We are just left with proving $f_{n_l} \rightarrow F$ in $L^1(I_\alpha)$ for $l \rightarrow \infty$. By lemma \ref{powergrowth}
\begin{small}
$$ \int_{I_\alpha} |f_{n_l} - F| \leq \int_{I_\alpha \setminus L_k} |f_{n_l}| + |F| + \int_{L_k} |f_{n_l} - F|  \leq 2 \int_{I_\alpha \setminus L_k} \frac{A}{|x|^\delta} dx + \int_{L_k} |f_{n_l} - F_k|$$
\end {small}
The first term tends to $0$ as $k \rightarrow \infty$ and the second does so for $l \rightarrow \infty$ as $k$ is fixed.
\end{proof}

\subsubsection{Spectral decomposition in $B_{K, \delta}$}

The goal of this section is to prove a spectral decomposition analogous to theorem \ref{spectraldec} in the space $B_{K, \delta}$, namely

\begin{theorem} \label{decBkdelta}
For every $\alpha \in (0,1]$, $0 < \delta < 1$ and $K$ sufficiently large, the transfer operator $\Phi_\alpha : B_{K, \delta}
\rightarrow B_{K, \delta}$ decomposes as
$$\Phi_\alpha = \Pi_\alpha + \Psi_\alpha$$
where $\Pi_\alpha$ and $\Psi_\alpha$ are bounded linear, commuting operators on $B_{K, \delta}$, $\rho(\Psi_\alpha) < 1$ and
$\Pi_\alpha$ is a projection onto a one-dimensional eigenspace.
\end{theorem}

The main ingredient to get the spectral decomposition is again a Lasota-Yorke type estimate:

\begin{proposition} \label{LYlog}
Let $\alpha \in \left(0, 1 \right]$, $0 < \delta < 1$. Then there exist $K \in \mathbb{N}$, $0 < \lambda < 1$, $C > 0$, $D > 0$
such that
$$\Vert \Phi^n_\alpha(f)\Vert_{K, \delta}  \leq C \lambda^n \Vert  f  \Vert_{K, \delta}  + D \Vert f \Vert_{L^1}\qquad \forall f \in B_{K, \delta}$$
\end{proposition}

\begin{proof} First consider the case $\alpha < 1$. By formula \eqref{transfer} and lemma \ref{BVprop}, 4.
$$ \var_{L_k^+} \Phi^n_\alpha(f) \leq \sum_{j \in \mathcal{P}_n} \var_{T^n_\alpha(I_j) \cap L_k^+} \frac{f \circ \sigma_j}{|(T^n_\alpha)' \circ \sigma_j| } +
2 \sup_{T^n_\alpha(I_j)\cap L_k^+} \left| \frac{f \circ \sigma_j}{|(T^n_\alpha)' \circ \sigma_j| } \right| \leq $$
and by lemma \ref{BVprop}, 1. and the fact that $T_\alpha^n: I_j \rightarrow T_\alpha^n(I_j)$ is a homeomorphism
\begin{small}
$$\leq \sum_{I_j \in \mathcal{P}_n} 3 \var_{I_j} \frac{f}{|(T^n_\alpha)'|} +
\frac{ 2 \int_{I_j}|f(x)|dx}{m(T_\alpha^n(I_j) \cap L_k^+)} \leq 3 \sum_{I_j \in \mathcal{P}_n} \var_{I_j} (f g_{n, \alpha}) +
\frac{2  \Vert f \Vert_1}{\inf_{j \in \mathcal{P}_n} \{ m(T_\alpha^n(I_j)\cap L_k^+) \} }$$
\end{small}
where the inf is taken over all non-empty intervals. Now, note that by lemma \ref{BVprop}, 3. and prop. \ref{classeC}, 2.
$$\sum_{I_j \in \mathcal{P}_n} \var_{I_j} (f g_{n, \alpha}) \leq
 \sum_{I_j \in \mathcal{P}_n} \var_{I_j} f \sup_{I_j} g_{n, \alpha} +
\frac{2 \Vert f \Vert_1 }{1-\gamma_\alpha}$$
hence we are left with only one term to estimate: in order to do so, we will split the sum in several parts, according to the filtration $L_k$:

\begin{small}
$$\sum_{I_j \in \mathcal{P}_n} \var_{I_j} f \sup_{I_j} g_{n, \alpha} \leq
\left\Vert g_{n, \alpha} \right\Vert_\infty \sum_{\stackrel{I_j \in \mathcal{P}_n}{ I_j \subseteq L_k }} \var_{I_j} f +
\sum_{\stackrel{h = 1}{\epsilon = \pm}}^\infty \sum_{\stackrel{I_j \in \mathcal{P}_n}{I_j \subseteq L^\epsilon_{(h+1)k}\setminus L^\epsilon_{hk}}} \var_{I_j}f \sup_{I_j} g_{n, \alpha} \leq $$
$$ \leq \gamma_{\alpha}^n \var_{L_k}f  + \sum_{\stackrel{h = 1}{\epsilon = \pm}}^\infty \var_{L^\epsilon_{(h+1)k}} f \sup_{L^\epsilon_{(h+1)k} \setminus L^\epsilon_{hk}} g_{n, \alpha} \leq
\gamma_\alpha^n  k^\delta \Vert f \Vert_{K, \delta} + \sum_{h =1}^\infty \Vert f \Vert_{K, \delta} [(h+1)k]^\delta \sup_{L_{(h+1)k} \setminus L_{hk}} g_{1, \alpha}  \leq $$
\end{small}
and since $L_{(h+1)k} \setminus L_{hk} = \left[ -\frac{1}{hk + \alpha}, -\frac{1}{(h+1)k+\alpha} \right)\cup \left( \frac{1}{(h+1)k + \alpha}, \frac{1}{hk+\alpha} \right]$
$$ \leq \Vert f \Vert_{K, \delta} k^\delta \left(  \gamma_\alpha^n + \sum_{h =1}^\infty \frac{ (h+1)^\delta}{h^2 k^2} \right) \leq \Vert f \Vert_{K, \delta} k^\delta \left( \gamma_\alpha^n + \frac{M}{K^2} \right)$$
for some universal constant $M$ for all $k\geq K$. The same estimate holds for $\var_{L_k^-} \Phi^n_\alpha(f)$.
Moreover, for fixed $n$ and $\alpha$ the set $\{ T_\alpha^n(I_j) \mid I_j \in \mathcal{P}_n \}$ is finite, and since $L_k^+$ and $L_k^-$ are increasing sequences of intervals,
$\inf \{ m(T_\alpha^n(I_j) \cap L_k^\pm) \ : \ I_j \in \mathcal{P}_n, \ k \geq K\}$ is bounded below by a positive constant,
and for every $\alpha$ one can choose $n$ and $K$ such that
$\lambda := 6\left( \gamma_\alpha^{n} + \frac{M}{K^2} \right)< 1$.
By combining all estimates, there exists a constant $D$ such that
$$ \Vert \Phi_\alpha^{n}(f) \Vert_{K, \delta} \leq \lambda \Vert f \Vert_{K, \delta} + D \Vert f \Vert_1 \qquad \forall f \in B_{K, \delta}$$
and by iteration the claim follows. The case $\alpha = 1$ follows similarly; in this case $\Vert g_{1, \alpha} \Vert_\infty = 1$, but
proposition \ref{classeC} is replaced by
$$ \Vert g_{n, \alpha} \Vert_\infty \leq 4 \left( \frac{\sqrt{5}-1}{2} \right)^{2n -4} \qquad \Vert g_{n, \alpha}' \Vert_\infty \leq 2$$
\end{proof}

