text:
\section{Introduction}
\label{chapter renormalization for q.p. forced maps}

This is the first of a series of papers (together with \cite{JRT11b,JRT11c})
proposing an extension of the one dimensional renormalization
theory for the case of quasi-periodic forced maps. Each of these
papers is self contained, but highly interrelated with the others. A more
detailed exposition can be found in \cite{Rab10}.
In this paper we give a concrete definition of the operator to the case of
quasi-periodic maps and we use it to prove the existence of
reducibility loss bifurcations when the coupling parameter goes to
zero, like the ones observed in \cite{JRT11p, JRT11p2}
for the Forced Logistic map.
In \cite{JRT11b} we will use the results developed here to study the
asymptotic behavior of these bifurcations when the period of the attracting set
goes to infinity. Our quasi-periodic extension of the renormalization
operator is not complete in the sense that
several conjectures must be assumed. In \cite{JRT11c} we include
the numerical evidence which support these conjectures and we show
that the theoretical results agree with the behavior observed numerically.

The classic one dimensional renormalization theory was motivated to explain
the cascades of period doubling bifurcations.
The paradigmatic example in the case of unimodal maps is the
Logistic Map,
but the properties of renormalization and universality
are also observable in a wider class of maps.
Concretely, given a typical one parametric family for unimodal maps
$\{f_\alpha\}_{\alpha\in I}$  one  observes
numerically that there exists a sequence of parameter
values $\{d_n\}_{n\in \mathbb{N}} \subset I$ such that
the attracting periodic orbit of the map undergoes a period doubling
bifurcation. Between one period doubling and the next one there
exists also a parameter value $s_n$, for which the critical point
of $f_{s_n}$ is a periodic orbit with period $2^n$. One can
also observe that
\begin{equation}
\label{universal limit sumicon}
\lim_{n\rightarrow \infty} \frac{d_n - d_{n-1}} {d_{n+1} - d_{n}} =
\lim_{n\rightarrow \infty} \frac{s_n - s_{n-1}} {s_{n+1} - s_{n}} =
\text{\boldmath $\delta$}  = \texttt{ 4.66920...}.
\end{equation}

Moreover, the constant $\text{\boldmath $\delta$}$ is universal,
in the sense that for any family  of unimodal maps with a quadratic
turning point having a cascade of period doubling bifurcations,
one obtains the same ratio $\text{\boldmath $\delta$}$. For technical
reason the discussion is typically focussed
around the values $s_n$.

The renormalization theory for unimodal one dimensional maps
was originated by the seminal works of Feigenbaum
(\cite{Fei78,Fei79}) and Collet and Tresser (\cite{CT78}) who independently
proposed the renormalization operator to explain the universal
behavior observed in the cascades of bifurcations of
one dimensional maps, see \cite{MvS93} for a review.
Let us do a quick summary of the theory. The (doubling) renormalization
operator, which is denoted by $\mathcal{R}$, is defined in the space of unimodal maps
as the self composition of the map composed with a change of
scale (see subsection \ref{subsection Set up 1-d renor} for more
details). There are some basic assumptions on the dynamics of the operator
$\mathcal{R}$ (known as the Feigenbaum conjectures) which give a suitable
explanation to the universality described before.
The first of these conjectures is that the operator has a fixed
point $\Phi$ and it is differentiable in a neighborhood of $\Phi.$
The second conjecture is that the spectrum of $D\mathcal{R}(\Phi)$ has a
a unique real eigenvalue $\text{\boldmath $\delta$}$ bigger than one, and the rest of
eigenvalues are strictly smaller than one. Then one has that
the unstable manifold $W^u(\Phi,\mathcal{R})$ has dimension one and the stable manifold
$W^s(\Phi,\mathcal{R})$ has codimension one.

On the other hand, one has that the values $\alpha=s_n$
where the critical point
of a map $f_{s_n}$ has period $2^n$ correspond to the
parameter values where the family $\{f_\alpha\}_{\alpha\in I\subset\mathbb{R}}$
intersects certain codimension one manifolds $\Sigma_n$.
Moreover one has that $\mathcal{R}(\Sigma_n) \subset \Sigma_{n-1}$ .
The third conjecture claims that these manifolds intersect
transversally $W^u(\Phi,\mathcal{R})$. Then one has that they accumulate to
$W^s(\Phi,\mathcal{R})$ with ratio $\text{\boldmath $\delta$}$ in a neighborhood $U$ of $\Phi$.
Then, for any one dimensional family $\{f_\alpha\}_{\alpha\in I}$ intersecting
$W^s(\Phi,\mathcal{R})\cap U$ transversally, the family has
a sequence $\{s_n\}_{n\in \mathbb{N}}$ of parameters where the critical
 point of the map $f_{s_n}$ is periodic with
period $2^n$. Hence, applying the $\lambda$-lemma, this parameter values
satisfy the asymptotic behavior
given by the equation (\ref{universal limit sumicon}), with $\text{\boldmath $\delta$}$ the
unstable eigenvalue of $D\mathcal{R}(\Phi)$.

The first proofs of the Feigenbaum conjectures were done  with
computer assistance (\cite{Lan82,EW87}). Later on completely conceptual proofs
appeared (\cite{Sul92,Lyu99}), all of them for the case of analytic maps.
For studies of the operator in the $C^r$ context see \cite{FEM06} and
\cite{CMMT09}. Our extension to the quasi-periodic case does
not cover all the theory exposed in the cited works.

\begin{figure}[t]
\centering
\resizebox{11cm}{!}{
\input{EsquemBifPar.pdf_t}
\phantom{aaaaa}
}
\caption
Schematic representation of the bifurcations diagram
of the Forced Logistic Map, see \cite{JRT11p} for the numeric
computation of this diagram.}
\label{Esquema BifPar}
\end{figure}

The paradigmatic example in this paper is the Forced Logistic Map (FLM for
short). Nevertheless the obtained results are applicable to a wider class
of maps. The FLM is a map in the cylinder $\mathbb{T}\times \mathbb{R}$
defined as
\begin{equation}
\label{FLM}
\left.
\begin{array}{rclcl}
\bar{\theta} & = & r_\omega(\theta) & = & \theta + \omega  ,\\
\bar{x} & = & f_{\alpha,\varepsilon} (\theta,x) & = & \alpha x(1-x)(1+ \varepsilon \cos(2\pi \theta)) ,
\end{array}
\right\}
\end{equation}
where $(\alpha,\varepsilon)$ are parameters and $\omega$ a fixed Diophantine
number.
The dynamics on the periodic component is a rigid rotation
and the dynamics on the real component is the Logistic Map
plus a perturbation depending on the periodic one. Sometimes the
FLM is also defined with $f_{\alpha,\varepsilon} (\theta,x) =
\alpha x(1-x) +  \varepsilon \cos(2\pi \theta)$. The results in this
paper applies to both cases.

The FLM map appears in the literature in different
contexts, usually related with the destruction of invariant curves,
see \cite{JRT11p} and references therein.
Concretely, we are interested on the truncation of the
period doubling cascade. As discussed above, the Logistic Map
exhibits an infinite cascade of period doubling bifurcations
which leads to chaotic behavior. For zero coupling ($\varepsilon=0$),
these periodic orbits become invariant curves of the FLM (provided the rotation
number $\omega$ is irrational). But when the coupling
parameter is different from zero, the number of period doubling
bifurcations of the invariant curves is finite.

We studied numerically this phenomenon in \cite{JRT11p}.
Concretely,  we computed some bifurcation diagrams in terms
of the dynamics of the attracting set, taking
into account different properties,  as the Lyapunov exponent
and, in the case of having a periodic invariant curve, its
period and its reducibility. In the case of analytic maps in
the cylinder, the reducibility
loss of an invariant curve can be characterized as a bifurcation
(see definition 2.3 in \cite{JRT11p}). But the reducibility loss
is not a bifurcation in the classical sense because there are not
visible changes in the phase space, only the spectral properties
of the transfer operator associated to the continuation of that
curve changes (see \cite{JT08}). Despite of this, we will consider
it as a bifurcation for the rest of this paper. The numerical
computations in the cited work reveal that the parameter
values for which the invariant curve doubles its period are
contained in regions of the parameter space where the invariant
curve is reducible. As before, let $s_n$ be the parameter values where
the critical point of the uncoupled family is periodic with
period $2^n$. The numerical computations also revealed that from every
parameter  value $(\alpha,\varepsilon) = (s_n, 0)$
two curves are  born. These curves correspond to a
reducibility-loss bifurcation of the $2^n$-periodic invariant curve.
The scenario is sketched in figure \ref{Esquema BifPar}.

Assume that these two curves can be locally expressed as
$(\alpha_n^+(\varepsilon),\varepsilon)$  and $(\alpha_n^-(\varepsilon), \varepsilon)$
with $\alpha_n^+(0)= \alpha_n^-(0)=s_n$.
In theorem \ref{thm existence of non-reducibility direction} we
prove that these curves really exist for suitable
families of maps. Moreover, we give the values of
$\frac{d}{d \varepsilon} \alpha ^{+}_n(0)$ and $\frac{d}{d \varepsilon} \alpha ^{-}_n(0)$
in terms of the iterates of the renormalization operator.
To achieve this result we need to assume that the quasi-periodic
renormalization operator is injective. This assumption will be
called Conjecture {\bf \ref{conjecture H2}} which is
stated in section \ref{section boundaris of reducibility}.
This conjecture will be supported numerically in \cite{JRT11c}.

The paper is structured as follows. In section \ref{section definition
and basic properties} we propose a definition for the q.p. renormalization
operator and we study the operator as a map on the Banach space of
q.p. forced unimodal maps. Among other results, we prove that the
fixed point of the one dimensional renormalization operator
extends to the quasi-periodic one and we compute and
study its derivative. In section \ref{chapter application} we consider
certain codimension one manifolds, which correspond to
the bifurcation manifold associated to the reducibility loss of the
$2^n$ periodic invariant curve of the system. We
relate these manifolds for different values of $n$ by
means of the renormalization operator. Then we consider
a generic two parametric family such that it becomes
a full family of renormalizable one dimensional maps when
one of its parameters is equal to zero. We use the q.p. renormalization
theory to prove the existence of reducibility loss bifurcations
for the family. We also include an appendix where we
analyze the minimum function as a functional operator from the space of functions
$f:\mathbb{T}\rightarrow \mathbb{R}$  to $\mathbb{R}$, which is necessary for our discussion.

\section{Definition of the operator and basic properties}
\label{section definition and basic properties}

Consider a q. p. forced map as follows,
\begin{equation}
\label{q.p. forced system interval}
\begin{array}{rccc}
F:& \mathbb{T}\times I &\rightarrow & \mathbb{T} \times I \\
  & \left( \begin{array}{c} \theta \\ x \end{array}\right)
  & \mapsto
  & \left( \begin{array}{c} \theta + \omega \\ f(\theta,x) \end{array}\right),
\end{array}
\end{equation}
where $I=[-1,1]$ and $f\in C^r(\mathbb{T}\times I ,I)$.
To define the renormalization operator we only require $r \geq 1$,
but in this work we focus on the simplest case of analytic functions.

Let us remark that, in this section,  no additional assumptions will be done on
$\omega$.
The aim of this section is to define the quasi-periodic renormalization
operator. As long as the dynamics of the map $F$ is not considered, it
is not necessary any additional requirement on $\omega$. In section
\ref{chapter application} we will require $\omega$ to be Diophantine. But for
this section it is advantageous to define
the operator for any $\omega\in \mathbb{T}$, since then the operator will depend
continuously on $\omega$.

The definition of the renormalization operator will be done from a
perturbative point of view. In other words, we will consider a
map $F$ like (\ref{q.p. forced system interval})
 such that $f(\theta,x)= f_0(x) + h(\theta,x)$, with $f_0$ a
unimodal map on $\mathcal{D}(\mathcal{R})$ the domain of the renormalization operator. In
this section we will see that if $h$ small enough
(in $\|\cdot\|_\infty$ norm), then we can define a ``renormalization'' of $f$.

In section \ref{subsection Set up 1-d renor} we introduce the setup
of the one dimensional renormalization operator that we
consider in this paper.
In section \ref{section definition of q.p. renormalization}
we define the
renormalization operator for q.p. forced maps and we will check that the
definition is consistent.
In section \ref{subsection study of TT} the basic properties of the operator
are studied. Concretely, we  check that a fixed point
of the one dimension renormalization operator extends to a fixed point
of the q.p. one and  we will also check that the operator is differentiable
in a neighborhood of the fixed point.
In subsection \ref{section The Fourier expansion of DT}
we study the derivative of the operator with respect
to the Fourier expansion of the function to which the operator is applied.

\subsection{Setup of the one dimensional renormalization operator. }
\label{subsection Set up 1-d renor}

We introduce here the precise definition of the
one dimensional renormalization operator.
The approach chosen here is due to its
simplicity, which makes easier to adapt to the quasi
periodic case. Concretely we follow \cite{Lan82}, but we do
a slight modification on the domain of the operator
for technical reasons. For a set up in a much
more general context see \cite{Sul92, MvS93, Lyu99}
and for more recent works on renormalization of one-dimensional maps
see \cite{FEM06} and \cite{CMMT09}.

For the understanding of this section it is advisable to have
some familiarity with the definition of the (doubling) renormalization
operator given in \cite{Lan82}. Note that the definition given there
is for even maps defined on the interval $[-1,1]$, such that the
turning point is $0$ and it is mapped to $1$.  Given a skew map
$F$ like  (\ref{q.p. forced system interval}), we
want to define the renormalization of the map in a similar way
to the one-dimensional case. That is, to give some
generic conditions on $F$ in such a way that
it has a two periodic invariant subset, and such that $F^2$ restricted
to this subset is affinely conjugate to a map in the same class of
functions of $F$. Note that the $\theta$-component of $F$, when
$\omega$ is irrational,  does not allow the map to have invariant
subsets in this component. Moreover we want
the renormalization of $F$ to have a rigid rotation in the
periodic component. Then the affine conjugacy
should be of the form $A(\theta, x)= (\theta, a x)$, with $a$ a real number.
Note that the skew map $A^{-1}\circ F\circ F \circ A$ has
rotation number equal to $2\omega$ and is defined by
the function $\frac{1}{a} f(\theta +\omega, f(\theta, a x))$,

Suppose that we have $g$
a renormalizable one dimensional map
and $h \in C^{r}(\mathbb{T}\times I, I)$ such that its supremum norm $\|h \|$
is small. Then we would like to consider a map $F$ like
(\ref{q.p. forced system interval})
defined by  the function
\[f(\theta,x) = g(x) + h(\theta,x).\]

Note that the definition of the renormalization operator
given in \cite{Lan82} is for even maps defined on the
interval $[-1,1]$, such that the turning point is $0$ and it is mapped to $1$.
If we want $F:\mathbb{T} \times I \rightarrow \mathbb{T} \times I$ to be well defined
we should have $f(\mathbb{T}\times I) \subset I$.
Although we allow $\|h\|$ to be small,
we should require  $h(\theta,0)$ to be negative for any $\theta\in\mathbb{T}$.
This would make the construction quite artificial and not
applicable to the general q.p. forced maps like the FLM.
A solution  to this problem
is to replace the interval $I=[-1,1]$ by a wider one
$I_\delta=[-1-\delta, 1+\delta]$, but then
we have to check that the one dimensional renormalization operator
can be extended to this new domain.

\begin{figure}[t]
\centering
\resizebox{7.5cm}{!}{\input{MapLanf3.pdf_t}}
\caption{Given a function $\psi \in \mathcal{M}_\delta$, in the picture
we show the geometrical meaning of the new constants $a'$, $b'$, $\delta$ and
$\gamma$.}
\label{MapLanf modified}
\end{figure}

Given a small value $\delta$, let  $\mathcal{M}_\delta$ denote the space
of continuously differentiable even maps $\psi$ of the interval
$I_\delta=[-1-\delta,1+\delta]$ into itself such that
\begin{enumerate}
\item $\psi(0)=1$,
\item $x \psi'(x) <0$ for $x\neq 0$.
\end{enumerate}

Set $a=\psi(1)$, $a'= (1+\delta)a$ and $b'=\psi(a')$. In
figure \ref{MapLanf modified} one can see an example of a function in
$\mathcal{M}_\delta$ where these values are shown. Now,
we define $\mathcal{D}(\mathcal{R}_\delta)$ as the set of $\psi \in \mathcal{M}_\delta$
such that
\begin{enumerate}
\item  $a<0$
\item  $1> b'>-a'$,
\item  $\psi(b') <- a'$.
\end{enumerate}

\begin{rem}
\label{remmark open domain 1d}
Note that the values $a$, $a'$ and $b'$ can be seen as continuous
functions from $\mathcal{M}_\delta$ to $\mathbb{R}$, and therefore the set $\mathcal{D}(\mathcal{R}_\delta)$
is open in $\mathcal{M}_\delta$ (with the $C^k$ topology).
\end{rem}

We define the renormalization operator, $\mathcal{R}_\delta :
\mathcal{D}(\mathcal{R}_\delta) \rightarrow \mathcal{M}_\delta$ as
\begin{equation}
\label{renormalization operator lanford}
\mathcal{R}_\delta(\psi) (x) =  \frac{1}{a} \psi \circ \psi (a x).
\end{equation}
where $a=\psi(1)$.

\begin{prop}
\label{proposition RRdelta well defined}
The operator $\mathcal{R}_\delta$ is well defined, in the sense that
$\mathcal{R}_\delta(\psi)$ belongs to $\mathcal{M}_\delta$ for any $\psi \in \mathcal{D}(\mathcal{R}_\delta)$
\end{prop}

\begin{prop}
\label{proposition fixed points extend to RRdelta}
Any fixed point $\Phi\in \mathcal{M}_0$ of $\mathcal{R}_0$ extends to a fixed
point of $\mathcal{R}_\delta$, as long as $\Phi\in \mathcal{D}(\mathcal{R}_\delta)$.
\end{prop}

Let us remark that in the proof of the existence of a fixed
point done in \cite{Lan82} it is claimed  that there exists
a function $\Phi$, which is analytic on the
domain $\left\{z \in \mathbb{C} | \thinspace |z^2-1|\leq \sqrt{8} \right\}$,
and such that its restriction to $I=[-1,1]$ is a fixed point of $\mathcal{R}$.
Then the fixed point $\Phi$ extends to a fixed point of
$\mathcal{R}_\delta$ as long as $1+\delta < \sqrt{8}$ and
$\Phi(1+\delta) > -(1+\delta)$. Concretely we have that there exist
$\delta_0$ (small enough) such that
$\Phi$ extends to a fixed point of $\mathcal{R}_\delta$ for any  $0<\delta<\delta_0$.
For the rest of this paper $\delta$ is fixed equal to $\delta_0$.

\subsubsection{Proofs}

\begin{proof}[Proof of proposition \ref{proposition RRdelta well defined}]
Given a function $\psi\in \mathcal{D}(\mathcal{R}_\delta)$, let $\Psi=\mathcal{R}_\delta(\psi)$.
Then
\[
\Psi(0) = \frac{1}{a} \psi \circ \psi (0)=\frac{1}{a} \psi(1) = 1.
\]

It is easy to check that
\begin{equation}
\label{derivada Psi}
x \Psi'(x) = x \psi'(ax) \psi'(\psi(ax))
\end{equation}

Then, using that $x\psi'(x) < 0$, for any $x\in I_\delta$ and $x\neq0$,
we have that $x\psi'(-ax) \geq 0$, for any $x\in I_\delta$ and $x\neq 0$.
On the other hand, for any $x \in I_\delta$ we have that $\psi(-ax) \in [b',1]$.
Using again $x\psi'(x) < 0$ and $0<a<a'<b'$ we have that $\psi'(\psi(ax)) < 0$
for any $x\in I_\delta$. Then we can conclude that $x \Psi'(x) < 0 $ for any
$x\in I_\delta$ and $x\neq 0$.

The last condition to check is that for any $x\in I_\delta$, the map
$\Psi$ maps $x$ inside the set $I_\delta$. Using
that $\Psi(0)=1$ and the monotonicity consequences of  $x\Psi'(x)<0$ we have
that $\Psi(x)<\Psi(0)=1$. We only have to check that $\Psi(1+\delta) >
-1 -\delta$. Since $\Psi(1+\delta)= -\frac{1}{a} \psi(b')$, then
the inequality holds from $\psi(b') < a' $.
\end{proof}

\begin{proof}[Proof of proposition
\ref{proposition fixed points extend to RRdelta}]
A fixed point of the renormalization
operator can be extended to the real line using recursively
the invariance equation  $\psi(x)=\frac{1}{a}\psi\circ \psi(ax)$ to
evaluate points from outside of $I$ (recall that $|a|<1$).
Moreover we have to check that $\psi\in \mathcal{D}(\mathcal{R}_\delta)$.
Using again that $|a|<1$, we have that $a=f(1)> -1$,
therefore for a sufficiently small $\delta$ we will have
$f(1+\delta) > -1-\delta$.
\end{proof}

\subsection{The renormalization operator for
quasi-periodically forced maps}
\label{section definition of q.p. renormalization}

In this section we define the quasi-periodic renormalization
operator and we check that there exists a non-empty set
of maps where it is well defined.

Consider the operator
\begin{equation}
\label{equation projection 0}
\begin{array}{rccc}
p_0:& C^r(\mathbb{T} \times I_\delta, I_\delta) &\rightarrow & C^r(I_\delta, I_\delta) \\
\displaystyle \rule{0pt}{3ex} & f(\theta,x) & \mapsto &  \rule{0pt}{3ex}
\displaystyle  \int_{0}^{1} f(\theta, x) d\theta .
\end{array}
\end{equation}
If we consider the natural inclusion of $C^r(I_\delta, I_\delta)$
into $C^r(\mathbb{T} \times I_\delta, I_\delta)$ then we have that $p_0$ is a projection
($(p_0)^2 = p_0$).

Consider $\mathcal{M}_{\delta}\subset C^r(I_\delta, I_\delta)$ defined in the
previous subsection. Then we can consider
the space $\mathcal{X}$ defined as:
\[
\mathcal{X}= \{ f \in C^r(\mathbb{T} \times I_\delta, I_\delta) | \thinspace
 p_0(f) \in \mathcal{M}_\delta\},
\]
and the decomposition $\mathcal{X}=\mathcal{X}_0 \oplus \mathcal{X}_0^c$ of it given by the projection
$p_0$, i.e $\mathcal{X}_0=\{f \in \mathcal{X} \thinspace |\text{ } p_0(f)=f\}$ and
$\mathcal{X}^c_0=\{f \in \mathcal{X} \thinspace | \text{ } p_0(f)=0\}$. Note
that from the definition of $\mathcal{X}$ it follows that $\mathcal{X}_0$ is an isomorphic
copy of $\mathcal{M}_\delta$.

\begin{prop}
\label{condition to belong  in X}
Let $f$ be a function in $\mathcal{X}_0$ and consider
$\gamma := f(1+\delta) +1+\delta $ (see figure \ref{MapLanf modified}).
Consider also $h(x,\theta) \in C^{r}(\mathbb{T}\times I_\delta, I_\delta)$ with
$p_0(h)=0$. If $\|h\|< \delta$ and $\|h\| < \gamma$, then $g=f+h$ belongs to $\mathcal{X}$.
\end{prop}

In other words, we have that small quasi-periodic perturbations of
an uncoupled one dimensional map of $\mathcal{M}_\delta$ belong to
the space considered here. With this space of functions we are able to
define the q.p. renormalization operator. The proof of proposition
\ref{condition to belong  in X} is at the end of the subsection.

\begin{defin}
Given a function $g\in \mathcal{X}$, we can
define
the {\bf renormalization} of $g$ as
\begin{equation}
\label{operator tau}
[\mathcal{T}_\omega(g)](\theta,x) :=  \frac{1}{\hat{a}} g(\theta + \omega,
g(\theta, \hat{a}x)),
\end{equation}
where $\displaystyle \hat{a} = \int_{0}^{1}
g(\theta, 1) d\theta$. Consider the set $\mathcal{D}(\mathcal{T})=
\{g \in \mathcal{X} \thinspace | \text{ } \mathcal{T}_\omega(g) \in \mathcal{X} \}$,
then the renormalization operator
$\mathcal{T}_\omega$ is defined from $\mathcal{D}(\mathcal{T})$ to $\mathcal{X}$.
\end{defin}

\begin{rem}
The choice of  $\hat{a} = \int_{0}^{1}
g(\theta, 1) d\theta$ is somehow arbitrary.
In a more general context, one might allow $\hat{a}$ to be indeed
a function of the angle, but the first problem arises with
the choice of this function. We  have chosen
$a$ to be constant in order to keep the problem as simple as possible.
For our purposes we find this approach sufficient, since we
are always considering maps which are perturbations of
one dimensional maps.
\end{rem}

Note that, with the definition given above, it might happen that
$D(\mathcal{T})=\emptyset$.  Then let us ensure that this is not the case.

The space $\mathcal{X}_0$, which is an isomorphic copy $\mathcal{M}_\delta$,
is a subspace of $\mathcal{X}$. Then let us define $\mathcal{D}_0(\mathcal{T})$ as
the set of functions in $\mathcal{X}_0$ that correspond
to functions of $\mathcal{D}(\mathcal{R}_\delta) \subset \mathcal{M}_\delta$.
For any function $g\in \mathcal{D}_0(\mathcal{T})$ we have
$\mathcal{T}_\omega(g) = \mathcal{R}_\delta(g) \in \mathcal{X}$, therefore
$\mathcal{D}_0(\mathcal{T})$ is a subset of $\mathcal{D}(\mathcal{T})$ and consequently
$\mathcal{D}(\mathcal{T})$ is not empty.  Indeed we have
the following result on the topology of $\mathcal{D}(\mathcal{T})$.

\begin{prop}
\label{proposition existence of open set domain of TT}
There exists an open set $W$ in $\mathcal{X}$  such
that $\mathcal{T}_\omega: W \rightarrow
C^r(\mathbb{T}\times I_\delta, I_\delta)$ is a well defined
continuous map.
Additionally consider $U=\left(p_0 \circ \mathcal{T}_\omega\right)^{-1} (\mathcal{M}_\delta)$, then
we have that there exists an open set $W'$ in
$U\cap \mathcal{X}$ such that $\mathcal{D}_0(\mathcal{T}) \subset W'\subset \mathcal{D}(\mathcal{T})$.
\end{prop}

\begin{rem}
\label{Remmark dependency of set on parameters}
Note that the sets $\mathcal{X}$, $\mathcal{D}(\mathcal{T})$, $\mathcal{D}_0(\mathcal{T})$
and the operator $\mathcal{T}_\omega$ depend
on the value $\delta$, but here we do not make it explicit
to keep the notation simple. On the other
hand, we made explicit the dependence of $\mathcal{T}_\omega$ on
$\omega$, but not in the set $\mathcal{D}(\mathcal{T})$ and $\mathcal{D}_0(\mathcal{T})$, which a priori
should also depend on $\omega$. The set  $\mathcal{D}_0(\mathcal{T})$ is
an isomorphic copy of $\mathcal{D}(\mathcal{R}_\delta)$, then it does not depend on
$\omega$. The set $\mathcal{D}(\mathcal{T})$ actually depends on $\omega$,
but it is omitted from the notation for simplicity.
\end{rem}

\begin{rem}
\label{remark doubling of the rotation number}
Consider a map $F$ like (\ref{q.p. forced system interval}). We have
that the map is determined by a function $f\in \mathcal{X}$ and a value
of $\omega\in \mathbb{T}$. Then we can define the renormalization of $(\omega,f)$
as the map determined by $(2\omega, \mathcal{T}_\omega f)$, which acts
from $\mathbb{T} \times \mathcal{D}(\mathcal{T})$ to $\mathbb{T} \times \mathcal{X}$. Note that the frequency
$\omega$ has been doubled. This is due to the fact that
the renormalization of a map is constructed
 as the affine transformation of the map iterated twice.
For convenience, in the remaining of this section we will study
$\mathcal{T}_\omega$ as an operator acting from $\mathcal{D}(\mathcal{T})$ to $\mathcal{X}$
depending on the parameter $\omega$. In section \ref{chapter application}
we will take into account the doubling of the
rotation number again.
\end{rem}

\subsubsection{Proofs}

\begin{proof}[Proof of proposition \ref{condition to belong  in X}]
Note that $p_0(g)=p_0(f) + p_0(h) = f\in \mathcal{M}_\delta$.
To prove that $g\in \mathcal{X}$, it is only necessary to check that
$g\in C^r(\mathbb{T}\times I_\delta, I_\delta)$. The map $g$ is $C^r$
for being the linear combination of $C^r$ maps. The
function $g$ belongs to  $C^r(\mathbb{T}\times I_\delta, I_\delta)$
if $g(\theta,x) \in I_\delta$ for any $\theta \in \mathbb{T}$ and $x \in I_\delta$.

We begin with the upper bound. As $f(x) \leq f(0) =1 $
for any $x\in I_\delta$, then
\[
g(\theta, x) = f(x) + h(\theta,x) \leq f(x) + \|h\| \leq f(0) +\varepsilon = 1+\varepsilon,
\]
for any $(\theta,x) \in  \mathbb{T}\times I_\delta$.

Now we check the lower bound. We have that
$\gamma$ only depends on $f$ and it is always greater or equal to 0.
On the other hand, we have $f(x) \geq f(1+\delta)$ for any $x\in I_\delta$. Then
\[
g(\theta, x) = f(x) + h(\theta,x) \geq f(x) - \|h\|  \geq
f(1+\delta) - \|h\| = -1-\delta+\gamma-\|h\|,
\]
which implies that $g(\theta,x) \geq -1-\delta$ if $\|h\|<\gamma$.
\end{proof}

\begin{proof}[Proof of proposition
\ref{proposition existence of open set domain of TT}]
First of all we need to build an open set where the
q.p. renormalization operator (\ref{operator tau}) is well
defined. With this aim we have the following lemma.

\begin{lem}
\label{lemma open domain}
There exists an open set $U_1\subset \mathcal{X}$ with $\mathcal{D}_0(\mathcal{T})
\subset U_1 $ such that the map
\[
\begin{array}{rccc}
F_1:& U_1\subset C^r(\mathbb{T} \times I_\delta, I_\delta) &\rightarrow & C^r(\mathbb{T}
\times I_\delta, I_\delta) \\
\displaystyle \rule{0pt}{3ex} & g= g(\theta,x) & \mapsto &  \rule{0pt}{3ex}
\displaystyle [F_1(g)](\theta,x) =  g(\theta,\hat{a} x),
\end{array}
\]
with $\displaystyle \hat{a}=\int_0^1 g(\theta,1) d\theta$, is well defined
and continuous.
\end{lem}

\begin{proof}
Given a function $g \in \mathcal{X}$ we have that $p_0(g)\in \mathcal{M}_\delta$.
Note that the value $\hat{a}$ as a functional
operator $\hat{a}: \mathbb{C}^r(\mathbb{T}\times I_\delta, I_\delta) \rightarrow I_\delta$
is equal to the evaluation map
at $x=1$ composed with the projection $p_0$. This is a
bounded linear operator, therefore we have that
$\hat{a}: C^r(\mathbb{T}\times I_\delta,I_\delta) \rightarrow I_\delta$ is
continuous.
Then we can consider $U_1= \hat{a}^{-1}((-1,1))$, which
is an open set because it is the preimage of
an open set by a continuous function.
For any $g\in U_1$ using $|\hat{a}|<1$
we have
\[
\sup_{(\theta,x)\in \mathbb{T}\times I_\delta} |g(\theta,\hat{a}x)|\leq
\sup_{(\theta,x)\in \mathbb{T}\times I_\delta} |g(\theta,x)|\leq 1+\delta.
\] Hence $g(\theta,\hat{a}x)$ is well defined
for any $g\in U_1$ and $(\theta,x)\in \mathbb{T}\times I_\delta$.
Moreover for any $g\in \mathcal{D}_0(\mathcal{T})$ we
have $\hat{a}(g)=\left[p_0(g)\right](1) \in (-1,0)$,
which proves that $\mathcal{D}_0(\mathcal{T})\subset U_1$.
\end{proof}

As discussed in the proof of lemma \ref{lemma open domain}
above we have that $\hat{a}: U_1\rightarrow (-1,1)$ defined as
$\hat{a}(g) = \int_0^1 g(\theta,1)d\theta$ is  a
continuous function.

Using the results on the smoothness of the composition map from \cite{Irw72}
and lemma \ref{lemma open domain} above,
we have that the map
\[
\begin{array}{rccc}
F_2:& U_1\subset C^r(\mathbb{T} \times I_\delta, I_\delta) &\rightarrow & C^r(\mathbb{T}
\times I_\delta, I_\delta) \\
\displaystyle \rule{0pt}{3ex} & g= g(x,\theta) & \mapsto &  \rule{0pt}{3ex}
\displaystyle [F_2(g)](\theta,x) = g(\theta+\omega, g(\theta,\hat{a} x)),
\end{array}
\]
is well defined and is also continuous.

On the other hand, let $U_2 =p_0^{-1} (\mathcal{D}_0(\mathcal{T}))$ and consider
\[
\begin{array}{rccc}
F_3:& U_2 \subset C^r(\mathbb{T} \times I_\delta, I_\delta) &\rightarrow & C^r(\mathbb{T}, \mathbb{R} ) \\
\displaystyle \rule{0pt}{3ex} & g= g(x,\theta) & \mapsto &  \rule{0pt}{3ex}
\displaystyle [F_3(g)](x) = \frac{1}{\hat{a}} x.
\end{array}
\]
For any $g\in U_2$ we have $\hat{a}(g)<0$, therefore
the map $F_3$ is well defined. Indeed, we have that it
is continuous with respect to $g$.

Finally note that $\mathcal{T}_\omega$ is obtained as the composition
$(F_3(g))\circ(F_2(g))$. Using the results from \cite{Irw72}
we have that $\mathcal{T}_\omega$ is
continuous (and it is well defined) as an operator
$\mathcal{T}_\omega: U_3 \subset C^r(\mathbb{T} \times I_\delta, I_\delta) \rightarrow
C^r(\mathbb{T} \times I_\delta, \mathbb{R} )$, where $U_3 = U_2 \cap U_1$.
Note that $\mathcal{D}_0(\mathcal{T}_\omega)$ is the image by the inclusion of $\mathcal{D}(\mathcal{R}_\delta)$.
Therefore we have $\mathcal{D}_0(\mathcal{T}_\omega)\subset U_2$ and, consequently,
$\mathcal{D}_0(\mathcal{T}_\omega)\subset U_3$.

Note that the image space of the operator is  $\mathcal{T}_\omega$ is not
the desired one. Since $\mathbb{T} \times I_\delta$ is compact, we have
that the map $N:C^r(\mathbb{T} \times I_\delta, \mathbb{R} )\rightarrow [0,\infty)$ defined
as $N(g)= \|g \|_{\infty}$ is continuous. Consider the set
$U_4:=N^{-1}([0,1+\delta))$ which is an open subset of $C^r(\mathbb{T} \times I_\delta, \mathbb{R} )$.
At this point we can define the set $W$  in the statement of the
proposition as
\[W= \mathcal{T}_\omega ^{-1}( U_4) \cap U_3.\]
Using this construction of the set $W$ we have $\mathcal{T}_\omega: W
\subset C^r(\mathbb{T} \times I_\delta, I_\delta) \rightarrow
C^r(\mathbb{T} \times I_\delta, I_\delta)$. Again we have $\mathcal{D}_0(\mathcal{T}_\omega)\subset W$
due to the fact that for any $g\in \mathcal{D}_0(\mathcal{T}_\omega)$ we
have $\|\mathcal{T}_\omega(g)\|_{\infty}= \|\mathcal{R}_\delta(g)\|_{\infty}  < 1+\delta$.
This concludes the proof of the first assertion in the proposition.

We check now the second assertion of the proposition.
Consider $U=\left(p_0 \circ \mathcal{T}_\omega\right)^{-1} (\mathcal{M}_\delta)$ and
$U_5=W\cap U$. Consider also the following auxiliary functions,
\[
\begin{array}{rccc}
F_4:& U_5 \subset C^r(\mathbb{T} \times I_\delta, I_\delta) &\rightarrow & [0,+\infty), \\
\displaystyle \rule{0pt}{3ex} & g= g(x,\theta) & \mapsto &  \rule{0pt}{3ex}
\displaystyle \|\mathcal{T}_\omega(g) - p_0(\mathcal{T}_\omega(g)) \|.
\end{array}
\]
and
\[
\begin{array}{rccc}
F_5:& U_5 \subset C^r(\mathbb{T} \times I_\delta, I_\delta) &\rightarrow & \mathbb{R}, \\
\displaystyle \rule{0pt}{3ex} & g= g(x,\theta) & \mapsto &  \rule{0pt}{3ex}
\displaystyle 1+\delta +  [\mathcal{T}_\omega(g)](1+\delta) -
\|\mathcal{T}_\omega(g) - p_0(\mathcal{T}_\omega(g)) \|).
\end{array}
\]

Now we can define the set $W'$  in the statement of the
proposition as
\[W':= U_5\cap F_4^{-1}([0,\delta))\cap F_5^{-1}((-\delta,\delta)).\]

First let us check that $\mathcal{D}_0(\mathcal{T}_\omega)\subset W \subset \mathcal{D}(\mathcal{T})$.
For any map $g\in \mathcal{D}_0(\mathcal{T}_\omega)$ we have that
$\mathcal{T}_\omega(g) =\mathcal{R}_\delta(g)= p_0(\mathcal{R}_\delta(g))= p_0(\mathcal{T}_\omega(g))$;
then it follows easily that $\mathcal{D}_0(\mathcal{T}_\omega)\subset F_4^{-1}([0,\delta))$.
Moreover as $\mathcal{T}_\omega(g) = \mathcal{R}_\delta(g)\in \mathcal{M}_\delta$, we have
$[\mathcal{R}_\delta(g)](1+\delta) < 1+\delta$,  which implies that
$g\in  F_5^{-1}(-\delta,\delta)$.