\textit{Proof of theorem \ref{decBkdelta}.} By propositions \ref{compactlog} and \ref{LYlog},
the transfer operators $\Phi_\alpha$ acting on $B_{K, \delta}$ satisify the hypotheses of
Ionescu-Tulcea and Marinescu's theorem \cite{ITM}, hence we have a spectral decomposition of $\Phi_\alpha$
with a finite number of spectral projectors onto eigenvalues of unit modulus. Moreover,
mixing of $T_\alpha$ still implies there is only one eigenvalue of modulus one and its eigenspace is one-dimensional. \qed

\medskip
Note that since $BV(I_\alpha) \subseteq B_{K, \delta}$, the invariant density $\rho_\alpha$ previously obtained is still a fixed
point of $\Phi_\alpha$, hence $\Pi_\alpha$ is nothing but projection onto $\mathbb{C}\rho_\alpha$.

\subsubsection{End of proof}

The proof of theorem \ref{CLTunbounded} now follows from standard application of the martingale central limit theorem.
We will refer to the version given in (\cite{Viana}, Thm. 2.11). In order to adapt it to our situation, we need the following two lemmas:

\begin{lemma} \label{Edecay}
Let $\alpha \in (0, 1]$, $0 < \delta < \frac{1}{2}$, and $K$ s.t. theorem \ref{decBkdelta} holds,
and consider $f \in B_{K, \delta}$ with $\int_{I_\alpha} f d\mu_\alpha = 0$. Denote by $\mathcal{F}_0$ the Borel $\sigma$-algebra
 on $I_\alpha$ and $\mathcal{F}_n := T_\alpha^{-n}(\mathcal{F}_0)$. Then
$$\sum_{n = 0}^{\infty} \Vert \mathbb{E}(f \mid \mathcal{F}_n) \Vert_{L^2(\mu_\alpha)} < +\infty$$
\end{lemma}

\begin{proof}
$$\Vert \mathbb{E}(f \mid \mathcal{F}_n) \Vert_{L^2(\mu_\alpha)} = \sup \left\{\int_{I_\alpha} (\psi \circ T_\alpha^n) f d\mu_\alpha: \psi \in L^2(\mu_\alpha), \Vert \psi \Vert_{L^2(\mu_\alpha)} = 1 \right\} = $$
$$= \sup\left\{\int_{I_\alpha} \psi \Phi_\alpha^n(f \rho_\alpha) dx : \psi \in L^\infty(\mu_\alpha), \Vert \psi \Vert_{L^2(\mu_\alpha)} = 1 \right\} \leq \frac{\Vert \Phi_\alpha^n(f \rho_\alpha) \Vert_{L^2(dx)}}{\sqrt{\inf \rho_\alpha}}$$
Now, by lemma \ref{powergrowth} and since $0 < \delta < \frac{1}{2}$, $\Vert \Phi_\alpha^n(f\rho_\alpha)\Vert_{L^2(dx)} \leq C \Vert \Phi^n_\alpha(f \rho_\alpha) \Vert_{K, \delta}$, and
by theorem \ref{decBkdelta} $\Phi^n_\alpha(f\rho_\alpha) = \Psi^n(f \rho_\alpha)$ goes to $0$ exponentially fast in $B_{K, \delta}$-norm as $n \rightarrow \infty$.
\end{proof}

\begin{lemma} \label{Bcohom}
Let $f \in B_{K, \delta}$ real-valued, non-constant such that $\int_{I_\alpha} f d\mu_\alpha = 0$. Then there exists no function $u \in B_{K, \delta}$ such that
$$f = u - u \circ T_\alpha \quad \mu_\alpha-a.e.$$
\end{lemma}

\begin{proof}
Notice that $\mu_\alpha$ and Lebesgue measure are abs. continuous w.r.t. each other, hence measure zero sets are the same.
Suppose there exists $u$ which satisfies the equation; then, $u \circ T_\alpha$ belongs to $B_{K, \delta}$.
However,
$$\var_{L_k^+}(u \circ T_\alpha) \geq \sum_{\stackrel{ I_j \textup{ full}}{j \leq k}} \var_{I_j} (u \circ T_\alpha)
= \sum_{\stackrel{ I_j \textup{ full}}{j \leq k}} \var_{(\alpha-1, \alpha)} u = (k - j_{min})\var_{(\alpha-1, \alpha)} u$$
On the other hand, $\var_{L_k^+}(u\circ T_\alpha) \leq k^{\delta}\Vert u\circ T_\alpha \Vert_{K, \delta}$ with $\delta < 1$, which contradicts
the previous estimate unless $\var_{(\alpha-1, \alpha)} u = 0$, i.e. $u$ is constant a.e.
\end{proof}