Finally, we have to check  $W \subset \mathcal{D}(\mathcal{T})$, which is equivalent to
prove that $\mathcal{T}_\omega(g)\in \mathcal{X}$ for any $g\in W$. Given $g\in W$,
we have
$\mathcal{T}_\omega(g) = p_0(\mathcal{T}_\omega(g)) + \mathcal{T}_\omega(g) - p_0(\mathcal{T}_\omega(g))$.
From $g\in U_5$ we have that $p_0(\mathcal{T}_\omega(g))\in \mathcal{M}_\delta$. Moreover
 from $g\in  F_4^{-1}([0,\delta))$ we have
$ \|\mathcal{T}_\omega(g) - p_0(\mathcal{T}_\omega(g)) \|< \delta$ and from
 $g\in  F_5^{-1}([0,\delta))$ we have $ \|\mathcal{T}_\omega(g) - p_0(\mathcal{T}_\omega(g)) \|<
1+\delta + \left[p_0(\mathcal{T}_\omega)\right](1+\delta) $. Note
that as $p_0\left(\mathcal{T}_\omega(g) - p_0(\mathcal{T}_\omega(g))\right)=0$,
we can apply proposition \ref{condition to belong  in X} and then
it follows that $g\in \mathcal{X}$.
\end{proof}

\subsection{Study of the operator $\mathcal{T}_\omega$}
\label{subsection study of TT}

Let us follow with the study of
the operator $\mathcal{T}_\omega$. In this section we start showing that the
fixed points of $\mathcal{R}_\delta$ extend to
fixed points of $\mathcal{T}_\omega$. Then we give a result on the
differentiability of $\mathcal{T}_\omega$, in the $C^r$ topology.
With this result it becomes evident that the $C^r$
topology is a bad choice for the study of the operator.
Lastly, we introduce the topology of real analytic
maps and we check that the operator is
well defined and differentiable if certain hypothesis (which
will be called {\bf H0}) is satisfied. Again, all the proofs have been moved to
the end of the section.
\begin{prop}
\label{proposition fixed points extend}
The operator $\mathcal{T}_\omega$ restricted to  the set $\mathcal{D}_0(\mathcal{T})$,
is isomorphically conjugate to $\mathcal{R}_\delta$. Concretely we
have that any fixed point of $\mathcal{R}_\delta$ extends
to a fixed point of $\mathcal{T}_\omega$.
\end{prop}

We have the following
result on the differentiability of $\mathcal{T}_\omega$.

\begin{thm}
\label{theorem renormalization oprator is differentiable}
Let $\mathcal{T}_\omega:\mathcal{D}(\mathcal{T}) \rightarrow \mathcal{X}$ be the renormalization operator in
the $C^r$-topology, and consider $\Phi$ a fixed point of
the operator. If $\Phi\in \mathcal{D}_0(\mathcal{T})\cap
C^{r+s}(\mathbb{T}\times I_\delta,I_\delta)$ then we have that
there exists $U$ an open neighborhood of $\Phi$ in
$\mathcal{D}_0(\mathcal{T})\cap C^{r+s} (\mathbb{T}\times I_\delta,I_\delta)$
such that $\mathcal{T}_\omega$
is a $C^s$ operator in $U$.
Moreover, if $s\geq 1$ for any point $\Psi\in U $ we have
that the Gateaux derivative of
$\mathcal{T}_\omega$ on $\Psi$ in the direction $h$ is given by
\begin{equation}
\label{diferencial de TT 0}
\begin{array}{rl}
\displaystyle [d \mathcal{T}_\omega(\Psi, h)](\theta,x) = &
\displaystyle  \phantom{+}\frac{1}{a} \psi'(\psi( a x )) h(\theta,
a x)  +  \frac{1}{a} h(\theta + \omega, \psi(a x)) \\
\rule{0pt}{4ex} & \displaystyle + \frac{b}{a}  \psi'(\psi(ax))\psi'(ax) x
- \frac{b}{a^2} \psi(\psi(ax)),
\end{array}
\end{equation}
where $\psi=p_0(\Psi)$,
$a=\psi(1)$ and $b= \displaystyle
\int_0^{1} h(\theta,1)d\theta$.
\end{thm}

Note that there is a ``loss of differentiability'', in the sense
that one needs to assume that the function $\Psi$ where
we differentiate the operator is in $C^{r+s}$ while the
operator acts in subsets of  $C^{r}(\mathbb{T}\times I_\delta, I_\delta)$.
This is due to the self composition in the renormalization operation.
To skip this problem, let us introduce the topology
of analytic functions instead of the $C^r$ one, for the
forthcoming study of the operator.

\begin{defin}
\label{definition space BB}
Let $\mathbb{W}$ be an open set in the complex plane containing the
interval $I_\delta$ and let
$\mathbb{B}_\rho = \{z = x + i y\in \mathbb{C} \text{ such that } |y| < \rho\}$.   The
we define the set $\mathcal{B}=\mathcal{B}(\mathbb{B}_\rho,\mathbb{W})$ as the space of functions
$f: \mathbb{B}_\rho \times  \mathbb{W} \rightarrow \mathbb{C}$ such that:
\begin{enumerate}
\item  $f$ is holomorphic in $\mathbb{B}_{\rho}\times \mathbb{W}$ and continuous
in the closure of  $\mathbb{B}_{\rho}\times \mathbb{W}$.

\item $f$ is real analytic (it maps real numbers to real numbers).

\item  $f$ is $1$-periodic in the first variable, i. e.
$f(\theta +1,z) = f(\theta,z)$ for any $(\theta,z) \in \mathbb{B}_{\rho}
\times\mathbb{W}$.
\end{enumerate}
\end{defin}

\begin{prop}
\label{proposition BB is banach}
The space $\mathcal{B}$ endowed with the supremum norm
$\left( \displaystyle \| f\| = \sup_{\mathbb{B}_{\rho_1}\times\mathbb{W}}
| f(\theta,z)| \right)$  is  a Banach space.
\end{prop}

We want to consider the quasi-periodic renormalization
operator $\mathcal{T}_\omega$ (see equation (\ref{operator tau}))
restricted to the domain
$ \mathcal{D}(\mathcal{T}) \cap \mathcal{B}$, but then it is necessary to check that
$\mathcal{T}_\omega$ is well defined in the complex domain.
For any function $f\in  \mathcal{D}(\mathcal{T}) \cap \mathcal{B}$
we should check that $f( \mathbb{B}_{\rho}\times a \mathbb{W})
\subset \mathbb{W}$ (where $a\mathbb{W}=\{z\in \mathbb{C} |
\thinspace a z\in \mathbb{W} \}  $). In the one
dimensional renormalization theory  the open set $\mathbb{W}$
is chosen such that this condition is satisfied. Concretely,
this condition is
typically checked with computer assistance, together with
other conditions to prove the existence of fixed points (see
\cite{Lan82,Lan92,KSW96}).
In our case we will assume that the following hypothesis is satisfied.

\begin{description}
\item[H0)] There exists an open set $\mathbb{W}\subset \mathbb{C}$ containing
$I_\delta$ and a function $\Phi\in \mathcal{B} \cap \mathcal{X}_0$ such
that $p_0(\Phi)$ is a fixed point of the
renormalization operator $\mathcal{R}_\delta$  and such that
the closure of both $a\mathbb{W}$ and $p_0(\Phi)(a \mathbb{W})$ is
contained in $\mathbb{W}$ (where $a:=\Phi(1)$)
\end{description}

Although the results on the existence of the fixed point of
renormalization operator done by Lanford in \cite{Lan82} are well
accepted, in the cited article there are no proofs and many details are omitted.
For the proof of the existence of the fixed point
it is also necessary to check this condition.
In \cite{Lan92}, some more details on how to prove the existence of
the fixed point are given. Actually,
it is claimed that the hypothesis {\bf H0} is true for the set
\[
\left\{ z\in \mathbb{C} \text { such that } |z^2 -1| < \frac{5}{2} \right\}.
\]

This set used by Lanford is more convenient in his study
since he works in the
set of even holomorphic functions. In the numerical computations
from \cite{JRT11c} we use as $\mathbb{W}$ the disc centered at
$\frac{1}{5}$ with radius $\frac{3}{2}$,
and we check hypothesis {\bf H0} numerically
(without rigorous bounds).

\begin{thm}
\label{theorem renormalization analitic set up}
Assume that {\bf H0} holds and let $\Phi$ be
the fixed point given by this assumption. Then we have that there exists
$U\subset \mathcal{D}(\mathcal{T})\cap\mathcal{B}$, an open neighborhood of $\Phi$,
such that $\mathcal{T}_\omega :U\rightarrow \mathcal{B}$ is well defined,
and $\mathcal{T}_\omega$ is
Fr\'echet differentiable for any  $\Psi \in U$ and the derivative is equal to
\begin{equation}
\label{diferencial de TT}
\begin{array}{rl}
\displaystyle [D \mathcal{T}_\omega(\Psi, h)](\theta,x) = &
\displaystyle  \phantom{+}\frac{1}{a} (\partial_x\Psi)
(\theta+\omega, \Psi(\theta,a x )) h(\theta, a x)  +
\frac{1}{a} h(\theta + \omega, \Psi(\theta, a x)) \\
\rule{0pt}{4ex} & \displaystyle + \frac{b}{a}  (\partial_x \Psi)
(\theta+\omega,\Psi(\theta,ax))
(\partial_x \Psi (\theta,ax)) x - \frac{b}{a^2} \Psi(\theta+\omega,\Psi(\theta,ax)),
\end{array}
\end{equation}
where $a=\displaystyle \int_0^{1}\Psi(\theta,1)d\theta$ and $b= \displaystyle
\int_0^{1} h(\theta,1)d\theta$.
\end{thm}

\subsubsection{Proofs}

\begin{proof}[Proof of proposition \ref{proposition fixed points extend}]
If a map belongs to $\mathcal{D}_0(\mathcal{T})$ then it does not
depend on $\theta$, therefore the operator $\mathcal{T}_\omega$ coincides
with $\mathcal{R}_\delta$ composed with the inclusion of $\mathcal{M}_\delta$ in $\mathcal{X}$.
\end{proof}

\begin{proof}[Proof of theorem
\ref{theorem renormalization oprator is differentiable}]
By definition (see equation (\ref{operator tau})) given a function
$g\in  C^r(\mathbb{T} \times I_\delta, I_\delta)$ we have that
$ [\mathcal{T}_\omega(g)](\theta,x) :=  \displaystyle\frac{1}{\hat{a}} g(\theta + \omega,
g(\theta, \hat{a}x))$ where $\displaystyle \hat{a} =
\int_{0}^{1}
g(\theta, 1) d\theta$. Note that $\hat{a}$ can be written as $\hat{a}= [p_0(g)](1)$. The function
$p_0:  C^{r+s}(\mathbb{T} \times I_\delta, I_\delta) \rightarrow  C^{r+s}(I_\delta, I_\delta)$
defined by (\ref{equation projection 0}) is $C^s$
(actually it is a linear bounded operator). On the other hand the evaluation
of a $C^r$ function in a concrete
value is also a $C^r$ function
(see proposition 2.4.17 from \cite{AMR88}).
Therefore $\hat{a}=\hat{a}(g)$ as a function of $g$  is $C^r$ as well.

Note that $\mathcal{T}_\omega(g)$
can be written as the composition of
different functions, which are $g$ itself,
a translation in the $\theta$ variable and a scalar multiplication by $a$ (and its inverse) in the $x$
variable. Each one of these functions are $C^r$ dependent with respect to $f$
except the composition of $f$ with itself which is only a $C^s$ map,
when we work in the $C^r$ topology
(see \cite{Irw72}). We can conclude that $\mathcal{T}_\omega$ is only a $C^s$ operator.

Now we compute explicitly the Gateaux
derivative. As $\Psi$ belongs to $\mathcal{X}_0$, we have $\Psi(x,\theta)=\psi(x)$
and consequently
\begin{eqnarray}
\label{operator derivative computation}
\displaystyle
\left[\mathcal{T}(\Psi + t h)\right] (\theta, x) & = &
\displaystyle
\frac{1}{\hat{a}} \left[ \Psi + t h \right] ( \theta +\omega,
\left[ \Psi + t h \right](\theta,\hat{a} x)) \nonumber  \\
& = &  \displaystyle  \frac{1}{\hat{a}}
\psi (\psi( \hat{a}x)) +
\frac{1}{\hat{a}} \psi'(\psi(\hat{a}x )) h(\theta,\hat{a}x) t \nonumber
 \\ & &  \displaystyle
+ \frac{1}{\hat{a}}   h(\theta + \omega, \psi(\hat{a}x)) t  + O(t^2),
\end{eqnarray}
where $\displaystyle \hat{a} =
\int_{0}^{1} [\Psi + t h](1,\theta) d\theta$.

Set $a= \psi(1)$ and $\displaystyle b=\int_{0}^{1}
h(1,\theta) d\theta$. We have that $\hat{a}= a+ tb$.
Therefore,
\begin{equation}
\label{Taylor 1 ente a}
\frac{1}{\hat a} = \frac{1}{a+tb}= \frac{1}{a} - \frac{1}{a^2} t b
+ O(t^2),
\end{equation}
and using the chain rule we have
\begin{equation}
\label{psiopsi de hat ax}
\psi\left(\psi( \hat{a} x) \right) = \psi\left(\psi( a x) \right)
 + \psi'\left(\psi(ax)\right) \psi'(ax) t   b x  + O(t^2).
\end{equation}

Combining equations (\ref{operator derivative computation}),
(\ref{Taylor 1 ente a}) and (\ref{psiopsi de hat ax}) follows that
the Gateaux derivate of $\mathcal{T}$ in $\Psi$ is the one given by
(\ref{diferencial de TT}).
\end{proof}

\begin{proof}[Proof of proposition \ref{proposition BB is banach}]
Consider $\mathcal{A}$ the space of
holomorphic functions in $\mathbb{B}_{\rho}\times \mathbb{W}$ and continuous
in the closure of  $\mathbb{B}_{\rho}\times \mathbb{W}$.
Using basic properties of the holomorphic functions
in several variables (see \cite{Hor73, Kra82}) is
easy to check that $\mathcal{A}$ is a Banach space. The space $\mathcal{B}$ is the set
of functions of $\mathcal{A}$ such that
\begin{itemize}
\item $f(\theta + 1 ,z) - f(\theta, z) =0 $ for any
$(\theta,z) $ in $\mathbb{B}_{\rho} \times\mathbb{W}$.
\item $f(\theta,x) - \overline{f(\theta,x)} = 0$ for any
$(\theta,x)$ in $\mathbb{R} \times I_\delta$.
\end{itemize}
Then $\mathcal{B}$ is the preimage of a closed subset
by a continuous function, therefore it is closed in $\mathcal{A}$
and consequently it is a Banach space.
\end{proof}

\begin{proof}[Proof of the theorem
\ref{theorem renormalization analitic set up}]
Given a function $f\in  \mathcal{D}(\mathcal{T}) \cap \mathcal{B}$, we have
that its image by $\mathcal{T}_\omega(f)$ is given by (\ref{operator tau}).
If we want it to be well defined we must
check that $f( \mathbb{B}_{\rho}\times a \mathbb{W})
\subset \mathbb{W}$ (where $a\mathbb{W}=\{z\in \mathbb{C} |
\thinspace a z\in \mathbb{W} \}  $).

We have that $\phi(\theta,x) = \left[p_0(\phi)\right] (x)$ for any $\theta\in \mathbb{B}_{\rho}$.
Using hypothesis {\bf H0}  we have that
$\text{Cl}\left(\Phi\left(\mathbb{B}_{\rho}\times a \mathbb{W}\right)\right)\subset \mathbb{W}$,
where $\text{Cl}(\cdot)$ denotes the closure of the set.
If we consider now a function $f$  in a
suitable neighborhood
of $\Phi$ we have that it still maps $ \mathbb{B}_{\rho}
\times a \mathbb{W}$ inside $\mathbb{W}$ (if
$f$ is close enough to $\Phi$ in the topology of $\mathcal{B}$).

To prove the differentiability of the map we will check it
directly from the definition of Fr\'echet derivative.

From $\text{Cl}\left(\Phi\left(\mathbb{B}_{\rho}\times a \mathbb{W}\right)\right)\subset \mathbb{W}$,
and the fact of $\mathbb{W}$ being bounded it follows that
$\text{Cl}\left(\Phi\left(\mathbb{B}_{\rho}\times a \mathbb{W}\right)\right)$ is compact.
Consider the following filtration of sets in the
complex plane
\[
\text{Cl}\left(\Phi\left(\mathbb{B}_{\rho}\times a \mathbb{W}\right)\right) = K_0 \subset V_0 \subset
K_1 \subset V_1 \subset  K_2 \subset V_2= \mathbb{W},
\]
with each $K_i$ compact and each $V_i$ open, for $i=0,1,2$.

Consider now $U_1\subset U$ the open neighborhood of $\Phi$ in $\mathcal{B}$ formed
by the $\Psi\in \mathcal{B}$  such that
\[
\Psi\left(\mathbb{B}_{\rho}\times a \mathbb{W}\right) \subset V_0.
\]
For any map $\Psi\in U_1$ we have that
$\text{Cl}\left(\Psi\left(\mathbb{B}_{\rho}\times \hat{a} \mathbb{W}\right)\right)\subset K_1$.

On the other hand, from  $K_2 \subset \mathbb{W}$ and the fact
that $K_2$ is compact and $\mathbb{W}$ open, it follows that
there exist a value $r>0$ such that for any $x_0 \in K_2$
the ball centered on $x_0$ with radius $r$ is
contained in $\mathbb{W}$. Then for any map $f\in \mathcal{B}$ we have
\[
\partial_x f(\theta,x_0) =\frac{1}{2\pi i} \int_{|z-x_0|=r}
\frac{f(\theta,z)}{(z-z_0)^2}  d\theta.
\]
Then it follows easily that, for any $f \in \mathcal{B}$ and
$x_0\in K_2$ we have
\[
| \partial_x f(\theta,x_0)| \leq \frac{1}{r} \| f \|_\infty.
\]
Modifying  the same argument, we can check that
\[
| \partial^2_{x^2} f(\theta,x_0)| \leq \frac{2}{r^2} \| f \|_\infty.
\]
Note that both bounds are uniform for any map in $U_1$.

Consider $\Psi \in U_1$, and $h \in \mathcal{B}$ with $\|h\|$ small. We
want to compute $\mathcal{T}_\omega(\Psi+h)$ up to $O(\|h\|^2)$.
First of all we have,
\begin{equation}
\label{equation analicity 0.0}
\mathcal{T}_\omega(\Psi+h)=
\frac{1}{\hat{a}(\Psi+h)} \left[
\begin{array}{r}
\Psi(\theta+\omega, \Psi(\theta,\hat{a}(\Psi+h)x) + h(\theta,\hat{a}(\Psi+h)x))  + \\
h(\theta+\omega, \Psi(\theta,\hat{a}(\Psi+h)x) + h(\theta,\hat{a}(\Psi+h)x))
\end{array}
\right].
\end{equation}

To simplify the notation consider
\[
a= \int_0^1 \Psi(\theta,1)d\theta , \quad
b= \int_0^1 h(\theta,1)d\theta.
\]
Then we have $\hat{a}(\Psi+h) = a + b$, and
\[
|b| \leq  \int_0^1 |h(\theta,1)|d\theta \leq \|h \|.
\]

Since $\Psi \in U_1$ we have that for any $h$ with $\|h\|$ sufficiently
small $\Psi + h \in U_1$, therefore we have that
$\Psi(\theta,(a+b)x) + h(\theta,(a+b)x)
\in V_1$. Using the complex Taylor expansion with respect to $x$ up
to second order we have
\begin{eqnarray}
\label{equation analicity 1.1}
\Psi(\theta+\omega, \Psi(\theta,(a+b)x) + h(\theta,(a+b)x)) &= &
\Psi(\theta+\omega,\Psi(\theta,(a+b)x)) +
  \\ & &
(\partial_x \Psi)(\theta+\omega,\Psi(\theta,(a+b)x))  h(\theta,(a+b)x)) + \nonumber \\
& &  R_2(\theta,x) \nonumber
\end{eqnarray}

with
\begin{equation}
\label{equation analicity 1.01}
| R_2(\theta,x) | \leq \frac{1}{r^2}\frac{1}{1-\frac{\|h\|}{r}} \| \Psi\| \|h\|^2 =
O(\|h\|^2).
\end{equation}

Analogously we have
\begin{eqnarray}
\label{equation analicity 1.2}
h(\theta+\omega, \Psi(\theta,(a+b)x) + h(\theta,(a+b)x)) &= &
h(\theta+\omega,\Psi(\theta,(a+b)x)) +   \\ & &
R_1(\theta,x), \nonumber
\end{eqnarray}
with
\begin{equation}
\label{equation analicity 2.0}
| R_1(\theta,x) | \leq  \frac{1}{r}\frac{1}{1-\frac{\|h\|}{r}} \| h\| \|h\| = O(\|h\|^2).
\end{equation}

Recall that $|b|=O(\|h\|)$  then applying Taylor expansion
and the uniform bound on $K_2$ it follows easily that
\begin{eqnarray}
\label{equation analicity 3.1}
\Psi(\theta,(a+b)x) &= &  \Psi(\theta,ax) + (\partial_x \Psi)
(\theta,ax) b x + O(\|h\|^2),  \\
\label{equation analicity 3.2}
h(\theta,(a+b)x) &= &  h(\theta,ax)  +  O(\|h\|^2).
\end{eqnarray}

Using that $\Psi\in U_1$  we have that
$ \Psi(\theta,ax) + (\partial_x \Psi) (\theta,ax) b x$
belongs to $V_1\subset K_2$ for $\|h\|$ sufficiently small. Now
we can combine this fact with the uniform bound on $K_2$  and
equation (\ref{equation analicity 3.1}) and (\ref{equation analicity 3.2}) to prove that
\begin{equation}
\nonumber
h(\theta+\omega,\Psi(\theta,(a+b)x)) = h(\theta+\omega,\Psi(\theta,ax)) + O(\|h\|^2).
\end{equation}
Using this together with equations (\ref{equation analicity 1.2}) and
(\ref{equation analicity 2.0}) we obtain
\begin{eqnarray}
\label{equation analicity 4.2}
h(\theta+\omega, \Psi(\theta,(a+b)x) + h(\theta,(a+b)x)) &= &
h(\theta+\omega,\Psi(\theta,a x)) +  O(\|  h\|^2) .
\end{eqnarray}

With the same argument it follows that
\[
(\partial_x \Psi)(\theta+\omega,\Psi(\theta,(a+b)x)) =
(\partial_x \Psi)(\theta+\omega,\Psi(\theta,ax)) + O(\|h\|) ,
 \]
and
\begin{eqnarray}
\Psi(\theta+\omega,\Psi(\theta,(a+b)x)) &=&
\Psi(\theta+\omega,\Psi(\theta,ax)) + \nonumber \\
& & (\partial_x\Psi) (\theta+\omega,\Psi(\theta,ax)) (\partial_x \Psi)(\theta,ax) bx \nonumber
+
 O(\|h\|^2).
\end{eqnarray}
Replacing the last two equations in equation (\ref{equation analicity 1.1}) and using the
bound given by (\ref{equation analicity 1.01}) yields to
\begin{eqnarray}
\label{equation analicity 4.1}
\Psi(\theta+\omega, \Psi(\theta,(a+b)x) + h(\theta,(a+b)x)) &= &
\Psi(\theta+\omega,\Psi(\theta,ax)) +
  \\ & &
(\partial_x \Psi) (\theta+\omega,\Psi(\theta,ax))
(\partial_x \Psi)(\theta,ax) bx \nonumber +
  \\ & &
(\partial_x \Psi)(\theta+\omega,\Psi(\theta,ax))  h(\theta,(a+b)x))
+ O(\|h\|^2). \nonumber
\end{eqnarray}

Finally, recall that $|b|= O(\|h\|)$, therefore
\[
\frac{1}{a+b}= \frac{1}{a} - \frac{b}{a^2} + O(\|h\|).
\]
When we replace this value and the ones of
equations (\ref{equation analicity 4.2}) and (\ref{equation analicity 4.1})
in (\ref{equation analicity 0.0}) it follows that
\[\left\|\mathcal{T}_\omega(\Psi+h) - \mathcal{T}_\omega(\Psi) - D\mathcal{T}_\omega(\Psi)h \right\|=
O\left(\|h\|^2\right),\]
which proves the differentiability of the operator in the analytic context.
\end{proof}

\subsection{Fourier expansion of $D\mathcal{T}_\omega(\Psi)$.}
\label{section The Fourier expansion of DT}

Let $\Psi$ be a function as in the hypothesis of theorem
\ref{theorem renormalization analitic set up}, but additionally
assume that $\Psi\in U\cap \mathcal{D}_0(\mathcal{T}_\omega)$.
In this section we study $D\mathcal{T}_\omega(\Psi)$, the
differential of the quasi-periodic renormalization operator. Concretely,
given $f\in \mathcal{B}$ we study the Fourier expansion of $D\mathcal{T}_\omega(\Psi) f$
in terms of the Fourier expansion of $ f$.
It will turns out that, fixed a positive
integer $k$, the spaces generated by functions of the type
$f(x) \cos(2\pi k \theta) + g(x) \sin(2 \pi k \theta)$ (with $f$ and
$g$ one dimensional real analytic functions) are invariant by
$D\mathcal{T}_\omega(\Psi)$. This allows us to reduce the study of
$D\mathcal{T}_\omega(\Psi)$ to a simpler operator $\mathcal{L}_\omega$. We finish
giving different spectral properties on the operator $\mathcal{L}_\omega$.
As usual the proofs have been moved to the end of the
section.