\textit{Proof of theorem \ref{CLTunbounded}.}
We can assume $\int_{I_\alpha} f d\mu_\alpha = 0$. By (\cite{Viana}, Thm. 2.11) and lemma \ref{Edecay},
the claim follows unless there exists $u \in L^2(\mu_\alpha)$ such that
$$f =  u - u \circ T_\alpha \qquad \mu_\alpha-a.e.$$
If there exists such $u$, one can assume that $\int u \ d\mu_\alpha = 0$, and then, by the proof of (\cite{Viana}, Thm. 2.11), $u$ is given by
$$ u = - \sum_{j = 1}^{\infty} \frac{\Phi_\alpha^j(f \rho_\alpha)}{\rho_\alpha}$$
where convergence of the series is in $L^2(\mu_\alpha)$.
By the spectral decomposition, $\sum_j \Phi^j(f \rho_\alpha)$ converges also in $B_{K, \delta} \subseteq L^2(\mu_\alpha)$. Moreover, since $\rho_\alpha$
is in $BV$ and is bounded from below, then $\frac{1}{\rho_\alpha}$ is in $BV$. Thus,
$u$ lies in $B_{K, \delta}$, and this contradicts lemma \ref{Bcohom} unless $f$ is constant.  \qed

\medskip
Now, the function $x \mapsto \log|x|$ belongs to every $B_{K, \delta}$, therefore

\begin{corollary}
For every $\alpha \in (0,1]$, the Birkhoff averages for the observable $\log|T_\alpha'(x)| = -2\log|x|$ distribute normally around the value $h(T_\alpha)$.
\end{corollary}

\section{Stability of standard deviation} \label{last}

Having established the convergence of Birkhoff sums to a Gaussian distribution, we are now interested in analyzing how the standard deviation of this Gaussian varies when $\alpha$ varies.
 The question is motivated by the numerical simulations in \cite{CMPT}, section 2. We prove the following

\begin{theorem} \label{contsigma}
Let $f : (-1, 1) \rightarrow \mathbb{R}$ of class $C^1$. For every $\alpha \in (0, 1)$ let us consider the variance
$$\sigma_\alpha^2 := \lim_{n \rightarrow \infty} \int_{I_\alpha} \left( \frac{S_n f - n \int_{I_\alpha} f d\mu_\alpha}{\sqrt{n}} \right)^2 d\mu_\alpha$$
Then, for every $\underline{\alpha} \in (0, 1)$
$$\lim_{\alpha \rightarrow \underline{\alpha}} \sigma_{\alpha}^2 = \sigma_{\underline{\alpha}}^2.$$
\end{theorem}

The variance $\sigma_\alpha^2$ of the limit distribution is the second derivative of
the eigenvalues $\lambda_0(\theta)$ of the perturbed transfer operators $\{\Phi_{\alpha, f, \theta}\}$ (see the discussion in section \ref{CLTBVsect}, and
in particular eq. 2. after prop. \ref{variance}). In order to prove the theorem, we will
prove uniform convergence in $\alpha$ of the eigenvalues, via application of theorem \ref{KellerLiv} to the
family $\{\Phi_{\alpha, f, \theta}\}_{\{|\alpha - \underline{\alpha}| < \epsilon, |\theta| < \epsilon ,
\Vert f - \underline{f} \Vert_\infty < \epsilon \}}$.

Hypothesis (1) of thm. \ref{KellerLiv} is easily proved:

\begin{lemma} \label{condition1}
For any $C > 0$, there exists $M > 0$ such that
$$\Vert \Phi_{\alpha, f, \theta}^n \Vert_1 \leq M^n \qquad \forall n \in \mathbb{N} \ \forall \alpha \in (0,1) \ \forall |\theta| < C$$
for every $f \in L^{\infty}(I_\alpha)$ s.t $\Vert f \Vert_\infty \leq C$.
\end{lemma}

\begin{proof}
 For $g \in BV$
$$\Vert \Phi_{\alpha, f, \theta}^n(g) \Vert_1 = \Vert \Phi_{\alpha}^n(e^{\theta S_n f}g) \Vert_1
\leq \Vert e^{\theta S_n f}g \Vert_1 \leq \Vert e^{\theta S_n f} \Vert_{\infty} \Vert g\Vert_1  \leq
e^{n |Re \theta| \Vert f \Vert_{\infty}} \Vert  g \Vert_1 $$
where we used that the unperturbed operators have unit norm on $L^1$.
\end{proof}

Hypothesis (4) follows directly from lemma \ref{Kellerdist}; the precise statement, whose proof we omit, is the following:

\begin{lemma} \label{continuitymodule}
Let $\underline{\alpha} \in (0,1)$. Then there exist $\epsilon, C > 0$ such that for any $f, \underline{f} \in BV([0,1])$ s.t. $\Vert f - \underline{f} \Vert_\infty < \epsilon$, $\forall |\theta| < \epsilon$, $\forall |\alpha - \underline{\alpha}| < \epsilon$
$$ \vert \! \vert \! \vert \Phi_{\underline{\alpha}, \underline{f}, \theta} - \Phi_{\alpha, f, \theta} \vert \! \vert \! \vert \leq  C\left( |\alpha -\underline{\alpha}|^{1/2} + \Vert f - \underline{f} \Vert_\infty\right)$$
\end{lemma}

We now check condition (2), using the estimates in section \ref{LY} to get a Lasota-Yorke inequality which is uniform in both $\alpha$ and $\theta$.

\begin{proposition}
Let $\underline{\alpha} \in (0, 1)$. There exist $0 < \lambda <1$, $\epsilon$, $C_2, C_3$ such that
$$ \var_{I_\alpha} \Phi^n_{\alpha, f, \theta}(g) \leq C_2 \lambda^n \var_{I_\alpha} g + C_3 \Vert g \Vert_1 \quad \forall n \in \mathbb{N} $$
for every $\alpha \in (\underline{\alpha}-\epsilon, \underline{\alpha} + \epsilon)$, for every $|\theta| < \epsilon$ and for every $f \in C^1(I_\alpha)$
with $\Vert f \Vert_{C^1} \leq 1$.

\end{proposition}

\begin{proof}
Let us fix $g \in BV$. We have

$$\var_{I_\alpha} \Phi_{\alpha, f, \theta}^n(g) = \var_{I_\alpha} \Phi_{\alpha}^n(e^{\theta S_n f}g)
\leq \var_{I_\alpha} (e^{\theta S_n f}g \cdot g_{n, \alpha}) = \sum_{j \in \mathcal{P}_n} \var_{\overline{I_j}} (e^{\theta S_n f}g \cdot g_{n, \alpha}) $$
Note that, since $g_{n, \alpha} \mid_{\partial I_j} = 0$,
$$\var_{\overline{I_j}} (e^{\theta S_n f}gg_{n, \alpha}) \leq
\var_{I_j} (e^{\theta S_n f}gg_{n, \alpha}) +
e^{n|\theta| \Vert f \Vert_\infty} \var_{\overline{I_j}} (gg_{n, \alpha})
$$
Now, by lemma \ref{BVprop}, 3
\begin{small}
$$\var_{I_j} (e^{\theta S_n f}g_{n, \alpha}g) = \var_{I_j} \frac{ e^{\theta S_n f}g}{|(T^n_{\alpha})'|} \leq
\sup_{I_j} \left| \left(\frac{e^{\theta S_n f}}{(T_\alpha^n)'} \right)' \right| \int_{I_j} |g| +
\sup_{I_j} \left| \frac{e^{\theta S_n f}}{(T_\alpha^n)'}\right| \var_{I_j} g$$
\end{small}
and by expanding the derivative
\begin{small}
$$ \left( \frac{e^{\theta S_n f}}{(T_\alpha^n)'} \right)' = \frac{ (e^{\theta S_n f})'}{(T_\alpha^n)'} + \left( \frac{1}{(T_\alpha^n)'} \right)' e^{\theta S_n f} = \frac{ e^{\theta S_n f} \theta \sum_{k = 0}^{n-1} (f' \circ T_\alpha^k)(T_\alpha^k)'}{(T_\alpha^n)'}
+ \left( \frac{1}{(T_\alpha^n)'} \right)' e^{\theta S_n f} = $$
$$ = e^{\theta S_n f} \left[ \theta \sum_{k = 0}^{n-1} \frac{(f' \circ T_\alpha^k)}{[(T_\alpha^{n-k})'\circ T_\alpha^{k}]} + \left( \frac{1}{(T_\alpha^n)'} \right)' \right]  \leq
 e^{n |\theta|  \Vert f \Vert_\infty} \frac{ |\theta| \Vert f' \Vert_{\infty} + 2}{1-\gamma_\alpha} $$
\end{small}
Moreover, by the estimates of proposition \ref{LY} (eq. \eqref{finalLY}), for each $\underline{\alpha} \in (0,1)$ and each $n$ there exist $\eta, D$ such that
$$\var_{I_\alpha} (g g_{n,\alpha}) \leq (2n+4)\gamma_\alpha^n \var_{I_\alpha} g + D \Vert g \Vert_1 \qquad \forall \alpha \in (\underline{\alpha}-\eta, \underline{\alpha} + \eta)$$
hence by combining all previous estimates
\small
$$\var_{I_\alpha} \Phi_{\alpha, f, \theta}^n(g) \leq (2n+5) e^{n |\theta|  \Vert f \Vert_\infty} \gamma_\alpha^n  \var_{I_\alpha} g +
 e^{n |\theta|  \Vert f \Vert_\infty} \left( \frac{ |\theta| \Vert f' \Vert_{\infty} + 2}{1-\gamma_\alpha} + D \right) \Vert g \Vert_1$$
\normalsize
and the claim follows by choosing some $n$ large enough and iterating.
\end{proof}

\begin{remark}
Notice this is the only place where we need $f \in C^1$. This is because, if $f \in BV$, $e^{\theta S_n f}$ will not in general be of bounded variation.
\end{remark}

We are now ready to draw consequences for the spectral decomposition: let us denote by $\lambda_{\alpha, f}(\theta)$ the eigenvalue of
$\Phi_{\alpha, f, \theta}$ which is closest to $1$.

\begin{lemma} \label{convergence}
Let $\underline{\alpha} \in (0, 1)$, and suppose we have a family $\{f_\alpha\}_{\alpha \in (0,1)}$ of functions $f_\alpha : [0,1] \rightarrow \mathbb{R}$ of class $C^1$ for every $\alpha$ and such that
\begin{itemize}
 \item[-] $\Vert f_\alpha - f_{\underline{\alpha}} \Vert_\infty \rightarrow 0 \ \textup{for }\alpha \rightarrow \underline{\alpha}$
\item[-] $\sup_{\alpha \in (0,1)} \Vert f'_\alpha \Vert_\infty < \infty$
\end{itemize}
Then there exists $\epsilon > 0$ such that $\lambda_{\alpha, f_\alpha}(\theta)$ converges to $\lambda_{\underline{\alpha}, f_{\underline{\alpha}}}(\theta)$ on $|\theta| < \epsilon$ uniformly in $\theta$ as $\alpha \rightarrow \underline{\alpha}$.
\end{lemma}

\begin{proof}
Let us fix $r \in (\lambda_0, 1)$ and $\delta$ such that $0 < \delta < \frac{1-r}{2}$. Then the projectors
\begin{equation} \label{proj}
\Pi_{\alpha, f_\alpha, \theta} := \frac{1}{2 \pi i} \oint_{\partial B(1, \delta)} (z- \Phi_{\alpha, f_\alpha, \theta})^{-1} dz
\end{equation}
are defined for $|\alpha - \underline{\alpha}| < \epsilon$ and $|\theta| < \epsilon$ for some $\epsilon$ and for $\delta$ sufficiently small rank($\Pi_{\alpha, f_\alpha, \theta}$) = rank($\Pi_{\underline{\alpha}, f_{\underline{\alpha}}, 0}$) = 1 (\cite{KellerLiverani}, cor. 3) so they are all projections on the $1$-dimensional eigenspace relative to the eigenvalue which is closest to $1$. By Dunford calculus we also have
\begin{equation} \label{lproj}
\lambda_{\alpha, f_\alpha}(\theta)\Pi_{\alpha, f_\alpha, \theta} = \Phi_{\alpha, f_\alpha, \theta} \Pi_{\alpha, f_\alpha, \theta} = \frac{1}{2 \pi i} \oint_{\partial B(1, \delta)} z (z - \Phi_{\alpha, f_\alpha, \theta})^{-1} dz
\end{equation}
By thm. \ref{KellerLiv} and proposition \ref{continuitymodule} there exists $C$ such that for $|\underline{\alpha}-\alpha| < \epsilon$ and $|\theta| < \epsilon$
$$ |\!|\!| (z-\Phi_{\alpha, f_\alpha, \theta})^{-1} - (z - \Phi_{\underline{\alpha}, f_{\underline{\alpha}}, \theta})^{-1} |\!|\!| \leq C \left( |\alpha - \underline{\alpha}|^{1/2} + \Vert f_\alpha - f_{\underline{\alpha}} \Vert_\infty \right)^{\eta}$$
with $\eta > 0$ fixed by thm. \ref{KellerLiv} so by eqns \ref{proj} and \ref{lproj}
$$|\lambda_{\alpha}(\theta) - \lambda_{\underline{\alpha}}(\theta)| = O(|\alpha-\underline{\alpha}|^{1/2} + \Vert f_{\alpha} - f_{\underline{\alpha}} \Vert_\infty )^{\eta}$$
uniformly in $\theta$ as $\alpha \rightarrow \underline{\alpha}$.
\end{proof}