Given a function $f\in \mathcal{B}$ we can consider its complex Fourier
expansion in the periodic variable
\begin{equation}
\label{fourier expansion complex}
f(\theta,z)= \sum_{k\in \mathbb{Z}} c_k(z) e^{2\pi k\theta i },
\end{equation}
where
\[
c_k(z)= \int_{0}^{1} f(\theta,z) e^{-2\pi k\theta i} d\theta.
\]
We can also consider its real Fourier expansion
\begin{equation}
\label{fourier expansion real}
f(\theta,z)= a_0(z) + \sum_{k>0} a_k(z) \cos(2\pi k\theta) +  b_k(z) \sin(2\pi k\theta).
\end{equation}
Here the coefficients are given as,
\[
a_0(z)= \int_{0}^{1} f(\theta,z) d\theta,
\]
\[
a_k(z)=  \frac{1}{2} \int_{0}^{1} f(\theta,z) \cos(2\pi k\theta)  d\theta,  \quad k> 0,
\]
\[
b_k(z)= \frac{1}{2} \int_{0}^{1} f(\theta,z) \sin(2\pi k\theta)  d\theta,  \quad k>0.
\]
We have the well known relation
between both expansions, given by
$c_k(z) = \displaystyle \frac{a_k(z) + i b_k(z)}{2}$ when $k>0$ and
$c_0(z) = a_0(z)$.

Note that each function $c_k$ is holomorphic in $\mathbb{W}$
but not real holomorphic (the image of a real number will not be a real number
in general). On the other hand the real Fourier coefficient $a_k(z)$
and $b_k(z)$ are real holomorphic in $\mathbb{W}$.

Let $\psi=p_0(\Psi)$ be representative of $\Psi$ as
a one dimensional map. Evaluating  (\ref{diferencial de TT})
on $c(z) e^{2\pi k \theta i}$ (for any $k\neq0$) we have
\begin{equation}
\label{invariancia node}
[D\mathcal{T}_\omega(\Psi)] \left(c_k(z) e^{2\pi k\theta i} \right)=
\frac{1}{a} \left([ \psi'\circ\psi]( a z ) c_k(az)
+  [c_k \circ \psi](a z) e^{2\pi k \omega i} \right) e^{2\pi k \theta i }.
\end{equation}

On the other hand, when (\ref{diferencial de TT}) is
 evaluated at $c_0(z)$ one has
\[[D\mathcal{T}(\Psi)] \left(c_0(z) \right)= D\mathcal{R}_\delta(\psi) c_0(z),\]
as should be expected since $c_0 = p_0(f)$.

Given $U$ an open subset of $\mathbb{C}$ we will denote by $\mathcal{RH}(U)$ the set of
real holomorphic functions in $U$ and continuous in its closure.

Consider the operators,
\[
\begin{array}{rccc}
L_1: & \mathcal{RH}(\mathbb{W}) & \rightarrow & \mathcal{RH}(\mathbb{W})  \\
  &   g(z)  & \mapsto &
\displaystyle  \frac{1}{a} \psi'\circ\psi(a z) g(az),
\end{array}
\]
and
\[
\begin{array}{rccc}
L_2: & \mathcal{RH}(\mathbb{W}) & \rightarrow & \mathcal{RH}(\mathbb{W})  \\   &   g(z)  & \mapsto &
\displaystyle  \frac{1}{a} g\circ\psi(a z),
\end{array}
\] with $\psi=p_0(\Psi)$ and
$a=\psi(1)$.

Given a function $f\in \mathcal{B}$, we can consider its Fourier expansion
(\ref{fourier expansion complex}) and apply (\ref{invariancia node}), hence
\begin{equation}
\label{equation Fourier expansion differential renormalization operator}
\left[D\mathcal{T}_\omega(\Psi) f\right](\theta,z) = D\mathcal{R}_\delta[c_0](z)
+  \sum_{k\in \mathbb{Z}\setminus\{0\}} \left([L_1(c_k)](z) +
[L_2(c_k)](z) e^{2\pi k\omega i}\right) e^{2\pi k\theta i }.
\end{equation}
Looking at this formula it can be said that $D\mathcal{T}_\omega$ ``diagonalizes''
with respect to the complex Fourier base.

We define
\[
U_k:= \big\{ f \in B |\text{ } f(\theta, x) = u(x) \cos(2\pi k \theta), \text{ for
some } u\in \mathcal{RH}(\mathbb{W})\big\},
\]
and
\[
V_k:= \big\{ f \in B |\text{ } f(\theta, x) = v(x) \sin(2\pi k \theta), \text{ for
some } v\in \mathcal{RH}(\mathbb{W})\big\}.
\]

On the other hand, given $\omega \in \mathbb{T}$, consider the following operator
\begin{equation}
\label{equation maps L_omega}
\begin{array}{rccc}
\mathcal{L}_\omega: &  \mathcal{RH}(\mathbb{W})\oplus  \mathcal{RH}(\mathbb{W}) & \rightarrow
&  \mathcal{RH}(\mathbb{W})\oplus  \mathcal{RH}(\mathbb{W})  \\ \\
& \left( \begin{array}{c} u \\v \end{array} \right) &
\mapsto &  \left( \begin{array}{c}  L_1(u) \\ L_1(v)
\end{array} \right) +
\left( \begin{array}{cc}  \cos(2\pi \omega) & - \sin(2\pi \omega)  \\
\sin(2\pi \omega) & \cos( 2\pi \omega)
\end{array} \right)  \left( \begin{array}{c}  L_2(u) \\
L_2(v)
\end{array} \right) .
\end{array}
\end{equation}

Then we have the following result.

\begin{prop}
\label{proposition invariance of Fourier node spaces}
The spaces $U_k \oplus V_k$ are
invariant by $D\mathcal{T}(\Psi)$ for any $k>0$. Moreover $D\mathcal{T}_\omega (\Psi)$ restricted to $U_k \oplus V_k$ is
conjugated to $\mathcal{L}_{k\omega}$.
\end{prop}

Due to this proposition we have that the understanding of the
derivative of the renormalization operator in $\mathcal{B}$ is equivalent
to the study of the operator $\mathcal{L}_\omega$ for any $\omega\in \mathbb{T}$.
Therefore we focus now on the study of  $\mathcal{L}_\omega$.

Given a value $\gamma \in \mathbb{T}$, consider the rotation $R_{\gamma}$ defined
as
\begin{equation}
\label{equation rotation rgamma}
\begin{array}{rccc}
R_\gamma: &  \mathcal{RH}(\mathbb{W})\oplus  \mathcal{RH}(\mathbb{W}) & \rightarrow
&  \mathcal{RH}(\mathbb{W})\oplus  \mathcal{RH}(\mathbb{W})  \\ \\
& \left( \begin{array}{c} u \\v \end{array} \right) &
\mapsto &
\left( \begin{array}{cc}  \cos( 2\pi \gamma) & - \sin(2\pi \gamma)  \\
\sin(2\pi \gamma) & \cos(2\pi \gamma)
\end{array} \right)  \left( \begin{array}{c} u \\ v
\end{array} \right).
\end{array}
\end{equation}
Then we have the following result.

\begin{prop}
\label{proposition commutation of LLomega and Rgamma}
For any $\omega, \gamma \in \mathbb{T}$ we have that $\mathcal{L}_\omega$ and
$R_\gamma$ commute.
\end{prop}

This proposition has the following spectral consequences on $\mathcal{L}_\omega$.
\begin{cor}
\label{corollary rotational simetry eigenvectors}
For any eigenvector $(u,v)$ of $\mathcal{L}_\omega$ we have that $R_\gamma (u,v)$
is also an eigenvector of the same eigenvalue for any $\gamma \in \mathbb{T}$.
\end{cor}

\begin{cor}
\label{corollary multiplicity eigenvalues}
All the eigenvalues of $\mathcal{L}_\omega$ (different from zero) are either real with
geometric multiplicity even, or a pair of complex conjugate eigenvalues.
\end{cor}

On the other hand we have the following result on the dependence
of the operator with respect to $\omega$

\begin{prop}
\label{analytically denpendence Lomega}
The operator $\mathcal{L}_\omega$ depends analytically
on $\omega$.
\end{prop}

This result allows us to apply theorems
III-6.17 and VII-1.7 of \cite{Kat66}. These results imply that, as long as
the eigenvalues of $\mathcal{L}_\omega$ do not cross each other, then the eigenvalues
and their associated eigenspaces depend analytically on the parameter $\omega$.

We want to prove the compactness of $\mathcal{L}_\omega$ as an operator.
For technical reasons this can not be proved with $\mathcal{L}_\omega$ as an
operator on $\mathcal{RH}(\mathbb{W})\oplus\mathcal{RH}(\mathbb{W})$, but it can be
proved on a closed subspace of $\mathcal{RH}(\mathbb{W})\oplus\mathcal{RH}(\mathbb{W})$.

\begin{prop}
\label{proposition L_omega is compact}
Consider $K\subset \mathbb{W}$ a compact set in the complex plane, such that
$\psi(a\mathbb{W})\subset K$ and $a\mathbb{W} \subset K$
where $\psi= p_0 \left( \Psi\right)$. Let us denote by
$B= \mathcal{RH}(\mathbb{W})\cap C^0(K,\mathbb{C})$, which is a
Banach subspace of $\mathcal{RH}(\mathbb{W})$.

Then the operator $\mathcal{L}_\omega$ restricted to the  subspace
$B\oplus B \subset \mathcal{RH}(\mathbb{W})\oplus\mathcal{RH}(\mathbb{W})$
is well defined (i.e. $\mathcal{L}_\omega:B\oplus B\rightarrow B\oplus B$)
and it is compact.
\end{prop}

Recall that the compacity of an operator implies that its
spectrum is either finite or countable with $0$ on its
closure (see for instance theorem III-6.26 of \cite{Kat66}).

\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{SpecC.pdf}
\includegraphics[width=7.5cm]{SpecR.pdf}
\includegraphics[width=7.5cm]{SpecI.pdf}
\caption{ Numerical approximation of the
spectrum of $\mathcal{L}_\omega$ for $\omega \in \mathbb{T}$.
On the top we have the projection in the complex plane of the spectrum when
$\omega$ varies in $\mathbb{T}$. In the bottom we have the
evolution of the real (left) and the imaginary (right)
part of the spectrum with respect to $\omega$.
}
\label{Operator Spectrum}
\end{center}
\end{figure}

To finish this section we have included in figure
\ref{Operator Spectrum} a numerical approximation of
the spectrum of the operator $\mathcal{L}_\omega$ depending on $\omega$.
In the figure it can be observed that the different properties
(and their spectral consequences) on the operator stated above are satisfied.
The details on the numerical computations involved to approximate
the spectrum are described in \cite{JRT11c}. Several numerical tests on
the reliability of the results are also included there.

\subsubsection{Proofs}

\begin{proof}[Proof of proposition
\ref{proposition invariance of Fourier node spaces}]
Let $f$ be a function in $U_k \oplus V_k$, then we have
$f(\theta,x) = u(z) \cos(2\pi k \theta) + v(z) \sin (2\pi k\theta)$. Consider the
function $c(z) = \frac{u(z) + i v(z)}{2}$. Using formula (\ref{invariancia node})
on the function $u(z) = c(z) + \bar{c}(z)$ and doing some algebra one can see that
\[
\begin{array}{rcl}
[D\mathcal{T}_\omega(\Psi)](u(z) \cos(2\pi k \theta)) &  = & \phantom{-}[L_1(u)](z)\cos(2\pi k \theta) +
 [L_2(u)](z) \cos(2\pi k \omega) \cos(2\pi k \theta)
\\ \rule{0pt}{3ex} & &  - [L_2(v)](z)  \sin(2\pi k \omega)  \sin(2\pi k \theta),
\end{array}
\]
and, doing a similar calculation for $v(z) = i (\overline{c(z)} - c(z))$
\[
\begin{array}{rcl}
[D\mathcal{T}_\omega(\Psi)](v(z) \sin(2\pi k \theta))  &  =
& \phantom{+} [L_1(v)](z)\sin(2\pi k \theta)  + [L_2(u)](z)\sin(2\pi k \omega) \cos(2 \pi k \theta)
\\ \rule{0pt}{3ex} & &  + [L_2(v)](z) \cos(2\pi k \omega) \sin(2\pi k \theta).
\end{array}
\]

Now notice that there is a natural isomorphism between $U_k$ and
$ \mathcal{RH}(\mathbb{W})$ given by the function
$p_c: \mathcal{RH}(\mathbb{W}) \rightarrow U_k $ defined as
$p_c(f)(x):= \int_0^{1} f(\theta,x) \cos(2\pi k\theta) d\theta $ and
the function $i_c:  U_k \rightarrow \mathcal{RH}(\mathbb{W})$ defined as
$i_c(f)(\theta,x) = f(x) \cos(2\pi k\theta)$. The same argument can
be applied to $V_k$ considering the functions
$p_s:V_k \rightarrow \mathcal{RH}(\mathbb{W})$  and $i_s:\mathcal{RH}(\mathbb{W}) \rightarrow V_k $
defined as before but with $\sin(2\pi k \theta)$ instead of
$\cos(2\pi k \theta)$.
Then these functions can be used to define the isomorphic conjugacy
between $D\mathcal{T}_\omega(\Psi)$ and $\mathcal{L}_{k\omega}$.
\end{proof}
\begin{proof}[Proof of proposition
\ref{proposition commutation of LLomega and Rgamma}]
It follows from $L_1$ and $L_2$ being linear and the fact that any pair of
rotations commute.
\end{proof}

\begin{proof}[Proof of corollary
\ref{corollary rotational simetry eigenvectors}]
Suppose that $(u,v)$ is an eigenvector of eigenvalue $\lambda$, we have
$\lambda (u,v) = \mathcal{L}_\omega (u,v)$. Composing in both parts by $R_\gamma$ and
using the last proposition the result follows.
\end{proof}

\begin{proof}[Proof of corollary
\ref{corollary multiplicity eigenvalues}]
In the case of a real eigenvalue, suppose it has geometric multiplicity odd, then
its eigenspace is generated by $n$ vectors $y_1, y_2, \dots, y_n$, with $n$ odd.
We can consider $R_\gamma y_i$ for any $i$, which will also be in the eigenspace of the eigenvalue.
Since the vector $R_\gamma y_i$ is linearly independent with $y_i$ but it
is in the eigenspace, we have that it is generated by the other eigenvectors.
Then one of the original vectors can be replaced by $R_\gamma y_i$. Rearranging the
vectors if necessary we can suppose that $y_2=R_\gamma y_1$. Doing this
process repeatedly we will end up with an even number of vectors.

In the case of a complex eigenvalue, using that the operator $\mathcal{L}_\omega$
is real, if it has a complex eigenvalue $\lambda$ with
eigenvector $v_r + i v_i$, then $\bar{\lambda}$ will also be an eigenvalue with
eigenvector  $v_r - i v_i$. Given a complex pair of conjugate eigenvalues, the
restriction of the operator to the corresponding eigenspace can be written as
a uniform scaling composed with a rotation. It can happen that the space
generated by these two vectors is invariant by the rotation
$R_\gamma$ introduced before. Then the
multiplicity of the eigenvalue can be odd. Actually, if the pair of complex
eigenvalues are simple, then there exists a $\gamma_0 \in \mathbb{T}$ such that the
rotation associated to the pair of eigenvectors is $R_{\gamma_0}$.
\end{proof}

\begin{proof}[Proof of proposition
\ref{analytically denpendence Lomega}]
It follows from the fact that $\mathcal{L}_\omega$ is the
sum of two bounded linear operators (which do not depend on $\omega$)
times an entire function on $\omega$.
\end{proof}

\begin{proof}[Proof of proposition
\ref{proposition L_omega is compact}]
Note that it is enough to prove that the operators
$L_1$ and $L_2$ are well defined and compact.

Given a map in $g\in B=\mathcal{RH}(\mathbb{W})\cap C^0(K_1,\mathbb{C})$, consider
$\left[L_1(g)\right](z)= \frac{1}{a} \psi'\circ\psi(a z) g(az)$.
Since $\text{Cl}\left(a\mathbb{W}\right)\subset K$ then the map
$g(a\cdot)$ is in $\mathcal{RH}(\mathbb{W})\cap C^0(K,\mathbb{C})$.
Therefore, the map $L_1(g)$ belongs to $B$ which means that
$L_1:B\rightarrow B$ is well defined. On the other hand we
have that for any $g\in B$, $L_2(g)$ is defined as
$\left[L_2(g)\right](z) = \frac{1}{a} g\circ \psi(az)$. As
the set $K$ has been considered such that
$ \psi(a \mathbb{W})\subset K$, $L_2$ is also well defined.

Consider $U$ the unit ball of $B$.
Since $B$ is a
Banach space, to prove that $L_i$ is compact it is enough to prove that
 $L_i(U)$ is relatively compact (for $i=1,2$).
This follows easily using proposition 9.13.1 of \cite{Die69}.
For each compact set
in $K'$ in $\mathbb{W}$ we have that the maps $L_i(U)$ are bounded,
then it follows that $L_i(U)$ is relatively compact in
$C^0(K',\mathbb{C})$. Concretely, we can take $K'=K$ and
 we have that $L_i(U)\subset\mathcal{RH}(\mathbb{W})$
is relatively compact in $C^0(K,\mathbb{C})$, therefore it is
relatively compact in $B$.
\end{proof}

\section{Reducibility loss and quasi-periodic renormalization}
\label{chapter application}

In this section we use the renormalization operator
to study the reducibility loss bifurcations of
a two parametric family of q.p. forced map.
Concretely, the main result of this section is a proof
of the existence of reducibility loss bifurcations for a  two-parametric
family of  q.p. forced map satisfying
suitable conditions, see theorem \ref{thm existence of
non-reducibility direction}. In particular this
theorem applies to the case of the Forced Logistic
Map considered in \cite{JRT11p}, but this will be
discussed in \cite{JRT11c}.
The proof is not complete, in the sense
that we need to assume the injectiveness of the renormalization operator
$\mathcal{T}_\omega$ (see conjecture {\bf \ref{conjecture H2}}).