\textit{Proof of theorem \ref{contsigma}.}
Let $f_\alpha :[0,1] \rightarrow \mathbb{R}$ be $f_\alpha(x) := f(x+\alpha-1) - \int_{\alpha-1}^\alpha f d\mu_\alpha$.
Since $\tilde{\rho}_\alpha \rightarrow \tilde{\rho}_{\underline{\alpha}}$ in $L^1$ and $f(x+\alpha-1) \rightarrow f(x + \underline{\alpha}-1)$
in $L^\infty$, we have $\int_{\alpha-1}^\alpha f d\mu_\alpha \rightarrow \int_{\underline{\alpha}-1}^{\underline{\alpha}} f d\mu_{\underline{\alpha}}$,
 and the family $\{f_\alpha\}$ satisfies the hypotheses of lemma \ref{convergence}, therefore
$\lambda_{\alpha, f_\alpha}(\theta)$ converges uniformly in a nbd of $\theta = 0$ to
$\lambda_{\underline{\alpha}, f_{\underline{\alpha}}}(\theta)$. Since all $\lambda_{\alpha, f_\alpha}(\theta)$ are analytic in
$\theta$ you also have convergence of all derivatives, in particular $\lambda_{\alpha, f_\alpha}''(0) \rightarrow \lambda_{\underline{\alpha}, f_{\underline{\alpha}}}''(0)$.
We now note that $\int_0^1 f_\alpha(x) \tilde{\rho}_\alpha(x) dx = 0$, which implies, as we have seen in section \ref{spectral},
 that $\lambda''_{\alpha, f_\alpha}(0) = \sigma_{\alpha}^2$.  \qed

\section{Appendix}

Let us recall a few well-known properties of total variation:

\begin{lemma} \label{BVprop}
Let $I \subseteq \mathbb{R}$ be a bounded interval, $J \subseteq I$ a subinterval and $f$ of bounded variation. Then:
\begin{enumerate}
\item $$\sup_{x \in J} |f(x)| \leq \var_{J} f + \frac{1}{m(J)} \int_J |f(x)|dx$$
\item If $g \in BV(J)$, $$\var_J (fg) \leq \sup_{x\in J} |f(x)| \var_J g + \sup_{x \in J} |g(x)| \var_J f$$
\item If $g$ is of class $C^1$ on $J$, $$\var_J(fg) \leq \var_J f \sup_{x \in J}|g(x)| + \sup_{x \in J}|g'(x)|\int_J |f(x)| dx$$
\item
$$\var_I(f \chi_{J}) \leq \var_J f + 2 \sup_{J} |f|$$
\end{enumerate}
\end{lemma}

Let us also prove the basic properties of $T_\alpha$ mentioned in section \ref{cyl}.
\textit{Proof of proposition \ref{classeC}.}
 \begin{enumerate}
  \item
$$  \sup_{j \in \mathcal{P}_1} \sup_{x \in I_j} \frac{1}{|(T_{\alpha})'(x)|} = \sup_{x \in [\alpha-1, \alpha]} x^2 \leq \max\{\alpha^2, (\alpha-1)^2 \} $$
The case for $n > 1$ follows from the chain rule for derivatives.
\item Let $K_n := \sup_{j \in \mathcal{P}_n} \sup_{x \in I_j} \left| g_{n, \alpha}' (x) \right| $. For $n = 1$,
$$K_1 = \sup_{j \in \mathcal{P}_1} \sup_{x \in I_j} \left| \left( \frac{1}{(T_{\alpha})'} \right)'(x) \right| = \sup_{x \in [\alpha-1, \alpha] } 2|x| \leq 2$$

Now,
$$(T_{\alpha}^{n+1})'(x) = (T_{\alpha}^n)'(T_{\alpha}(x))T_{\alpha}'(x)$$
$$(T_{\alpha}^{n+1})''(x) = (T_{\alpha}^n)''(T_{\alpha}(x))[T_{\alpha}'(x)]^2 + (T_{\alpha}^n)'(T_{\alpha}(x))T_{\alpha}''(x)$$
For every $x$ in the interior of some interval $I_j \in \mathcal{P}_{n+1}$,
$$\left|\frac{(T_{\alpha}^{n+1})''(x)}{[(T_{\alpha}^{n+1})'(x)]^2}\right| \leq \left|\frac{(T_{\alpha}^n)''(T_{\alpha}(x))(T_{\alpha}'(x))^2}{[(T_{\alpha}^n)'(T_{\alpha}(x))T_{\alpha}'(x)]^2} + \frac{(T_{\alpha}^n)'(T_{\alpha}(x))T_{\alpha}''(x)}{[(T_{\alpha}^n)'(T_{\alpha}(x))T_{\alpha}'(x)]^2}\right| \leq $$
$$\leq \left|\frac{(T_{\alpha}^n)''(T_{\alpha}(x))}{[(T_{\alpha}^n)'(T_{\alpha}(x))]^2}\right| + \left|\frac{T_{\alpha}''(x)}{[T_{\alpha}'(x)]^2}\right| \frac{1}{|(T_{\alpha}^n)'(T_{\alpha}(x))|} \leq K_n + K_1 {\gamma_\alpha^n}$$
hence $K_{n+1} \leq K_n + 2{\gamma_\alpha^n}$ and by induction $K_{n} \leq \sum_{k = 0}^{n-1} 2{\gamma_\alpha^k} \leq \frac{2}{1-\gamma_\alpha}$.
\item
By induction on $n$: let $I^{-}_{j_M}$ be the interval of the partition $\mathcal{P}_1$ which contains $\alpha -1$ and $I^{+}_{j_m}$ be the one which contains $\alpha$. \\ For $n = 1$, $T_{\alpha}(I_j)= I_{\alpha}$ for $I_j \neq I^{-}_{j_M}, I^{+}_{j_m}$, hence

$$\{ T_{\alpha}(I_j) | I_j \in \mathcal{P}_1 \} \subseteq \{ I_{\alpha}, T_{\alpha}(I^{-}_{j_M}), T_{\alpha}(I^{+}_{j_m}) \} $$