In section \ref{section boundaris of reducibility} we consider
certain sets $\Upsilon^+_n(\omega)$ and $\Upsilon^-_n(\omega)$
of codimension one in the space $\mathcal{B}$, which correspond to
the reducibility loss of the attracting $2^n$-periodic orbit.
We show that the intersection of these sets with the subset of uncoupled
maps corresponds to the sets $\Sigma_n$ of maps (in the one dimensional
case) such that its attracting $2^n$ periodic orbit is super-attracting.
The main result of this section relates the sets $\Upsilon^+_n(\omega)$
(respectively $\Upsilon^-_n(\omega)$)
for different values of $n$ through the renormalization operator $\mathcal{T}_\omega$.

In section \ref{Section consequences for a two parametric family of maps}
we consider a generic two parametric family of maps satisfying
certain hypotheses. We intersect the family with the previous
$\Upsilon^+_n(\omega)$ and $\Upsilon^-_n(\omega)$.
Differently to the one dimensional analogous, the intersection
gives a one dimensional curve
in the family, which corresponds to a
reducibility loss bifurcation. Using the (quasi-periodic)
renormalization operator and the results in section
\ref{section boundaris of reducibility}  we prove
the existence of reducibility loss bifurcation
curves on the parameter space of the family considered.
In theorem \ref{thm existence of
non-reducibility direction} we show that, given
a two parametric family which uncouples, there exist two
reducibility loss bifurcation curves  around the points
such that the uncoupled map has a super-attracting periodic orbit.
This fact was observed numerically in \cite{JRT11p}.

We also discus the weakening of the hypothesis of
theorem \ref{thm existence of non-reducibility direction}
and we give explicit expressions of the slopes of the  reducibility loss bifurcations
in terms of the family of maps and its iterates by the
(quasi-periodic) renormalization operator.

As in previous sections the proofs have been moved to the
end of each subsection.
\subsection{Boundaries of reducibility}
\label{section boundaris of reducibility}

In this section we  work in the analytic framework,
concretely we consider maps belonging to $\mathcal{B}$ the space
of the q.p forced one dimensional unimodal maps (as defined in
subsection  \ref{subsection study of TT}). Again,
let us consider the splitting  $\mathcal{B}=\mathcal{B}_0\oplus \mathcal{B}_{0}^{c}$
given by the projection $p_0[f](x):= \int_{0}^{1}
f(\theta,x)d\theta$, in other words the spaces given by  $\mathcal{B}_0=p_0(\mathcal{B})$ and
$\mathcal{B}_0^c = (\mathrm{Id} - p_0)(\mathcal{B})$, where $\mathrm{Id}$ is the
identity map. The renormalization operator for
q.p. forced maps is  denoted by $\mathcal{T}_{\omega}$, the
renormalization operator for one dimensional maps by $\mathcal{R}$ and
the fixed point by $\Phi$ independently of the operator (recall that the
fixed points of $\mathcal{R}$ extend automatically to fixed points of $\mathcal{T}_\omega$). Given a
map $F$ like (\ref{q.p. forced system interval}) with $f\in\mathcal{B}$ and $\omega\in\mathbb{T}$ we
denote by $f^{n}:\mathbb{T} \times \mathbb{R} \rightarrow \mathbb{R}$ the $x$-projection
of $F^n(x,\theta)$. Equivalently, $f^n$ can be defined through the recurrence
\begin{equation}
\label{definicio falan}
f^n(\theta, x) = f( \theta +(n-1) \omega, f^{n-1}(\theta,x)).
\end{equation}

In this subsection, differently to the previous one,
whenever  $\omega$ is used, it is assumed to
be Diophantine.
Let us denote by $\Omega$
the set of Diophantine numbers,
this is $\Omega= \Omega_{\gamma,\tau}$ the set of $\omega\in \mathbb{T}$
such that there exists $\gamma >0$ and $\tau \geq 1$
such that
\[
|q w - p| \geq \frac{\gamma}{|q|^{\tau}}, \quad
\text{ for all } (p,q) \in \mathbb{Z} \times (\mathbb{Z} \setminus \{0\}).
\]

Additionally, we assume that the following conjecture is true.

\begin{conj}
\label{conjecture H2}
The operator $\mathcal{T}_{\omega}$ (for any $\omega\in\Omega$) is an injective
function when restricted to the domain $\mathcal{B}\cap \mathcal{D}(\mathcal{T})$. Moreover,
there exist $U$ an open set of $\mathcal{D}(\mathcal{T})$ containing $W^u(\Phi,\mathcal{R})
\cup W^s(\Phi,\mathcal{R})$\footnote{Here $W^s(\Phi,\mathcal{R})$ and $W^u(\Phi,\mathcal{R})$
are considered as the inclusion in $\mathcal{B}$ of the
stable and the unstable manifolds of the fixed point $\Phi$ (given by
{\bf H0}) by the map $\mathcal{R}$ in the topology of $\mathcal{B}_0$ (the inclusion
of one parametric maps in $\mathcal{B}$).}
where the operator $\mathcal{T}_\omega$ is differentiable.
\end{conj}

The first part of the conjecture is proved for the
one dimensional case in \cite{MvS93}. The proof consists
of, given two maps with the same image by the operator, first to show
that their renormalization interval is the same and then to expand
the image of the maps around their fixed point and then deduce
that the original maps are the same maps. With our approach to the
quasi-periodic case, as we do not have an equivalent concept to
renormalization interval, the
same argument is no longer applicable. Despite the analogy
with the one dimensional case, there is, a priori, no
evidence about the conjecture.  A posteriori, we
have that the results obtained assuming that the conjecture is true
are coherent with the dynamics of the Forced Logistic Map.
In \cite{JRT11c} we compute numerically the slopes
of the reducibility loss bifurcations of the FLM by
two independent methods. The first method is computing the slope using
the dynamical characterization of the bifurcations. The
second one is using the formulas given in corollary
\ref{corollary non-reducibility directions}.
Both coincide up to a reasonable accuracy.

The second part of the conjecture is only introduced to simplify
the forthcoming discussion, but it can be avoided if necessary.
See remark \ref{remark avoiding second part of conjecture H2)}
for details.

Whenever the conjecture {\bf \ref{conjecture H2}} is needed for a
result it is explicitly stated in the hypotheses.

Let $K_0>0$ be a fixed constant value. Then we can consider the sets
\[
\Upsilon^+_{n} (\omega)= \left\{ f \in \mathcal{B} \left| \begin{array}{c}
\displaystyle \text{ There exists } x_0 \in \mathcal{RH}(\mathbb{B}_\rho,\mathbb{W})
\text{ with } x_0(\theta + 2^n \omega) = f^{2^n} (\theta,x_0(\theta))
\text{ s. t. } \\
\rule{0pt}{3ex}
\displaystyle
\int_{0}^{1} \ln|  D_x f^{2^n}(\theta,x_0(\theta) )| d\theta < - K_0
\text{ and }\min_{\theta\in \mathbb{T}} D_x \left( f^{2^n}\right)(\theta, x_0(\theta))=0.
\end{array} \right. \right\},
\]
and
\[
 \Upsilon^-_{n}(\omega) = \left\{ f \in \mathcal{B} \left| \begin{array}{c}
\displaystyle \text{ there exists } x_0 \in  \mathcal{RH}(\mathbb{B}_\rho,\mathbb{W})
\text{ with } x_0(\theta + 2^n \omega) = f^{2^n} (\theta,x_0(\theta))
\text{ s. t. } \\
\rule{0pt}{3ex}
\displaystyle
 \int_{0}^{1} \ln|  D_x f^{2^n}(\theta,x_0(\theta) )| d\theta < -K_0
\text{ and }\max_{\theta\in \mathbb{T}} D_x \left( f^{2^n}\right) (\theta, x_0(\theta))=0.
\end{array} \right. \right\}.
\]

Note that (due to the two first conditions) these sets are
contained in the set of all the maps in $\mathcal{B}$ which have
a $2^n$-periodic attracting curve. We require
the integral being less than $-K_0$ instead of being less than $0$
for technical reasons. The third condition is imposed with the
aim that these sets correspond to the bifurcation manifold
associated to the reducibility loss. We have the following properties
which give a good characterization of these sets.

\begin{prop}
\label{prop relacio lambda upsilon}
Let $\Sigma_{n}$ be the inclusion in $\mathcal{B}$ of the
set of one dimensional unimodal maps with
a super-attracting $2^n$ periodic orbit.
We have  that
\[\Upsilon^+_{n}(\omega)\cap \mathcal{B}_0 = \Upsilon^-_{n}(\omega) \cap \mathcal{B}_0 = \Sigma_{n}, \]
for any $\omega\in \Omega$.
\end{prop}

\begin{prop}
\label{prop upsilon banach manifold}
Let $f\in\Upsilon^+_n(\omega)$ (respectively $f\in\Upsilon^-_n(\omega)$) and  let $x$ be its $2^n$-periodic curve. If
$D_x \left(f^{2^n}\right)(\theta, x(\theta))$ has a unique non-degenerate absolute
minimum\footnote{It can have several local minima but the absolute minimum has to be
unique and not degenerate.} (respectively maximum), then
$\Upsilon^+_n(\omega)$  (respectively  $f\in\Upsilon^-_n(\omega)$)
is a codimension one
manifold in a neighborhood of $f$.
\end{prop}

\begin{prop}
\label{prop upsilon is reducibility loss}
Let $\{f_\mu\}_{\mu\in A}$ be a one parametric family of maps
such that:
\begin{enumerate}
\item There exist a parameter value $\mu_0$ for which the family crosses
$\Upsilon_n^+(\omega)$ (respectively $\Upsilon_n^-(\omega)$) transversely at $\mu=\mu_0$.
\item
Consider $x_{\mu_0}$ the $2^n$ periodic
invariant curve of $f_{\mu_0}$ given by the definition of
$\Upsilon_n^+(\omega)$ (respectively  $\Upsilon_n^-(\omega)$)
such that
$D_x\left(f^{2^n}_{\mu_0}\right)(\theta,  x_{\mu_0}(\theta))$ has a
unique non-degenerate minimum (respectively maximum).
\end{enumerate}
Then we have that the invariant periodic curve $x_{\mu_0}$ extends to a
periodic invariant curve\footnote{To extend the periodic invariant curve
we need to reduce $\rho$ (the width of the band of analyticity
with respect to $\theta$) to be small enough in terms of $K_0$, but this
reduction of $\rho$ is
done only once.} $x_{\mu}$ of $f_\mu$ for any $\mu$ in
an open neighborhood of $\mu_0$. Additionally, this invariant curve
undergoes a reducibility loss bifurcation at $\mu=\mu_0$.
\end{prop}

Let us introduce some notation to state the next result. Consider the map,
\begin{equation}
\label{equation map T}
\begin{array}{rccc}
T:& \mathbb{T} \times \mathcal{D}(\mathcal{T}) &\rightarrow &  \mathbb{T} \times  \mathcal{B}  \\
\displaystyle \rule{0pt}{3ex} & (\omega,f) & \mapsto &  \rule{0pt}{3ex}
\displaystyle  (2\omega, \mathcal{T}_\omega (f)),
\end{array}
\end{equation}
where $\mathcal{T}_\omega$ is the renormalization operator for q.p. forced maps,
as in section \ref{section definition of q.p. renormalization},
and the set $\mathcal{D}(\mathcal{T}) \subset \mathcal{B}$ is its domain of definition.
Recall that to have $\mathcal{T}_\omega$ well defined it is not necessary
$\omega\in\Omega$ (i.e. $\omega$ Diophantine).
Additionally, if $\omega\in \Omega$ then
we have that $2^k\omega \in \Omega$ for any $k\in \mathbb{Z}$, therefore
$T(\Omega \times \mathcal{D}(\mathcal{T})) \subset \Omega \times \mathcal{B}$.
On the other hand we have that the
sets $\Upsilon_n^+(\omega)$ and $\Upsilon_n^-(\omega)$
are only defined for $\omega \in \Omega$.

\begin{defin}
\label{definition times renormalizable}
We will say that a pair $(\omega, f) \in \mathbb{T} \times \mathcal{B}$ is {\bf $n$-times
renormalizable} if $T^k(\omega,f) \in \mathbb{T} \times \mathcal{D}(\mathcal{T}) $ for
$k=0,\dots, n-1$.
\end{defin}

Consider a pair $(\omega,f_0)\in \Omega\times\mathcal{B}$  with a $2$-periodic
invariant attracting curve $x_0$ with rotation number $\omega$.
 Assume that the Lyapunov exponent of the curve is less that $-K_0$,
with $K_0>0$ a fixed value, in other words,
\[
 \int_{0}^{1} \ln|  D_x f_0^{2}(\theta,x_0(\theta) )| d\theta < -K_0 . \]
In forthcoming lemma \ref{Lemma proof prop upsilon banach} we will prove that
the persistence of
the invariant curve extends to a neighborhood of $f_0$
(if the width $\rho$ of the band $\mathbb{B}_\rho$ around the torus
$\mathbb{T}$ is small enough with respect to $K_0$).
Let $V\subset \mathcal{B}$ be this neighborhood, and let $x:
V \rightarrow \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})$, be the period $2$ invariant curve
associated to $f$. Here $\mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})$ denotes the space of functions
$f: \mathbb{B}_\rho\rightarrow \mathbb{C}$ which are real analytic in $\mathbb{B}_\rho$ and
continuous in its closure. Then we can define
the map $G_1 $ as
\begin{equation}
\label{equation definition G1}
\begin{array}{rccc}
G_1:& \Omega\times V &\rightarrow & \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C}) \\
\rule{0pt}{3ex} &  (\omega,g) & \mapsto &
D_x g \big(\theta+\omega,g(\theta, \left[x(\omega,g)\right](\theta))\big)
D_x g \big(\theta,\left[x(\omega,g)\right](\theta)\big).
\end{array}
\end{equation}
On the other hand, consider the minimum and the
maximum as operator between spaces of functions:
\begin{equation}
\label{equation definition of minim}
\begin{array}{rccc}
m:& \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C}) &\rightarrow & \mathbb{R} \\
\rule{0pt}{3ex} & g & \mapsto & \displaystyle \min_{\theta \in \mathbb{T}} g(\theta).
\end{array}
\end{equation}
and
\begin{equation}
\label{equation definition of maximum}
\begin{array}{rccc}
M:& \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C}) &\rightarrow & \mathbb{R} \\
\rule{0pt}{3ex} & g & \mapsto & \displaystyle \max_{\theta \in \mathbb{T}} g(\theta).
\end{array}
\end{equation}

We have the following theorem, which relates the manifolds
$\Upsilon_n^+(\omega)$ and $\Upsilon_n^-(\omega)$ for different $n$ through the
renormalization operator.

\begin{thm}
\label{theorem local chard of upsilon n}
Let $\omega\in\Omega$ and $f\in \Upsilon_n^+(\omega)$, respectively
$f\in \Upsilon_n^-(\omega)$,  be
a function such that the pair $(\omega,f)$ is $n-1$ times renormalizable.
Additionally assume that conjecture {\bf \ref{conjecture H2}} is true.
Then there exist $U$ a neighborhood (in $\mathcal{B}$)  of $f$ such that
\[
U\cap \Upsilon_n^+(\omega) = \{ f\in U | \thinspace G_1^+(T^{n-1} (\omega,f) ) =0\},
\]
respectively
\[
U\cap \Upsilon_n^-(\omega) = \{ f\in U | \thinspace G_1^-(T^{n-1} (\omega,f) ) =0\},
\]
where
\[
G^+_1(\omega, g) := m\circ G_1(\omega,g),
\]
and respectively
\[
G^-_1(\omega, g) := M\circ G_1(\omega,g),
\]
with $G_1$, $m$ and $M$ defined in equations (\ref{equation definition G1}),
(\ref{equation definition of minim})  and (\ref{equation definition of maximum}).
\end{thm}

{\bf Proofs}

\begin{proof}[Proof of proposition \ref{prop relacio lambda upsilon}]
We will do the proof only for the case $\Upsilon^+_n(\omega)$. The case
$\Upsilon^-_n(\omega)$ is completely analogous.

If we have a map $f_0\in \Sigma_n\subset \mathcal{B}_0$,
it has a super-attracting periodic orbit $x_0$, then we
have that its Lyapunov exponent is $-\infty$, since
$D_x f^{2^n} (\theta,x_0) \equiv 0$. Therefore $\Sigma_n \subset \Upsilon_n^+(\omega) \cap \mathcal{B}_0 $.

Consider $f\in \Upsilon_{n}^+\cap \mathcal{B}_0$. Since
$f$ is in $\mathcal{B}_0$ it does not depend on $\theta$. Consider
also $h=f^{2^n}$,
which neither depends on $\theta$.  On the other hand
using that $f$ is in $\Upsilon_{n}^+(\omega)$ we have that there exists a function
$x\in \mathcal{RH}(\mathbb{B}_\rho,\mathbb{W})$ satisfying the following invariance equation
\[x(\theta +2^n \omega) = h(x(\theta)). \]
Differentiating the invariance equation we have
\[ x'(\theta+ 2^n \omega) =   h'(x(\theta)) x'(\theta). \]
From the fact that $f \in \Upsilon^+_{n}(\omega)$ it follows that there exist a $\theta_0$ such
that $h'(x(\theta_0))= 0$. Using the last equation we have that $x'$ is
zero in a dense subset of $\mathbb{T}$ and using the continuity of
$x'$ we have that $x'\equiv 0$, therefore $x$ is constant.
Finally note that as $x=h(x)=f^{2n}(x)$ and
$D_x f^{2^n}(x)=0$, we have that  $f$ belongs to $\Sigma_n$.
\end{proof}

\begin{proof}[Proof of proposition \ref{prop upsilon banach manifold}]
As before we prove only the case of $\Upsilon_n^+$. We start with
a preliminary lemma.
\begin{lem}
\label{Lemma proof prop upsilon banach}
Consider $g_0\in \mathcal{B}=\mathcal{B}(\mathbb{B}_\rho,\mathbb{W})$ and $x_0$ an invariant curve of $g_0$.
Assume that
\begin{equation}
\label{eqn lemma prop upsilon banach}
\int_{0}^{1} \ln|  D_x g_0(\theta,x_0(\theta) )| d\theta < - K < 0.
\end{equation}
Then, for a sufficiently small value of $\rho$, there exist a neighborhood
$U$ of $g_0$ and a smooth function $x:U\rightarrow \mathcal{B}$ such that
$x(g)$ is an invariant curve of $g$ for any $g\in U$ and
$x(g_0)=x_0$.
\end{lem}
\begin{proof}
This lemma corresponds to the analytic version of the continuation
problem of an invariant curve. In \cite{JT08} it is studied
the $C^r$ version of this problem.
The authors prove that the curve can be continued if $1$ does not belong
to the spectrum of the transfer operator $\mathcal{L}$ associated
 to the problem (see section 3.3 of \cite{JT08}). Then it is
shown that the spectrum of $\mathcal{L}$ is contained in the disk
of radius $b=\exp\left(\int_{0}^{1} \ln|
D_x g_0(\theta,x_0(\theta) )|d\theta\right)$.

Let $\mathbb{B}_\rho$ be a band of width $\rho$ around the real torus $\mathbb{T}$,
and $\mathcal{H}(\mathbb{B}_\rho,\mathbb{C})$ denote the space of holomorphic maps from
$\mathbb{B}_\rho$ to $\mathbb{C}$ and continuous on the closure of $\mathbb{B}$.
Note that transfer operator $\mathcal{L}$ can be considered both, in
$C^r(\mathbb{T},\mathbb{R})$  endowed  with the standard $C^r$-norm,
or in $\mathcal{H}(\mathbb{B}_\rho,\mathbb{C})$ endowed with the supremum norm. To
distinguish the spectrum of the transfer operator with
respect to the norm considered we will denote
each of the respective cases by $ \text{Spec}(\mathcal{L}, C^r)$ or
$\text{Spec}(\mathcal{L},\mathcal{H})$. Using theorem 9.2 of \cite{HL07},
we have that
\[
\partial \text{Spec}(\mathcal{L},\mathcal{H}) \subset  \text{Spec}(\mathcal{L},C^r) + O(\rho),
\]
for $\rho>0$ small enough.