Let $n > 1$; consider an element of the partition $\mathcal{P}_{n+1}$, which will be of the form $I_{j_0} \cap T^{-1}(I_{j_1}) \cap \dots \cap T^{-n}(I_{j_n}) \neq \emptyset$, with $I_{j_0}, \dots, I_{j_n} \in \mathcal{P}_1$. If we let $L := I_{j_1} \cap \dots \cap T^{-(n-1)}(I_{j_n})$, we have $L \neq \emptyset$ and $L \in \mathcal{P}_n$. Moreover, one verifies that
$$T_{\alpha}^{n+1}(I_{j_0} \cap T_{\alpha}^{-1}(I_{j_1}) \cap \dots \cap T_{\alpha}^{-n}(I_{j_n})) \subseteq T_{\alpha}^n(I_{j_1} \cap \dots \cap T_{\alpha}^{-(n-1)}(I_{j_n})) = T_{\alpha}^n(L)$$
At this point we have two cases:
\begin{itemize}
 \item if $T_{\alpha}(I_{j_0}) \supseteq L$ then
$$T_{\alpha}^{n+1}(I_{j_0} \cap T_{\alpha}^{-1}(I_{j_1}) \cap \dots \cap T_{\alpha}^{-n}(I_{j_n})) = T_{\alpha}^n(L)$$
\item otherwise we have $T_{\alpha}(I_{j_0}) \nsupseteq L$ but $T_{\alpha}(I_{j_0}) \cap L \neq \emptyset$ (if the intersection is empty, so it is the interval we started with); since $T_{\alpha}(I_j) = I$ for $I_j \neq I^{-}_{j_M}, I^{+}_{j_m}$, this implies $I_{j_0} \in \{I^{-}_{j_M}, I^{+}_{j_m}\}$. Moreover, because $T_{\alpha}(I^{-}_{j_M})$ and $T_ {\alpha}(I^{+}_{j_m})$ are intervals with supremum equal to $\alpha$, there exists at most one interval $I_{\mu}$ of the partition $\mathcal{P}_n$ s.t. $T_{\alpha}(I^{+}_{j_m}) \cap I_{\mu} \neq \emptyset$ and $T_{\alpha}(I^{+}_{j_m}) \nsupseteq I_{\mu}$; in the same way there exists only one interval $I_{\nu}$ of the partition $\mathcal{P}_n$ such that $T_{\alpha}(I^{-}_{j_M}) \cap I_{\nu} \neq \emptyset$ and $T_{\alpha}(I^{-}_{j_M}) \nsupseteq I_{\nu}$, therefore either $L = I_{\mu}$ or $L = I_{\nu}$.
\end{itemize}
In conclusion $\{ T_{\alpha}^{n+1}(I_{j}) \mid I_j \in \mathcal{P}_{n+1} \}$ is contained in
\begin{small}
$$\{ T_{\alpha}^n(I_j) \mid I_j \in \mathcal{P}_{n} \} \cup T_{\alpha}^{n+1}(I^{+}_{j_m} \cap T_{\alpha}^{-1}(I_{\mu})) \cup T_{\alpha}^{n+1}(I^{-}_{j_M} \cap T_{\alpha}^{-1}(I_{\nu}))$$
\end{small}
hence at every step the cardinality can only increase by at most $2$.
\item Recall that
$$g_{1, \alpha}(x) := \left\{
\begin{array}{ll} x^2 & \textup{if }x \textup{ belongs to some }I_j \\
		  0 & \textup{otherwise}

\end{array}
\right.$$
hence the claim follows from sommability of the series $\sum \frac{1}{k^2}$.
\end{enumerate}
\qed

redpajama_set_name: RedPajamaArXiv


text:
Zena Carp
Zena Carp, 81, of Bloomfield, beloved wife for 61 years of Alan Carp, passed away peacefully on Friday, September 20, 2019 at Hartford Hospital with family by her side. Born December 16, 1937, in New York, she was the daughter of the late Nathan Seril and Sarah Wedeen. Zena graduated from the prestigious Hunter College High School before attending Ohio State University. She was the membership director of the Greater Hartford Jewish Community Center for many years and she also enjoyed positions in numerous retail shops. Zena was known for her love of cooking and entertaining. She had a passion for Broadway shows; was a voracious reader; enjoyed the weekly activities of completing the Sunday NY Times crossword puzzle (in ink) and playing Mah Jong with her many friends. She always looked forward to the next travel destination and just last year took a European cruise with her lifelong friend, Michaeline. Her greatest joy was her family. In addition to her husband, she leaves behind two daughters – Sheryl Carp and son-in-law Ken McIntyre of New York, and Laura Rosenblatt of Simsbury; her adored grandchildren (who affectionately called her Mimi) Jared, Aaron and Rachel; sisters Barbara Seril and Pearl Pell, both of New York. She was predeceased by her sister June Watkins. Zena shared a special relationship with her nephew, Steven Watkins and his wife, Pam. Jared's girlfriend, Jessica Tucker, has been like family and held a special place in Zena's heart. Zena's family would like to extend their greatest appreciation to her live-in caregiver, Sudeen Williams, whose extraordinary care and compassion were a source of great comfort to both Zena and her family. They would also like to express gratitude to Amy and Darcy, Zena's nurses at the Farmington Valley VNA, as well as the exceptional care provided by the staff at Hartford Hospital. Graveside funeral services will be held at Fairview Cemetery, 200 Whitman Ave., Section 13, West Hartford, TODAY, Sunday, September 22, 2019 at 11 am. Following interment, the family will receive friends at Alan and Zena's home in Bloomfield. In lieu of flowers, donations may be made to a charity of one's choice. May her memory be a source of strength and a blessing. Funeral arrangements provided by Hebrew Funeral Association, West Hartford.
Hall Grad Leaves Journalism, Becomes Farmer

redpajama_set_name: RedPajamaCommonCrawl


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text:
Diagnosing Nutrients and Soil Problems
Correcting Nutrient Problems
Correcting Soil Problems
Following are some basic concepts about fertilizing olive trees. For more in-depth information, we recommend the books and articles by Paul Vossen, as well as the book by G. Steven Sibbett and Louise Ferguson, all listed in our Sources below.
How do olive trees respond to over-fertilization?When it comes to fertilizing olive trees, more is not always better. In fact, more fertilizing often creates negative results. "Olive trees are not big feeders", says Paul Vossen. "They are semi-wild, hardy, tough plants that will tolerate poor growing conditions, especially low fertility, better than most plants. They also tolerate a very wide range of soil pH." In fact, olive trees tend to grow too vigorously in very fertile soils. They grow too tall and produce little fruit. Often, when pruned to limit their growth, they respond by sending long and vigorous non-fruiting shoots. The oil quality may be poorer in very fertile soils.
Paul Vossen also observes that a common mistake of new growers is to over-fertilize when their trees are not growing adequately, rather than make sure that irrigation and weed control are adequate. Olive trees respond much better to correct irrigation (enough but not too much) and weed control than to fertilization. There are many myths and exaggerated claims (sometimes encouraged by fertilizer manufacturers) that fertilizing will magically cure or prevent all sorts of problems and result in better quality olives and olive oil. It is often the opposite, however. In order to know when fertilizer is needed, observation and analysis are vital.
DIAGNOSING NUTRIENT AND SOIL PROBLEMS
Three general categories of observations can be used: visual symptoms, plant or tissue analysis, and soil and water analysis.
Visual Symptoms
Diagnosis from visual symptoms alone can be difficult. A tree may have already suffered in growth or in yield by the time visual symptoms appear. Problems other than nutrition may cause similar problems. Borderline deficiencies or deficiencies involving more than one element may be difficult to diagnose. It is best used in conjunction with plant or soil analysis.
Plant or Tissue Analysis
Olive leaves are sampled and chemically analyzed for mineral deficiencies and toxicities. A leaf's mineral composition depends on its maturity, current climactic conditions, availability of mineral elements in the soil, cultural practices, and other factors. The mineral nutrient level in the leaf integrates all these factors and thus reflects the nutrient status of the tree.
Although there are exceptions, the general rule is that nutrient deficiencies are best measured from late June through early August. The concentration of mineral nutrients in leaves changes as the leaves first emerge and then expand to full size. For many elements, the smallest change in concentration occurs during that period. Olive leaf samples should be taken then because critical levels have been established for that time period.
Leaf samples should be taken from the middle of non-bearing, current season shoots. A sample of 80 to 100 is sufficient. Ideally, a sample should be taken from similar trees. Samples from different varieties, or from parts of the orchard with different soils, microclimates, or irrigation systems should be sampled separately. Samples should consist of a few leaves of as many similar trees as possible, selected at random throughout the orchard. Avoid any leaves that are abnormal in appearance or from abnormal trees unless this is the specific problem to be solved. In this case, the abnormal leaves or trees should become a separate sample.
Olive trees, like most tree crops, have extensive root systems that occupy a larger volume of soil than those of most annual crops. Soil can vary widely within such an area. Thus it may be difficult to take a soil sample that accurately represents the root area and nutrient levels that the root extracts. In general, soil analysis is not accurate enough to diagnose fertility in olive trees.
It is a good idea, however, to do soil and water analyses before planting a new orchard as discussed in our Soil and Water Analysis page. It is also useful in mature orchards if a soil problem is suspected. It can help diagnose toxicities from excessive concentration of sodium, chlorine, and boron, as well as other problems, which helps in planning corrective procedures.
In general, soil samples can be taken at any time, as soil characteristics and nutrient levels are relatively stable. There are a few exceptions. Nitrogen levels can be depleted after a long wet season. In many soils, nitrates, chlorides, and boron can be leached by winter rains and irrigation. The sampling procedures depend on the problem and area involved. Is one tree affected or many? Is there a visible pattern? Ideally, 3 to 10 spots in a site should be sampled. Because soils differ in composition at different depths, the top 6 to 12" (15 to 30cm) should be a separate sample, as well as each subsequent foot downward. Samples taken from different distances from the trunk may be combined, but different soil depths should be separate. Samples should represent the effective rooting zone. A soil auger may be used to obtain samples. Generally, about a quart (1L) of soil per sample is adequate. The testing lab will often provide an interpretation of the results as well as suggestions for corrective action.
A lack of nitrogen (N) is the only common deficiency in olives. Potassium (K) and boron (B) deficiencies are quite uncommon. Deficiencies of other nutrients are rare.
Nitrogen levels directly affect fruit set, yield, and shoot growth. It is important to remember, however, that both nitrogen deficiencies and excesses can hurt yields. The goal is to maintain leaf nitrogen levels of 1.5 to 2.0%. Deficiency level is less than 1.4%. In deficient trees, the leaves are small and yellowish. Shoot growth is less than 8". These symptoms often occur in soils that are cold and wet during the winter when nitrogen is not as readily available, but disappear in the early summer. Nitrogen can be applied either with organic or with conventional fertilizers.
Organic applications can be done either with certified organic materials such as feather meal, blood meal, or compost, or with a leguminous cover crop. Rates of applications should be approximately 40 to 50 pounds of actual nitrogen per acre, per year, in a mature orchard, taking into account the rate of nitrogen in the fertilizer. Organic fertilizers such as compost can take quite a while to decompose and release the nitrogen in a form that the plants can absorb. They decompose over a period of about 15 years with the most intensive release during the first year or two. Annual applications over many years can build up high nitrogen levels.
Conventional nitrogen fertilizers such as urea, ammonium nitrate, ammonium sulfate, potassium nitrate, or calcium nitrate also work. The important thing about nitrogen is to make periodic applications in accordance with leaf analysis and adequate shoot growth of between 8 and 20" (203 to 508mm). It is also important to note that nitrogen can carry over in the soil for several years, especially in heavy clay. In sandy soils, it tends to leach and can cause ground water and run-off pollution, so make sure not to apply too much. It may not need to be applied each year.