The notation $A\subset B + O(\rho)$ means that
there exists a constant $C>0$ such that for any $a\in A$
there exists $b\in B$ with $d(a,b)\leq C\rho$.

Using equation (\ref{eqn lemma prop upsilon banach}) we have
that $b<1$. Then there exists a sufficiently small  $\rho$
such that $b+\rho<1$. Then using that $\partial \text{Spec}(\mathcal{L},\mathcal{H})$
is contained in the disc of radius $b+\rho<1$  one can
extend the persistence of invariant attracting curves to the analytic case.
\end{proof}

Consider an arbitrary function $f_0\in \Upsilon_n^+(\omega)$. We have that
there exists  $x_0: \mathbb{B}_\rho\rightarrow \mathbb{W}$ which is a $2^n$-invariant
 attracting curve of $f_0$. We can consider the auxiliary function
\[
\begin{array}{rccc}
F:& \mathcal{B}\times \mathcal{RH}(\mathbb{B}_\rho,\mathbb{W}) &\rightarrow & \mathcal{RH}(\mathbb{B}_\rho,\mathbb{W})  \\
\rule{0pt}{3ex} & (g,x )  & \mapsto & [F(g,x)](\theta):= x(\theta + 2^{n} \omega) -
g^{2^n}(\theta,x(\theta)).
\end{array}
\]

We have that the Lyapunov exponent of the curve is less than $-K_0$.
Using the lemma \ref{Lemma proof prop upsilon banach} we
have that, if the width $\rho$ of the band $\mathbb{B}_\rho$ around the torus
$\mathbb{T}$ is small enough with respect to $K_0$,
then  there exists a neighborhood $U_n$ of
$f_0$ in $\mathcal{B}$ and a function
\[
\begin{array}{rccc}
x:& U_n &\rightarrow & \mathcal{RH}(\mathbb{B}_\rho,\mathbb{W}) \\
\rule{0pt}{3ex} &  g & \mapsto & x(g),
\end{array}
\]
with $x(f_0)=x_0$ and such that $x(f)$ is a $2^n$ periodic curve of
$f$ for any $f\in U_n$. Moreover, due to the continuity
of the Lyapunov exponent, we can suppose that the Lyapunov
exponent of $x(g)$ is negative for any $g\in U_n$, replacing
$U_n$ by a smaller neighborhood if necessary.

Now we consider the auxiliary function
\begin{equation}
\label{equation definition tildeGn}
\begin{array}{rccc}
\tilde{G}_n:& U_n &\rightarrow & \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C}) \\
\rule{0pt}{3ex} &  f & \mapsto & [\tilde{G}_n(f)](\theta):=
D_x \left(f^{2^n}\right) \left(\theta, [x(f)](\theta)\right),
\end{array}
\end{equation}
and consider also the minimum operator defined as in
(\ref{equation definition of minim}).

By hypothesis we have that $ \left[D_x \left(f_0^{2^n}\right)\right](\theta, [x(f_0)](\theta))$ has
a unique minimum, therefore we can apply proposition
\ref{proposition diferentiability of minimum} (in appendix \ref{appendix
minimum function}). The uniqueness of the minimum extends
to $U_n$, replacing it again by a smaller neighborhood if necessary. Therefore
we have that the map $G_n:U_n \rightarrow \mathbb{R}$ defined as $m\circ \tilde{G}_n$
is differentiable. Let us remark that $G_n$ depends indeed on $\omega$, but the
differentiability is only needed with respect to $g$.
Then we have that
\begin{equation}
\label{equation local chard of Upsilon}
\Upsilon_n^{+}(\omega) \cap U_0= \{ f \in U |\thinspace G_n (\omega,f)=0\},
\end{equation}
which completes the proof.
\end{proof}

\begin{proof}[Proof of proposition \ref{prop upsilon is reducibility loss}]
As before we only consider the case involving $\Upsilon_n^{+}(\omega)$.
Consider $f_{\mu_0}$ the intersection of the family with the set
$\Upsilon_n^{+}(\omega)$.
Using the second hypothesis of the proposition we are under the
same hypothesis of proposition
\ref{prop upsilon banach manifold}. Following the proof of this proposition
we have that there exists $U_0$ an open neighborhood of $f_{\mu_0}$ such that
$\Upsilon_n^{+}(\omega) \cap U_0$ is given by equation
(\ref{equation local chard of Upsilon}) with $G_n= m\circ \tilde{G}_n$ and
the maps $m$ and
$\tilde{G}_n$ are given by equations (\ref{equation definition of minim})
and (\ref{equation definition tildeGn}). Moreover we also have
that  $G_n$  is differentiable in $U_0$.

Using that the family $f_\mu$ crosses transversely the manifold
$\Upsilon_n^{+}(\omega) $
we have that\\  $\partial_\mu G_n(f_\mu)_{|\mu=\mu_0} \neq 0$. Actually we can assume
that $\partial_\mu G_n(f_\mu)_{|\mu=\mu_0} < 0$, otherwise we can
replace $\mu$ by $\tilde{\mu}= 2\mu_0 - \mu$ and consider the family
$f_{\tilde{\mu}}$ instead of $f_\mu$.  Recall also that
\[
G_n(f_{\mu})  = \min_{\theta\in \mathbb{T}} D_x \left( f^{2^n}_{\mu}\right)(\theta, x_{\mu}(\theta)).
\]
Then for  $\mu < \mu_0$  we have $G_n(f_\mu)>0$ and therefore we have
that $D_x \left( f^{2^n}_{\mu_0}\right)(\theta, x_{\mu_0}(\theta))>0$
for any $\theta\in \mathbb{T}$. Using corollary 1 of \cite{JT08}
we have that $x_{\mu}$ is reducible.
Moreover, due to second hypothesis of the proposition we have that
$D_x \left( f^{2^n}_{\mu_0}\right)(\theta, x_{\mu_0}(\theta))$
has a double zero $\theta_0$. Finally using the transversality hypothesis
we have that $\partial_\mu \left(D_x \left( f^{2^n}_{\mu_0}\right)
(\theta_0, x_{\mu_0}(\theta_0))\right) \neq 0$. This proves that $x_{\mu}$ undergoes
 a reducibility loss bifurcation.
\end{proof}

\begin{proof}[Proof of theorem \ref{theorem local chard of upsilon n}]
Once again, we only consider the case involving $\Upsilon_n^{+}(\omega)$, since
the other case is completely analogous. Let us introduce the following lemma
for the proof of theorem.

\begin{lem}
\label{lemma invariancia upsilons}
Let $\mathrm{Im} (\mathcal{T}_\omega)$ be the image of the operator $\mathcal{T}_\omega$.
Assume that we have $\omega\in \Omega$, then we
have that
\begin{equation}
\mathcal{T}_\omega(\Upsilon_{n}^+ (\omega) \cap \mathcal{D}(\mathcal{T})) =
 \Upsilon_{n-1}^{+}(2\omega) \cap \mathrm{Im} (\mathcal{T}_\omega) .
\end{equation}
\end{lem}
\begin{proof}
Given $f\in \mathcal{T}_\omega(\Upsilon^+_{n}(\omega) \cap \mathcal{D}(\mathcal{T}))$, we
have that there exists a function $g \in \Upsilon^+_n(\omega)$ such that
\begin{equation}
\label{equation demo prop invariancia lambdas}
f(\theta, x) =\frac{1}{a}g(\theta+\omega,g(\theta,ax)),
\end{equation}
with $a= \int_0^{1}g(\theta, 1) d\theta$.
Note that the rotation number when we compose $g$ with itself
is $\omega$, while the rotation number when we compose
$f$ with itself is $2\omega$; this is not obvious from the notation
but it is important to have it in mind for this proof.

To prove the first inclusion it is enough to check that $f$ is
in $\Upsilon^+_{n-1}(2\omega)$ since $f$ is trivially in $\mathrm{Im}(\mathcal{T}_\omega)$.

Using that $g$ is in $\Upsilon^+_{n}(\omega)$ we have that there exists
$x_0: \mathbb{B}_\rho \rightarrow \mathbb{W}$ with
\[x_0(\theta + 2^n \omega) = g^{2^n} (\theta,x_0(\theta)), \]
and such that
\[\min_{\theta\in \mathbb{T}} D_x \left( g^{2^n}\right)(\theta, x_0(\theta))=0.\]

Using equation (\ref{equation demo prop invariancia lambdas}) it
is easy to check that
\begin{equation}
\label{equation demo prop invariancia lambdas 2}
f^{2^{n-1}} (\theta,x) = \frac{1}{a} g^{2^n}(\theta, ax),
\end{equation}
and
\begin{equation}
\label{equation demo prop invariancia lambdas 3}
D_x \left(f^{2^{n-1}} \right)  (\theta,x) = D_x\left(g^{2^n}\right)
(\theta, ax),
\end{equation}
for any $\theta\in\mathbb{B}_\rho$ and $x\in \mathbb{W}$.

Consider $x_1$ the function defined as
$x_1(\theta)=\frac{1}{a} x_0(\theta)$. From the last two equalities it follows that
$x_1(\theta + 2^{n-1} (2 \omega)) = f^{2^{n-1}} (\theta,x_1(\theta))$, and
\[\min_{\theta\in \mathbb{T}}  D_x \left( f^{2^{n-1}}\right)(\theta, x_1(\theta)) =
\min_{\theta\in \mathbb{T}}  D_x \left( g^{2^{n}}\right)(\theta, x_0(\theta)) = 0.
\]
Therefore $f$ is in $\Upsilon^+_{n-1}(2\omega)$.

Let us see the converse inclusion. Consider
$f\in  \Upsilon^+_{n-1}(2\omega) \cap \mathrm{Im} (\mathcal{T}_\omega)$.
Since $f$ is in $\mathrm{Im} (\mathcal{T}_\omega)$  we have that
there exists $g\in \mathcal{D}(\mathcal{T})$  with $f=\mathcal{T}_\omega(g)$. Therefore
we only have to prove that $g$ is in $\Upsilon^+_n(\omega)$.
Using $f=\mathcal{T}_\omega(g)$ one has that equation
(\ref{equation demo prop invariancia lambdas}) is satisfied again and
this implies that
equations  (\ref{equation demo prop invariancia lambdas 2})  and
 (\ref{equation demo prop invariancia lambdas 3}) also hold.
From $f \in \Upsilon^+_{n-1}(2\omega) $ we have that
there exists a function  $x_1:\mathbb{B}_\rho \rightarrow \mathbb{W}$ with
$x_1(\theta + 2^{n-1} (2 \omega)) = f^{2^{n-1}} (\theta,x_1(\theta))$ and
\[
\min_{\theta\in \mathbb{T}}  D_x \left( f^{2^{n-1}}\right)(\theta, x_1(\theta))=0.
\]
Consider now $x_0(\theta) := a x_1(\theta)$, then using equation
(\ref{equation demo prop invariancia lambdas 2})  we have that
\[ a x_1(\theta + 2^{n-1} (2 \omega)) =  g^{2^n}(\theta, ax_1(\theta)),\]
for any $\theta \in \mathbb{T}$. Using
(\ref{equation demo prop invariancia lambdas 3})  we obtain
\[
\min_{\theta\in \mathbb{T}}  D_x \left( g^{2^{n}}\right)(\theta, x_1(\theta)) = 0,
\]
what completes the proof of the lemma.
\end{proof}

The proof will follow by induction. Note that the case $n=1$ is satisfied trivially.
Then we can assume that the case $n-1$ is true and check the case $n$.

We have that $(f,\omega)$ is renormalizable,
then $f\in D(\mathcal{T})\cap \Upsilon^+_n(\omega)$. Using conjecture
{\bf \ref{conjecture H2}} we have that a point belongs to $\Upsilon^+_n(\omega)\cap \mathcal{D}(\mathcal{T})$ if,
and only if, $\mathcal{T}_\omega(g)$ belongs to
$\mathcal{T}_\omega(\Upsilon^+_n(\omega)\cap \mathcal{D}(\mathcal{T}))$. Using lemma
\ref{lemma invariancia upsilons} we have that
$\mathcal{T}_\omega(\Upsilon^+_n(\omega)\cap \mathcal{D}(\mathcal{T}))
=  \Upsilon^+_{n-1}(2\omega) \cap \mathrm{Im} (\mathcal{T}_\omega)$.

At this point we need to consider the case $n=2$ independently.
In the case $n=2$ we have that $\mathcal{T}_\omega(\Upsilon^+_2(\omega)\cap \mathcal{D}(\mathcal{T}))
=  \Upsilon^+_{1}(2\omega) \cap \mathrm{Im} (\mathcal{T}_\omega)$, then we have
that there exists $U_1$ a neighborhood of $\mathcal{T}_\omega(f)$ such
that
\[
U_{1} \cap \Upsilon^+_{1}(2 \omega) = \{g \in U | \thinspace G_1( 2 \omega,g) =0 \}.
\]
Consider $U_2= \mathcal{T}_\omega^{-1}(U_1)$, using that $\mathcal{T}_{\omega}$ is
continuous, we have that $U_2$ is an open neighborhood of $f$. Then we have
\[U_2 \cap \Upsilon^+_2(\omega)
= \mathcal{T}_{\omega} ^{-1} (U_{1}\cap \Upsilon^+_{1}(2\omega) ) =
\{ f \in U_{1} | \thinspace G_1( 2 \omega,\mathcal{T}_\omega(f) ) = G_1(T(\omega,f)) =0 \},
\] which finishes the proof for the case $n=2$.

In the case $n>2$ we have that
the pair $(2\omega, \mathcal{T}_\omega(f))$  is $n-2$ times renormalizable
We apply now the induction hypothesis, then we have that
there exists $U_{n-1}$ a neighborhood of $\mathcal{T}_\omega(f)$ such that
\[
U_{n-1} \cap \Upsilon^+_{n-1}(2 \omega) = \{g \in U | \thinspace G_1(T^{n-2} (2 \omega,g))  =0 \}.
\]

Consider $U_n=\mathcal{T}_{\omega}^{-1}(U_{n-1})$, since $\mathcal{T}_{\omega}$ is a
continuous function, we have that $U_n$ is an open neighborhood of $f$ and
then we have
\[U_n \cap \Upsilon^+_n(\omega)
= \mathcal{T}_{\omega} ^{-1} (U_{n-1}\cap \Upsilon^+_{n-1}(2\omega) ) =
\{ f \in U_{n-1} | \thinspace G_1(T^{n-2} (2 \omega,\mathcal{T}_\omega(f)) ) =0 \}.\]
Using that $T^{n-2} (2 \omega,\mathcal{T}_\omega(f)) = T^{n-1} ( \omega,f)$
the proof is finished.
\end{proof}

\subsection{Consequences for a two parametric family of maps}
\label{Section consequences for a two parametric family of maps}

Consider a two parametric family of maps like
(\ref{q.p. forced system interval}). For the rest of this
section we assume that $\omega$ is a
fixed Diophantine number ($\omega\in \Omega$). Then the family of
maps is determined by two parametric family of maps
 $\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A}$
contained in $\mathcal{B}$ (concretely they are unimodal q.p. forced maps),
where $A = [a,b]\times[0,c]$ and $a$, $b$ and $c$ are real numbers
(with $a<b$ and $0<c$).

We assume that the dependency on the parameters is analytic, then
the family can be thought as an analytic map
$c:A \rightarrow \mathcal{B}$. In this subsection we prove (under
suitable hypotheses) the existence of reducibility loss bifurcations like the
ones observed in the numerical computations of the Forced Logistic
Map \cite{JRT11p, JRT11c}.

We will consider families of maps satisfying the following hypothesis.
\begin{description}
\item[H1)]  The family $\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A}$ uncouples for
$\varepsilon=0$, in the sense that the family $\{c(\alpha,0)\}_{\alpha\in[a,b]}$
does not depend on $\theta$ and it
is a one parametric family of unimodal maps has a full cascade of
period doubling bifurcations. We assume that the family
$\{c(\alpha,0)\}_{\alpha\in[a,b]}$ crosses transversally the
stable manifold of $\Phi$ the fixed point
of the renormalization operator and each of
the manifolds $\Sigma_n$ for any $n\geq 1$, where
$\Sigma_{n}$ is the inclusion in $\mathcal{B}$ of the
set of one dimensional unimodal maps with
a super-attracting $2^n$ periodic orbit.
\end{description}

In other words, we assume that the family $c(\alpha,\varepsilon)$
can be written as,
\[
c(\alpha,\varepsilon)=c_0(\alpha) + \varepsilon c_1(\alpha,\varepsilon),
\]
with $\{c_0(\alpha)\}_{\alpha\in[a,b]}\subset \mathcal{B}_0$  having a full
cascade of period doubling bifurcations.

Given a family $\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A}$ satisfying
hypothesis {\bf H1} let $\alpha_n$ be the parameter value for
which the uncoupled family $\{c(\alpha,0)\}_{\alpha\in[a,b]}$
intersects the manifold $\Sigma_n$. Note that the critical
point of the map $c(\alpha_n,0)$ is a $2^n$-periodic orbit.
The main goal of this subsection is to prove that for every
parameter value $(\alpha_n,0)$ there are two curves in
the parameter space, one corresponding to a reducibility
loss bifurcation and the other one corresponding to a reducibility
recover. These curves are born at the point $(\alpha_n,0)$ of
the parameter space.

Consider a map $f_0\in \mathcal{B}$ and $\omega \in \Omega$,
such that $f$ has a
periodic invariant curve $x_0$ of rotation number $\omega$
with a  Lyapunov exponent less or equal than certain
$-K_0<0$. Recalling the arguments in
the proof of proposition \ref{prop upsilon banach manifold} we have that
there exist a neighborhood $V\subset \mathcal{B}$ of $f_0$ and a map
$x\in \mathcal{RH}(\mathbb{B}_\rho, \mathbb{W})$ such that $x(f)$ is a periodic
invariant curve of $f$  for any $f\in U_0$. Concretely,
if we have a map $f_0\in \mathcal{B}$  with a $2$-periodic
invariant attracting curve, we can define the map $G_1:\Omega\times V \rightarrow
\mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})$ as in (\ref{equation definition G1}).

On the other hand, we can consider the counterpart of the map
$G_1$ in the uncoupled case. Given a map $f_0\in \mathcal{B}_0$, consider
$U\subset \mathcal{B}_0$ a neighborhood of $f_0$ in the $\mathcal{B}_0$ topology.
Assume that $f_0$ has a attracting $2$-periodic
orbit $x_0\in I$. We have  that $x$ depends analytically on the map,
 therefore it induces a map $x:U \rightarrow \mathbb{W}$. Then if we take $U$ small
enough we can suppose that there exists an analytic map $x:U\rightarrow \mathbb{W}$
such that $x[f]$ is a periodic orbit of period 2.

Now consider the map
\begin{equation}
\label{equation definition widehatG1}
\begin{array}{rccc}
\widehat{G}_1:& U\subset \mathcal{B}_0 &\rightarrow & \mathbb{C} \\
\rule{0pt}{3ex} &  f & \mapsto & D_x f \big(f(x[f])\big) D_x f \big( x[f] \big).
\end{array}
\end{equation}
Let us remark that the zeros of this map define locally the manifold $\Sigma_1$.
On the other hand it corresponds to the map $G_1$ restricted to
the space $\mathcal{B}_0$, despite the fact that $\widehat{G}_1(f)$ has to be
seen as an element of $\mathcal{RH}(\mathbb{B}_\rho,\mathbb{W})$.

At this point we need to introduce an additional hypothesis on the family
$\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A}$. Consider $w_k=2^k w_0$ for any
$k\geq 0$ and $f^{(n)}_k=\mathcal{R}^k (c(\alpha_n,0))$. We have that  $f^{(n)}_0$
tends to $W^s(\mathcal{R},\Phi)$ when $n$ grow. Then  $\{f^{(n)}_k\}_{0\leq k <n}$
attains to $W^s(\mathcal{R},\Phi) \cup W^u(\mathcal{R},\Phi)$ and consequently there
exist $n_0$ s. t. $\{f^{(n)}_k\}_{0\leq k <n} \subset U$ ,
where $U$ is the neighborhood given in conjecture {\bf \ref{conjecture H2}}.
Therefore the operator $\mathcal{T}_\omega$ is differentiable in the
orbit $\{f^{(n)}_k\}_{0\leq k <n} \subset U$.  Consider the following
hypothesis.

\begin{description}
\item[H2)]  The family $\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A}$ is such that
\[
D G_1 \left(\omega_{n-1}, f^{(n)}_{n-1}\right)
D\mathcal{T}_{\omega_{n-2}}\left(f^{(n)}_{n-2}\right) \cdots
D\mathcal{T}_{\omega_0}\left(f^{(n)}_0\right) \partial_\varepsilon c(\alpha_n,0),
\]
has a unique non-degenerate minimum (respectively maximum) as a function from $\mathbb{T}$ to $\mathbb{R}$,
for any $n\geq n_0$.
\end{description}

Note that $c(\alpha_n,0)\in \Sigma_n$,
therefore $f^{(n)}_{n-1} \in \Sigma_1$,
consequently the function $G_1$ is defined at the point $f^{(n)}_{n-1}$.
Hypothesis {\bf H2} is rather technical and
not very intuitive. Further on in this section we show
that it is actually satisfied by maps like the Forced Logistic Map.

We have the following result, which ensures the existence of
reducibility-loss bifurcations curves in the $(\alpha,\varepsilon)$-plane
of parameters near the points $(\alpha_n, 0)$. This is one of the main
results of this chapter. On the one hand it gives the existence of
reducibility-loss bifurcations, but on the other hand it also gives
explicit expression of these bifurcations in term of
the renormalization operator $\mathcal{T}_\omega$.

\begin{thm}
\label{thm existence of non-reducibility direction}
Consider a family of maps $\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A}$ as before
 such that hypotheses  {\bf H1} and {\bf H2} are satisfied
 and consider $\alpha_n$ the parameter values where the uncoupled
family intersects the manifolds $\Sigma_n$ as above.
Suppose
that the rotation number of the system is equal to $\omega_0\in \Omega$.
Assume also that conjecture {\bf \ref{conjecture H2}} is true.
Then there exists $n_0$ such that, for any $n\geq n_0$,
two curves are  born from every parameter  value $(\alpha_n, 0)$,
locally expressed as $(\alpha_n^+(\varepsilon),\varepsilon)$ and
$(\alpha_n^-(\varepsilon), \varepsilon)$, such that they correspond to a
reducibility-loss bifurcation of the $2^n$-periodic invariant curve.

Consider also the sequences
\begin{equation}
\label{equation sequences corollary directions}
\begin{array}{rcll}
\omega_k  & = & 2 \omega_{k-1}  & \text{ for }  k=1,..., n-1. \\
\rule{0ex}{4ex}
f^{(n)}_k & = & \mathcal{R}\left(f_{k-1}^{(n)}\right)
& \text{ for }  k=1,..., n-1. \\
\rule{0ex}{4ex}
u^{(n)}_k & = & D \mathcal{R} \left(f^{(n)}_{k-1}\right) u^{(n)}_{k-1}
&  \text{ for } k = 1, ..., n-1.\\
\rule{0ex}{4ex}
v^{(n)}_k & = & D \mathcal{T}_{\omega_{k-1}}  \left(f^{(n)}_{k-1}\right) v^{(n)}_{k-1}
&  \text{ for } k = 1, ..., n-1.
\end{array}
\end{equation}
with
\begin{equation}
\label{equation sequences directions initial}
f^{(n)}_0  =  c(\alpha_n,0), \quad
u^{(n)}_0  =  \partial_\alpha c(\alpha_n,0), \quad
 v^{(n)}_0 =  \partial_\varepsilon c(\alpha_n,0).
\end{equation}

We also have that
\begin{equation}
\label{equation alpha n +}
\frac{d}{d \varepsilon} \alpha ^{+}_n(0) =
- \frac{m \left(
DG_1 \left(\omega_{n-1}, f^{(n)}_{n-1}\right) v_{n-1}^{(n)}
\right)
}{\rule{0ex}{3.5ex}
D \widehat{G}_1 \left(f^{(n)}_{n-1}\right) u_{n-1}^{(n)}} ,
\end{equation}
and
\begin{equation}
\label{equation alpha n -}
\frac{d}{d \varepsilon} \alpha ^{-}_n(0) =
- \frac{M \left(
D G_1 \left(\omega_{n-1}, f^{(n)}_{n-1}\right) v_{n-1}^{(n)}
\right)
}{\rule{0ex}{3.5ex}
D \widehat{G}_1 \left(f^{(n)}_{n-1}\right) u_{n-1}^{(n)}} ,
\end{equation}
where $m$, $M$, $G_1$ and $\widehat{G}_1$ are given by equations
(\ref{equation definition of minim}), (\ref{equation definition of maximum}),
(\ref{equation definition G1}) and (\ref{equation definition widehatG1}).
\end{thm}

\begin{rem}
\label{remark avoiding second part of conjecture H2)}
If the second part of conjecture {\bf \ref{conjecture H2}} is omitted, then we
can adapt the result to be asymptotically valid. Recall that we
have an open neighborhood of the fixed point $\Phi$ where the
renormalization operator is differentiable. The uncoupled
family
$\mathcal{R}^n\left(\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A} \right)$ is
contracted towards $W^u(\Phi,\mathcal{R})$, then replacing the family of maps
$\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A}$
by $\mathcal{R}^n\left(\{c(\alpha,0)\}_{(\alpha,\varepsilon)\in A} \right)$ it
would be close enough to $\Phi$ to be differentiable.
On the other hand the manifolds $\Sigma_n$ accumulate to $\Phi$ when $n$
goes to $\infty$, then the functions $G_1$ and $\widehat{G}_1$ associated
to the manifold $\Sigma_1$ must be replaced by suitable
function $G_n$ and $\widehat{G}_n$ associated to the
manifolds $\Sigma_n$.  These would give place to a more restrictive
result, but it would be valid for the
asymptotic estimates that are done in \cite{JRT11b}.
\end{rem}

Now we can go back to hypothesis {\bf H2}, which is not intuitive,
but we can introduce a stronger condition which is much more easy to
check. Moreover this condition is automatically satisfied by maps like
the Forced Logistic Map.

Consider a family of maps
$\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A}$ as before,
satisfying hypothesis  {\bf H1}. Consider the following hypothesis on the map.

\begin{description}
\item[H2')]  The family $\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A}$ is such that
the quasi-periodic perturbation $\partial_\varepsilon c(\alpha,0)$ belongs to the set
\begin{equation}
\label{equation space B1}
\mathcal{B}_1 : = \big\{ f \in \mathcal{B} |\text{ } f(\theta, x) = u(x) \cos(2\pi \theta) +
v(x) \sin(2\pi\theta),
\text{ for some } u,v\in \mathcal{RH}(\mathbb{W})\big\},
\end{equation}
for any value of $\alpha$ (with $(\alpha,0)\in A$).
Here $\mathcal{RH}(\mathbb{W})$ denotes the real holomorphic maps on the set $\mathbb{W}$,
and  $\mathbb{W}$ is the set given by hypothesis  {\bf H0}
(see section \ref{section definition and basic properties} for more details).
\end{description}

Then we have the following result.

\begin{prop}
\label{propostion H2' implies H2}
If a family $\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon) \in A}$
satisfies {\bf H1} and {\bf H2'} then it satisfies
{\bf H2}.
\end{prop}

On the other hand, we have the following propositions which allow
us to compute explicitly the derivative of $G_1$ and $\hat{G}_1$.

\begin{prop}
\label{proposition derivative x(f)}
Assume that we have $f_1 \in \Sigma_1$. Consider $V$ a neighborhood of
$f_1$ (in the topology of $\mathcal{B}$) and $x:V\rightarrow \mathcal{RH}(\mathbb{B}_\rho,\mathbb{W})$
the map such that $x(f)$ is a two periodic invariant curve of $f$ (with
rotation number $\omega$). Then we have
\[ \left[ D_f x(f_1) h  \right] (\theta) =
D_x f_1(1) h(\theta -2 \omega, 0) + h(\theta-\omega,1).\]
\end{prop}

\begin{prop}
\label{proposition derivative G1}
Assume that we have $f_1 \in \Sigma_1$. Consider $V$ a neighborhood of
$f_1$ (in the topology of $\mathcal{B}$). Consider also the map $G_1$ defined
in (\ref{equation definition G1})
where $x(g)$ is the two periodic invariant curve of the map.
Then we have
\begin{equation}
\label{equation defivative G1}
\left[ D_g G_1 (f_1) h\right](\theta)  =
D_x f_1(1) \Big[ D^2_{x^2} f_1(0) \Big( D_x f_1(1) h(\theta-2\omega, 0) +
h(\theta -\omega,1) \Big)  +  D_x h (\theta,0) \Big].
\end{equation}
\end{prop}

Using the last propositions one can be more explicit on the directions
of the reducibility-loss bifurcations given by formulas
(\ref{equation alpha n +}) and (\ref{equation alpha n -})

\begin{cor}
\label{corollary non-reducibility directions}
Assume that the same hypotheses of
theorem \ref{thm existence of non-reducibility direction} are satisfied and
consider additionally  the sequences (\ref{equation sequences
corollary directions}) for the same initial terms
(\ref{equation sequences directions initial}) of the theorem.
Then we have
\begin{equation}
\label{equation alpha n + corollary}
\frac{d}{d \varepsilon} \alpha ^{+}_n(0) =
- \frac{ \displaystyle
 \min_{\theta \in \mathbb{T}}
\left[ \begin{array}{r}
D^2_{x^2}  f_{n-1}^{(n)}(0) \Big( D_x  f_{n-1}^{(n)}(1)
v_{n-1}^{(n)}(\theta-2\omega_{n-1}, 0) + v_{n-1}^{(n)} (\theta -\omega_{n-1},1) \Big) \\  +
D_x v_{n-1}^{(n)} (\theta,0)  \end{array}
\right]}
{
D_{x^2}  f_{n-1}^{(n)}(0) \Big( D_x  f_{n-1}^{(n)}(1)
u_{n-1}^{(n)}(0) + u_{n-1}^{(n)} (1) \Big)  +
D_x u_{n-1}^{(n)} (0)
}.
\end{equation}
and
\begin{equation}
\label{equation alpha n - corollary}
\frac{d}{d \varepsilon} \alpha ^{-}_n(0) =
- \frac{ \displaystyle
 \max_{\theta \in \mathbb{T}}
\left[ \begin{array}{r}
D^2_{x^2}  f_{n-1}^{(n)}(0) \Big( D_x  f_{n-1}^{(n)}(1)
v_{n-1}^{(n)}(\theta-2\omega_{n-1}, 0) + v_{n-1}^{(n)} (\theta -\omega_{n-1},1) \Big) \\
 +  D_x v_{n-1}^{(n)} (\theta,0) \end{array}
\right]}
{
D_{x^2}  f_{n-1}^{(n)}(0) \Big( D_x  f_{n-1}^{(n)}(1)
u_{n-1}^{(n)}(0) + u_{n-1}^{(n)} (1) \Big)  +
D_x u_{n-1}^{(n)} (0)
},
\end{equation}
\end{cor}

These explicit formulas will be used in \cite{JRT11c} to compute numerically
the directions $\frac{d}{d \varepsilon} \alpha ^{+}_n(0)$
with the use of a discretization of the operators $\mathcal{R}$ and $\mathcal{T}_\omega$.

{\bf Proofs}

\begin{proof}[Proof of theorem \ref{thm existence of non-reducibility direction}]
We consider only the case involving $\frac{d}{d \varepsilon} \alpha ^{+}_n(0)$ since
the other is completely analogous. When the hypothesis {\bf H1} is satisfied
there exists a sequence of parameter values $\{\alpha_n\}_{n\in Z_+}$, such that
the critical point of the map $c(\alpha_n,0)$ is a $2^n$-periodic orbit.
In other words, we have that $c(\alpha_n,0)\in \Sigma_n$. Moreover
the map $c(\alpha_n,0)$ is (at least) $n-1$ times renormalizable in the
one dimensional sense. Now, due to the perturbative construction of
the q.p. renormalization operator $\mathcal{T}_\omega$ we have that there
exists a neighborhood $U_n$ (on $\mathcal{B}$) of $c(\alpha_n,0)$ such that
any map in $U_n$ is $n-1$ times renormalizable (in the q.p. sense).

Using theorem \ref{theorem local chard of upsilon n}
we have that $\Upsilon_n^+(\omega)$ is locally given as
\[
\Upsilon^+_n(\omega) \cap U_n = \{g \in U_n | \thinspace G^+_1(T^{n-1}(\omega, g)) =0 \},
\]
where $G^+_1=m\circ G_1$, with $m$ the minimum function
 (\ref{equation definition of minim})
and $G_1$
is given by (\ref{equation definition G1}).

Consider now a neighborhood $A_n=(a_n,b_n)\times[0,c_n)$ of the parameter value
$(\alpha_n,0)$, which is small enough to have
$\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A_n} \subset U_n$. We can define
the following map
\[
\begin{array}{rccc}
g_n:& (a_n,b_n)\times[0,c_n)  &\rightarrow & \mathbb{R} \\
\rule{0pt}{3ex} &  (\alpha,\varepsilon) & \mapsto & m\circ G_1(T^{n-1}(\omega,c(\alpha,\varepsilon))).
\end{array}
\]

Then we have that
\[\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A_n} \cap
\Upsilon^+_n = \{
(\alpha,\varepsilon)\in  A_n | \thinspace g_n(\alpha,\varepsilon) =0 \}.
\]

The proof of the theorem follows applying the implicit function
theorem to the function $g_n$ at the point $(\alpha_n,0)$.
With this aim, let us describe $T^{n-1}(\omega,c(\alpha,\varepsilon))$ with some more detail.

Recall that the family $\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon)\in A}$
 uncouples, therefore for any $(\alpha,\varepsilon)\in A$
we can write $c(\alpha,\varepsilon) = c_0(\alpha) + \varepsilon c_1(\alpha,\varepsilon)$.

From now on assume that $(\alpha,\varepsilon)$ are such that
$f_k = \mathcal{R}^k\left(c(\alpha,\varepsilon)\right)$ is contained
in the set $U$ of conjecture {\bf \ref{conjecture H2}}. Using the differentiability
of $T$ (with respect its component on $\mathcal{B}$) we have that
\[
T^{n-1}(\omega_0, c(\alpha,\varepsilon)) =
(\omega_{n-1} , \mathcal{R}^{n-1}(c_0(\alpha) ) + \varepsilon H_0(\omega_0,\alpha,\varepsilon)),
\]
with
\[
H_0(\omega_0, \alpha,0) =
 D\mathcal{T}_{\omega_{n-2}}(f_{n-2}) \cdots D\mathcal{T}_{\omega_0}(f_0) \partial_\varepsilon c(\alpha_n,0),
\]
where $\omega_k= 2^k\omega_0$ and $f_k = \mathcal{R}^k(c(\alpha,0))$, for $k=0,\dots,n-1$.

Applying now the differentiability of $G_1$ we have
\[
G_1(T^{n-1}(\omega_0, c(\alpha,\varepsilon)))=
G_1(\omega_{n-1}, \mathcal{R}^{n-1}(c_0(\alpha))) +
\varepsilon H_1(\omega_0, \alpha,\varepsilon),
\]
with
\[
H_1(\omega_0,\alpha,0) = DG_1 (\omega_{n-1}, f_{n-1}) H_0(\omega,\alpha,0),
\]
where $\omega_{n-1}=2^{n-1} \omega_0$ and $f_{n-1} = \mathcal{R}^{n-1}(c(\alpha,0))$.

Note that $\mathcal{R}^n(c_0(\alpha))$ is an uncoupled map, therefore
$G_1\left(\omega_{n-1}, \mathcal{R}^{n-1}(c_0(\alpha))\right) $ as a function of
$C^\omega(\mathbb{T},I)$ is constant, and then its minimum is equal to this constant.
Actually we have that
$\displaystyle m\left( G_1(\omega_{n-1}, \mathcal{R}^{n-1}(c_0(\alpha)))\right) =
\widehat{G}_1(\mathcal{R}^{n-1}(c_0(\alpha))$. On the other hand, note
that $\varepsilon\geq 0 $ for any $(\alpha,\varepsilon)\in A_n$.
Using these two facts to $m$, the minimum operator
(\ref{equation definition of minim}), we have
\begin{eqnarray}
g_n(\omega,\alpha,\varepsilon)  & = &
m\left( G_1(\omega_{n-1}, \mathcal{R}^{n-1}(c_0(\alpha))) +
\varepsilon H_1(\omega_0, \alpha,\varepsilon) \right)   \nonumber \\
& = & \rule{0ex}{2ex}
\label{equation intermediate expresion of gn}
\widehat{G}_1(\mathcal{R}^{n-1}(c_0(\alpha)))+m\left(\varepsilon H_1(\omega_0, \alpha,\varepsilon)\right)
\\
& = & \rule{0ex}{2ex}
\label{equation final expresion of gn}
\widehat{G}_1(\mathcal{R}^{n-1}(c_0(\alpha)))+\varepsilon \left( m\circ H_1(\omega_0,\alpha,\varepsilon)\right).
\end{eqnarray}
Where $\widehat{G}_{1}$ is defined like in
(\ref{equation definition widehatG1}) in a suitable
neighborhood of $\mathcal{R}^{n}(c(\alpha))$.

Our aim is to apply the IFT to $g_n$ at the point $(\alpha,\varepsilon)
=(\alpha_n,\varepsilon)$. Note that $c(\alpha_n,0)$ accumulates to $W^s(\Phi,\mathcal{R})$
when $n$ grows. Then the sequence $\{f_k^{(n)}\}_{0\leq k<n} $ attains to
$W^s(\Phi,\mathcal{R})\cup W^u(\Phi,\mathcal{R})$ when $n$ grows. Therefore
there exists $n_0$ such that, for any $n\geq n_0$, $f_k^{(n)}\in U$
for $k=0,\dots,n-1$, where $U$ is the set given by conjecture {\bf \ref{conjecture H2}}.
Then the function $g_n(\omega,\alpha,\varepsilon)$ is differentiable in a
neighborhood of $(\alpha_n,0)$ if  $H_1(\omega_0, \alpha_n,0))$ has a
unique non-degenerate minimum. Note that
\begin{equation}
\label{equation function H1}
H_1(\omega_0, \alpha_n,0) = DG_1 (\omega_{n-1}, f_{n-1})
D\mathcal{T}_{\omega_{n-2}}(f_{n-2}) \cdots D\mathcal{T}_{\omega_0}(f_0) \partial_\varepsilon c(\alpha_n,0),
\end{equation}
which actually corresponds to the hypothesis {\bf H2}, which is satisfied.

Recall that if $c(\alpha_n,0)\in \Sigma_n$ then
we have $g_n(\omega,\alpha_n,0)=0$. Therefore,
to apply the Implicit Function Theorem (IFT for short) to $g_n$,  we need to
check that $\partial_\alpha g_n(\omega,\alpha_n,0) \neq 0$.
Note also that $\widehat{G}_1$ is the function which gives locally the
manifold $\Sigma_1$, therefore
the condition  $\partial_\alpha g_n(\omega,\alpha_n,0) \neq 0$ is
equivalent to require that ${\mathcal{R}^{n-1}(c_0(\alpha))}_{\alpha\in (a_n,b_n)}$
intersects transversally the manifold $\Sigma_1$ at
the point $\mathcal{R}^{n-1}(c_0(\alpha_n))$.
This condition is satisfied due the hypothesis {\bf H1} which
requires the family
$\{c(\alpha)\}_{\alpha\in[a,b]}$  to cross transversely each manifold
$\Sigma_n$ for any $n$.