For dry farmed trees, it should be put on just before a rain in mid to late winter. Under irrigated conditions, it can be applied periodically throughout the growing season and watered in. For drip irrigation, it can be placed right under or through emitters.
Note that olive trees cannot tell if the nitrogen comes from an organic source or from a conventional fertilizer. From a chemical point of view, it is all the same to the tree. There are different reasons to choose one or the other, however. Organic materials have the characteristic (which can be desirable or not) of being released slowly. They are less likely to leach into ground or surface water. Conventional sources are generally cheaper and easy to handle. But the differences between the two are getting more subtle with the availability of slow release conventional fertilizers and high analysis organic fertilizers which are easy to apply. Costs do vary considerably, however.
Low levels of potassium in olive leaves and deficiency symptoms have been observed only in rare cases. Note that soil levels of potassium do not correlate well with leaf nutrient levels or deficiency symptoms. Potassium levels below the critical leaf level can occur long before any symptom appears. Adequate concentration is over 0.8%. There is a deficiency at less than 0.4%. Deficient leaves are light green, with tip burn, and there may be dead areas in the trees. Similar symptoms sometimes appear because of poor soil drainage.
Most potassium fertilizers are mined form natural sources and classified as organic. The rate of application to correct a deficiency is generally 10 to 20 lbs of potassium per tree at the drip line. It can be dug in to be more effective or applied right under the drip emitters or through the drip system in soluble forms. Composts and other organic fertilizers all contain some potassium, so regular application would most likely never allow a deficiency to arise.
Boron deficiencies are rare as well. In California, they have been observed in Butte County. Olive trees are much less sensitive to high boron levels than most other crops. Adequate boron concentration is 19-150ppm. The tree is deficient below 14ppm and there is an excess concentration above 185ppm. In deficient trees, the fruit is misshapen ("monkey face"). The growth is short and branched. There is limb dieback, rough bark, and small leaves with top dieback.
Deficiencies can be corrected by applying a pound of borax to each tree, which will last many years. It is important not to apply too much because of the risk of toxicity.
Phosphorus deficiencies have been observed in Europe. Adequate concentration is 0.1-0.3%.
Also see this UC Davis Overview of the new California Fertilization Guidelines funded by a grant from the California Department of Food and Agricultural Fertilizer Research and Education Program (CDFA-FREP).
A Note about Foliar Sprays
Beware of foliar sprays. All the necessary nutrients for olive trees are available via ground application, and the effect is much more lasting. Research trials have proven that all nutrients can be taken up by olive roots. Foliar sprays of nitrogen, boron, and potassium will be rapidly taken up by the tree. Nutrient levels will increase dramatically in the leaves, but only for a short while, usually just a few weeks. There is no measurable effect on shot berries, fruit set, shoot growth, number of flowers, number of perfect flowers, fruit yield, or fruit size. All of this has been measured in University of California trials. Foliar sprays also have had no effect on alternate bearing in various experiments around the world. Foliar applied compost tea will not control or prevent peacock spot on olives, nor has it been shown to have any other positive effects. The best plan is to have adequate nutrition for the tree through ground application. Foliar sprays can sometimes be useful if a quick and short-lasting effect is needed.
Various soil problems identified through soil analyses can be corrected or at least diminished. Following are the most common soil problems and corrective measures with amendments.
Correcting Soil pH
As noted above, olive trees are quite tolerant when it comes to soil pH. When soils are overly acidic, lime is commonly used to correct the pH. The amount required varies with soil texture. The approximate amount of finely ground limestone needed to raise the pH of a 7-inch (18.cm) layer of soil by one pH unit from an initial pH of 4.5 or 5.5 ranges from about ½ ton per acre for sandy soil to about 2 tons for a clay loam. Usually, only the surface becomes acidic enough to require liming.
Compensating for Low Cation Exchange Capacity
The cation exchange capacity (CEC) indicates the ability of the soil to hold many mineral nutrients against leaching. These nutrients are absorbed by tree roots from the soil solution. Soils with a higher CEC are typically more fertile, as they have a greater capacity for mineral nutrients. The CEC depends directly on a soil's clay and humus content. At one end of the spectrum, a sandy or loamy sand soil has a CEC of 2 to 7 milliequivalents (meq)/100g. Clay or peat soils have a CEC greater than 40 meq/100g. As a rule of thumb, fertilizers should be applied more frequently and in smaller amounts to soils with lower CEC.
Correcting Sodic (Alkali) Soils
Soils that contain excessive amounts of exchangeable sodium in proportion to calcium and magnesium are termed sodic or alkali soils. They are characterized by a dispersion of soil particles that reduces the soil permeability to water and air. By definition, a sodic soil has an exchangeable sodium percentage (ESP) of greater than 15. This means that 15% of the soil exchange capacity is associated with sodium, and the rest with calcium, magnesium, and other cations. Olive trees are affected when ESP levels reach 20 to 40. Soils with this problem are generally planted with annual crops.
This can be corrected by the application of gypsum. How much can be determined by a lab analysis. After the gypsum is applied, the displaced sodium must be leached below the root zone. Organic materials such as manure, cover crop, or crop residues may help improve the soil structure for leaching. In established orchards, heavy irrigation during the dormant period minimizes the damage to tree roots from lack of aeration.
There is a close association between the composition and concentration of soil salts and salts in irrigation water. When used for irrigation, water with high sodium relative to calcium and magnesium is likely to result in a sodic soil. This is something to watch for.
Correcting Water Penetration Problems
Irrigating with water that is very pure (low salt) may slow infiltration into sandy loam or fine-textured soils. It can drop to less than 0.1" (2.5mm) per hour, making it difficult to satisfy the orchard's water requirements. Gypsum application can help correct this water penetration problem. The infiltration rate can be increased by as much as fivefold by applying 1 to 2 tons of gypsum per acre in late spring or early summer, just before peak evapotranspiration. Gypsum is generally beneficial for three to five irrigations. Infiltration can also be improved by mixing gypsum directly into the irrigation water with specially designed equipment. Manure applied to the wetted areas can add soluble salts to the irrigation water.
Paul Vossen: Fertilizing Olive Trees
Paul Vossen: What is a "Chemical" Fertilizer
Recommended Guidelines from UC Davis