Then the IFT ensures us that there exists a neighborhood
$\tilde{A}= (\tilde{a}_n,\tilde{b}_n)\times[0,\tilde{c_n})$ and a function $\alpha_n(\omega) : [0,d_n)\rightarrow R)$
such that
\[
\{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon) \in \tilde{A}} \cap \Upsilon_n(\omega)
= \{c(\alpha_n(\omega,\varepsilon),\varepsilon)\}_{\varepsilon\in[0,d_n)}.
\]
Moreover we have that
\[
\frac{d}{d \varepsilon} \alpha ^{+}_n(0) = -
\frac{\partial_{\varepsilon} g_n(\omega,\alpha_n,0) }
{\partial_{\alpha} g_n(\omega,\alpha_n,0) }.
\]

Finally we can use equation (\ref{equation final expresion of gn}) and
(\ref{equation function H1}) to compute $\partial_{\varepsilon}
g_n(\omega,\alpha_n,0)$ and $\partial_{\alpha} g_n(\omega,\alpha_n,0)$. Then,
it follows
\[
\frac{d}{d \varepsilon} \alpha ^{+}_n(0) =
- \frac{ \displaystyle
 m\left(
D G_1 \left(\omega_{n-1}, f^{(n)}_{n-1}\right)
D\mathcal{T}_{\omega_{n-2}}\left(f^{(n)}_{n-2}\right) \cdots
D\mathcal{T}_{\omega_0}\left(f^{(n)}_0 \right) \partial_\varepsilon c(\alpha_n,0)
\right)}
{\rule{0ex}{3.5ex}
D \widehat{G}_1 \left(f^{(n)}_{n-1}\right)
D\mathcal{R} \left(f^{(n)}_{n-2}\right) \cdots D\mathcal{R} \left(f^{(n)}_0\right)
\partial_\alpha c(\alpha_n,0)},
\]
which indeed is  equivalent to (\ref{equation alpha n +}).
\end{proof}

\begin{rem}
In theorem \ref{thm existence of non-reducibility direction}
the parameter space has been set up in such a way
that $\varepsilon$ is greater or equal zero for any values of
the parameter. In the proof above the IFT
is applied to
a neighborhood of  $(\alpha,\varepsilon) = (\alpha_n,0)$. But the theorem is
not completely applicable in its usual form (see for example \cite{Die69},
because we do not have the
derivative defined w.r.t. directions with $\varepsilon$ negative. To bypass
this difficulty, we can consider the map $g_n(\omega,\alpha,\varepsilon)$ written as
in (\ref{equation final expresion of gn}). Then we can
extend the map symmetrically as
\[g_n(\omega,\alpha,\varepsilon) =  \widehat{G}_1(\mathcal{R}^{n-1}(c_0(\alpha)))+\varepsilon \left( m\circ H_1(\omega_0,\alpha,- \varepsilon)\right).
\]
for any $\varepsilon<0$. This extension of the map is enough to have
a $C^1$ map and to apply the IFT. If more differentiability
is required, then other extensions should be considered.
\end{rem}

\begin{figure}[t]
\centering
\input{EsquemUpsilons2.pdf_t}
\caption{Representation of the intersection of the sets
$\Upsilon_n^+ (\omega)$ and $\Upsilon_n^- (\omega)$
with a two parametric family of maps. The plane
represented correspond to the plane of parameters $(\alpha,\varepsilon)$
of the family.}
\label{Esquema Varietats Upsilon}
\end{figure}

\begin{rem} Recall that the domain of the parameters has been considered of the
form $A= [a,b]\times[0,c]$. One might replace this domain for one of
the form $\tilde{A}= [a,b]\times[-c,c]$. Then one can redo the
proof of theorem \ref{thm existence of non-reducibility direction} with the
new set up. But then, in the step from (\ref{equation intermediate expresion of gn})
to (\ref{equation final expresion of gn}) one can not apply the IFT, because
the function $g_n(\omega, \alpha,\varepsilon)$ is not differentiable. This is because the
minimum is replaced by a maximum for $\varepsilon<0$ which makes the function only $C^0$.
What one can do is to split $\tilde{A}= [a,b]\times [-c,c]$ into
$\tilde{A}_-=[a,b]\times[-c,0]$ and $\tilde{A}_+=[a,b]\times[0,c]$. Then
one can apply theorem \ref{thm existence of non-reducibility direction}
twice, and then we obtain four different bifurcation curves emerging form
the same point $(\alpha_n,0)$. What actually happens is that
we have two smooth curves, crossing at $(\alpha_n,0)$, but the curves defined
by $\Upsilon^+_n(\omega) \cap \{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon) \in \tilde{A}}$ and
$\Upsilon^-_n(\omega) \cap \{c(\alpha,\varepsilon)\}_{(\alpha,\varepsilon) \in \tilde{A}}$
swap their position when one crosses $\varepsilon=0$. This is illustrated
in figure \ref{Esquema Varietats Upsilon}.

\end{rem}

\begin{proof}[Proof of proposition \ref{propostion H2' implies H2}]
Applying proposition \ref{proposition invariance of Fourier node spaces}
on section \ref{section definition and basic properties}, we have
that the space $\mathcal{B}_1$ is invariant by $D\mathcal{T}_\omega(f)$ for any $\omega \in \mathbb{T}$ and
$f \in \mathcal{B}_0$ in a neighborhood of the fixed point. Consequently, we have that
$v_{n-1}= D\mathcal{T}_{\omega_{n-2}}\left(f^{(n)}_{n-2}\right) \cdots
D\mathcal{T}_{\omega_0}\left(f^{(n)}_0 \right) \partial_\varepsilon c(\alpha_n,0)$
belongs to $\mathcal{B}_1$, where $\omega_k=2^k \omega_0$ and
$f_k=\mathcal{R}^k(c(\alpha,0))$. Finally, note that when we consider
$DG_1(\omega_{n-1}, f_{n-1}) v_{n-1}$ we
obtain a periodic function of the form $A \cos(2\pi\theta) +
B \sin(2\pi\theta)$, which will have a unique non-degenerate
minimum (and maximum) if $A$ and $B$ are not simultaneously zero.
\end{proof}

\begin{proof}[Proof of proposition \ref{proposition derivative x(f)}]
We have that the map $x:V\rightarrow \mathcal{RH}(\mathbb{B}_\rho, \mathbb{W})$ is obtained
applying the IFT to the map
\[
\begin{array}{rccc}
F_1:& V \times \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C}) &\rightarrow & \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})  \\
\rule{0pt}{3ex} & (g,x )  & \mapsto & [F(g,x)](\theta):=
g(\theta + \omega, g(\theta, x(\theta))) - x(\theta + 2 \omega).
\end{array}
\]
at the point $(f_1, x(f_1))$. From the IFT we also know that
\begin{equation}
\label{equation derivative x via IFT}
D_h x[f] h = - (D_x F_1(f,x[f]))^{-1} \circ (D_f F_1(f,x[f])) h.
\end{equation}

Differentiating $F_1$ we have
\[
\begin{array}{rccc}
D_x F_1(g,x[g]):& \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C}) & \rightarrow & \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C}) \\
\rule{0pt}{3ex} & l   & \mapsto &
D_x g\big(\theta +\omega, g(\theta, x[g](\theta))\big)
D_x g\big(\theta, x[g](\theta)\big) l(\theta) - l(\theta + 2 \omega).
\end{array}
\]
and
\[
\begin{array}{rccc}
D_f F_1(g,x[g]):& \mathcal{RH}(\mathbb{B}_\rho\times \mathbb{W},\mathbb{C})  &\rightarrow & \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})  \\
\rule{0pt}{4.5ex} & h  & \mapsto &
\begin{array}{c}
D_x g\big(\theta + \omega, g(\theta, x[g](\theta))\big) h (\theta, x[g](\theta)) + \\
\rule{0pt}{2.5ex}h\big(\theta +\omega, g(\theta, x[g](\theta))\big). \end{array}
\end{array}
\]

From the fact that $f_1\in \mathcal{B}_0$ it follows that its critical point is
$x=0$ and $f_1(0)=1$. From $f\in \Sigma_1$ we have that the critical point is
a two periodic orbit, therefore $x(f_1) = 0$ and $f(f(0))=f(1)=0$.  Using these
properties, given $h\in \mathcal{RH}(\mathbb{B}_\rho\times \mathbb{W},\mathbb{C})$ we have
\[
\Big[D_f F(f_1, x[f_1]) h \Big](\theta) = D_x f_1(1) h(\theta,0) + h(\theta+\omega,1).
\]
On the other hand, given $\ell \in  \mathcal{RH}(\mathbb{B}_\rho\times \mathbb{W},\mathbb{C}) $ we have that
\[
\Big[D_x F(f_1, x[f_1]) \ell \Big](\theta) = - \ell(\theta +2\omega).
\]
Note that this operator is easily invertible, and its inverse is given by
\[ \Big[ (D_x F_1(f,x[f]))^{-1} \ell  \Big](\theta) = -\ell(\theta -2\omega). \]

Now we can use (\ref{equation derivative x via IFT}) and the last two equations
to conclude that
\[
\Big[D_h x(f_1) h\Big](\theta) = D_x f_1(1) h(\theta -2 \omega,0) + h(\theta-\omega,1).
\]
\end{proof}

\begin{proof}[Proof of proposition \ref{proposition derivative G1}]
Using the chain rule on map (\ref{equation definition G1}) is it
not hard to see that
\[
\begin{array}{rcl}
\Big[ D_g G_1(g) h \Big] (\theta) & = &
D_x g(\theta + \omega, g(\theta,x(\theta))) \Big(
D^2_{x^2} g(\theta, x[g](\theta))\big[ D_h x(g) h\big](\theta)
+ D_x h (\theta, x[g](\theta))
\Big) \\
 & & + D_x g(\theta, x[g](\theta) ) H(g,x[g],h) (\theta),
\end{array}
\]
where $H(g,x[g],h)$ is an expression on $g$, $x[g]$ and $h$ that
can be explicitly computed but here it has no interest since
it will be cancelled out further on.

From the fact that $f_1\in \mathcal{B}_0$, it follows that its critical point is at
$x=0$ (i.e. $D_x f_1 (0) =0$) and  $f_1(0)=1$.
From $f_1\in \Sigma_1$ we have that the critical point is
a two periodic orbit, therefore $x(f_1) = 0$ and $f_1(f_1(0))=f_1(1)=0$. Using this
 properties and the equation above we have
\[
\Big[ D_g G_1(f_1) h \Big] (\theta) =
D_x f_1 (1) \Big(
D^2_{x^2} f_1(0) \big[ D_f x(f_1) h\big](\theta)   + D_x h(\theta, 0)
\Big).
\]
Finally, we can use proposition \ref{proposition derivative x(f)} to
compute $\big[ D_f x(f_1) h\big](\theta)$, then the stated result holds.
\end{proof}

\begin{proof}[Proof of corollary \ref{corollary non-reducibility directions}]
From theorem
\ref{thm existence of non-reducibility direction} it follows that
\begin{equation}
\label{equation alpha n + proof of corollary}
\frac{d}{d \varepsilon} \alpha ^{+}_n(0) =
- \frac{ \displaystyle  m\left(
D G_1 \left(\omega_{n-1}, f^{(n)}_{n-1}\right) v_{n-1}^{(n)} \right)}
{  D \widehat{G}_1 \left(f^{(n)}_{n-1}\right) u_{n-1}^{(n)}}.
\end{equation}

We can apply now proposition \ref{proposition derivative G1} to compute
the derivative of $G_1$. Note that  $\widehat{G}_1$ can be seen as
$G_1$ restricted to $\mathcal{B}_0$, then the proposition \ref{proposition derivative G1}
is also applicable to $\widehat{G}_1$. Then we have
\begin{align*}
D G_1 \left(\omega_{n-1}, f^{(n)}_{n-1}\right) v_{n-1}^{(n)} = &
 D_x  f_{n-1}^{(n)}(1)
\Big[ D_{x^2}  f_{n-1}^{(n)}(0)
\Big( D_x  f_{n-1}^{(n)}(1)  v_{n-1}^{(n)}(\theta-2\omega_{n-1}, 0)  \\
&
+ v_{n-1}^{(n)} (\theta -\omega_{n-1},1)
\Big)   +
D_x v_{n-1}^{(n)} (\theta,0),
\Big]
\end{align*}
and
\[
D \widehat{G}_1 \left(f^{(n)}_{n-1}\right) u_{n-1}^{(n)} =
 D_x  f_{n-1}^{(n)}(1)
\Big[D_{x^2}  f_{n-1}^{(n)}(0) \Big( D_x  f_{n-1}^{(n)}(1)
u_{n-1}^{(n)}(0) + u_{n-1}^{(n)} (1) \Big)  +
D_x u_{n-1}^{(n)} (0) \Big].
\]

Finally let us remark that $f_{n-1}^{(n)}$ belongs to $\mathcal{B}_0$, therefore we have
that $D_x f_{n-1}^{(n)}(x)x <0$ for any $x\in I\setminus\{0\}$. Concretely we have
$D_x f_{n-1}^{(n)}(1)<0$. If we replace the values of
 $ D G_1 \left(\omega_{n-1}, f^{(n)}_{n-1}\right) v_{n-1}^{(n)}$
and $ D \widehat{G}_1 \left(f^{(n)}_{n-1}\right) u_{n-1}^{(n)} $ in
(\ref{equation alpha n + proof of corollary}), then when we
simplify the value $D_x f_{n-1}^{(n)}(1)<0$ the minimum becomes a maximum.
\end{proof}

\begin{appendix}

\section{The minimum function}
\label{appendix minimum function}

In this appendix we give some basic properties of the minimum of a function
$f:\mathbb{T} \rightarrow \mathbb{R}$ as an operator, with $\mathbb{T}= \mathbb{R}/\mathbb{Z} $ the one dimensional real torus.
To work in the same topology that in the rest of the paper
we will consider $f\in \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})$ the space of real analytic
functions from $\mathbb{B}_\rho$ to $\mathbb{C}$, and continuous on the
closure of $\mathbb{B}_\rho$, with $\mathbb{B}_\rho$  a band of width $\rho$ around
the real torus $\mathbb{T}$.
In other words we want to study the operator
\[
\begin{array}{rccc}
m:&  \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C}) &\rightarrow & \mathbb{R} \\
\rule{0pt}{3ex} & g & \mapsto & \displaystyle \min_{\theta \in \mathbb{T}} g(\theta).
\end{array}
\]

More concretely we focus on the differentiability of the
map $m$. Note that in the space of holomorphic functions, it has
no sense to consider the minimum of a function. Nevertheless,
for maps in $\mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})$
we have that the image of real numbers are real numbers,
then we can consider the minimum in the real torus.

Concerning to differentiability of the minimum as an operator, one has
the following result.

\begin{prop}
\label{proposition diferentiability of minimum}
Let $g_0\in \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})$ be a function such that its global minimum
in $\mathbb{T}$ is attained only at one value $\theta_0\in \mathbb{T}$ and it is not degenerate, i.e.
$g''(\theta_0)>0$. Then there is a function $G:  \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C}) \rightarrow \mathbb{C}$  such
that $G(g)=m(g)$ for any $g\in  \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})$ and $G$ is infinitely many times
differentiable in a small neighborhood of $g_0$.
In other words we have that $m$ can be extended to a differentiable
function $G:\mathcal{RH}(\mathbb{B}_\rho,\mathbb{C}) \rightarrow \mathbb{C}$ around $g_0$.

Moreover, the derivative of $G$ in $g_0$ is given by
\[
DG(g_0) \thinspace g_1 = g_1(\theta_0).
\]
\end{prop}

\begin{proof}
To prove the proposition we construct the map $G$ using
the IFT.

Consider the auxiliary function
\[
\begin{array}{rccc}
F:& \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})\times \mathbb{T} &\rightarrow & \mathbb{C} \\
\rule{0pt}{3ex} & (g,\theta)  & \mapsto & g'(\theta).
\end{array}
\]
We have that the function $F$ corresponds to the derivative of $g$ composed
with the evaluation in the point $\theta$. Both the derivative and the evaluation map
are $C^\infty$ functions (they are bounded
linear operators, therefore they are infinitely
differentiable),
then  its composition, which is $F$,  is also $C^\infty$.

Consider $g_0$ as in the hypothesis of the proposition. Then we have that
there exist a $\theta_0\in \mathbb{T}$ such that $(g_0,\theta_0)$ is a zero of
$F$. By hypothesis we have that $\theta_0$  is not a degenerate minimum, then
\[
D_\theta F(g_0,\theta_0) = g_0''(\theta_0) >0,
\]
We can apply now the IFT, consequently we have that there exist an open neighborhood
$U\subset  \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})$ of $g_0$  and a $C^\infty$ function
\begin{equation}
\label{function theta g}
\begin{array}{rccc}
\theta:& U &\rightarrow & \mathbb{T} \\
\rule{0pt}{3ex} & g & \mapsto & \theta(g),
\end{array}
\end{equation}
such that $F(g,\theta(g))=0$ for any $g\in U$. Concretely  we have
that $\theta(g)$ is a local minimum for any $g\in U$. Recall that by
hypothesis we have also that $\theta_0$ is a global minimum of $g_0$ and
that it is unique. Then, reducing $U$ to a smaller neighborhood if necessary, we
have that $\theta(g)$ is also the unique global minimum of $g$, for
any $g\in U$.

Let us consider the evaluation map $\operatorname{Ev}:  \mathcal{RH}(\mathbb{B}_{\rho'},\mathbb{C}) \times
\mathbb{T} \rightarrow \mathbb{C}$ the evaluation map (in a point of the real torus
$\mathbb{T}\subset \mathbb{B}_{\rho'}$) with $\rho'$ a value
$0<\rho'<\rho$. Let $\theta$ be the function
(\ref{function theta g}) defined by  the IFT. For any $g\in U$ we
can define the map $G$ in the statement of the proposition
\ref{proposition diferentiability of minimum}
as $G(g) = \operatorname{Ev}(g',\theta(g))$. With this definition we have that $G(g)= m(g)$ for
any $g\in U$. Therefore the minimum
is a  $C^\infty$ function in a neighborhood of $g_0$. Note that
in the definition of the evaluation map $\operatorname{Ev}$ we have considered
$\mathcal{RH}(\mathbb{B}_{\rho'},\mathbb{C})$ as its domain with $0<\rho'<\rho$. This is needed in
order to ensure that $g'$ is a bounded function, then one has $G$ is
a bounded operator.

To finish we compute the derivative of the minimum function. Is not difficult to
see that $D_\theta Ev(f,\theta) = f'(\theta)$ and $ D_g Ev(f,\theta) g_1 = g_1(\theta)$.
Using the chain rule we have
\[
DG(g) g_1 = D_g Ev(g,\theta(g)) g_1 + D_\theta Ev(g,\theta(g)) D_g \theta(g) g_1.
\]
Recall that $g_0'(\theta(g_0))= g_0'(\theta_0)=0$, then we have
$D_\theta Ev(g,\theta(g)) = g'(\theta(g)) =0 $. Then it follows
\[
DG(g_0)\thinspace g_1 = g_1(\theta_0).
\]
\end{proof}

Let us discuss the case when the hypotheses of the proposition
\ref{proposition diferentiability of minimum} are not satisfied.
The two main hypotheses of the proposition are the
non-degeneracy of the minimum and the uniqueness of it. When the non-degeneracy
condition is suppressed the proposition still being true, since the
argument done before can be adapted changing the auxiliary function, although the
proof becomes quite more technical.

On the other hand, the uniqueness condition of the
minimum is always necessary, because when
it is not satisfied the map ceases to be Frechet differentiable.

We just want to see that it is not differentiable. For simplicity
we will take de derivative in the reals. Let us consider the
most degenerate case, which is when the function $g_0$ is constant.
Then given  $g_1\in \mathcal{RH}(\mathbb{B}_\rho,\mathbb{C})$
we want to compute $Dm(g_0, g_1)$, the Gateaux derivative of $m$
at $g_0$ with respect to
the $g_1$ direction. By definition we have
\[
Dm(g_0, g_1 )= \lim_{t\rightarrow 0} \frac{m(g_0+t g_1) - m(g_0)}{t}
\] whenever the limit exist.

Indeed, when the function $g_0(\theta)$ is constant we have that
\[\min_{\theta\in T} (g_0 + t g_1) = g_0 +  \min_{\theta\in T} t g_1 =
\left\{
\begin{array}{rl}
\displaystyle g_0 + t\min_{\theta\in T} g_1  & \text{ if } t\geq 0 \\
\rule{0pt}{4ex} \displaystyle
g_0 + t\max_{\theta\in T} g_1  & \text{ if } t\leq 0
\end{array}
\right.\]

Then the Gateaux derivative not only depends on the sign of $t$ when taking the
limit, moreover when the sign is fixed the limit which we obtain is not even
a linear operator. It is clear that in this case the minimum
operator is not differentiable.

\end{appendix}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv