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BODY:
	\documentstyle[12pt]{article}
	\begin{document}
	\title{Directional Differentiability of the Rotation Number for the
		Almost Periodic Schr\"{o}dinger Equation}
	\author{Rafael Obaya \and  Miguel Paramio}
	\date{}
	\maketitle

		\newtheorem{defi}{Definition}[section]
		\newtheorem{theo}{Theorem}[section]
		\newtheorem{prop}{Proposition}[section]
		\newtheorem{lemm}{Lemma}[section]
		\newtheorem{coro}{Corolario}[section]

	\section{Introduction}
Let us consider the one-dimensional Schr\"{o}dinger equation
\begin{equation}\label{prim}
	L_{E}(x)=-x^{''}+g_0(t)x-Ex=0
\end{equation}
 with almost periodic potential $g_0$. The set $A$ of energies where  the Lyapunov
exponent
vanishes is also known to be the essential support of the absolutely continuous
part of the spectral measure. This paper deals with the variation of the 
rotation number on $A$, in particular with the differentiability and Lipschitz
character of this map.\\ 
\indent Since $g_0(t)$ is a bounded function then the boundaries $ \pm \infty $
are of the limit point type according to Weyl's classification. It follows 
that for $E$ 
complex with $Im(E)>0 $, even if $E$ belongs to the resolvent, $L_E$ has a basis
of solutions $u_{+}(t) $, $u_{-}(t) $ whithin $L^{2}(R_{+})$ and $L^{2}(R_{-})$. 
This allows to define and represent the rotation number as harmonic function on
the upper half plane. This is the starting point of a theory developped by Johnson, Moser,
Kotani, Simon and others. These authors obtain properties of the rotation 
number on the real axis taking limits following the lines $E+i \varepsilon $
as $\varepsilon$ tends to  $0^+$. From this point of view the rotation number
agrees with the imaginary part of the extension to the real axis of a Herglotz
function. We give a new proof of some known facts and present several new 
results using a different technique which has some interest in itself.\\
\indent Let us recall here some of the main notions and previous results  needed  to 
formulate our version. For the remaining of the paper $ \Omega$ will stand for 
the hull of $g_0$. The map $T: \Omega \times 
R \longrightarrow R; \,\,(\xi,t) \longrightarrow \xi_t$, where $\xi_t(s)= \xi
(t+s)$, describes a continuous flow on $ \Omega$. Thus $(\Omega,T)$ is a unique 
ergodic and almost periodic flow. Actually $ \Omega$ is a compact group and the
Haar measure its unique invariant measure. The map $J: \Omega \longrightarrow
R$ assigns its value at $t=0$ to each function of $ \Omega$.\\
\indent Let $E$ be fixed. Pastur \cite{past}  has shown that all the equations
\begin{equation}
-x^{''} + \xi (t)x = Ex \,\,\,\, , \,\,\,\, \xi \in \Omega
\end{equation}
admit a commom description of the spectrum. Taking polar-symplectic coordinates
$ \varphi = \tan^{-1} (x/x') \, \, , \,\,\, r = 1/2\;(x^2+(x')^2)$
we find the relations
\begin{eqnarray}\label{fluj}
\dot{\varphi} &=& f_E( \xi_t,\varphi ) = \cos^2 \varphi- (J(\xi_t) - E)\sin^2 
\varphi \\
\dot{r} &=& -\frac{\partial f_E}{\partial \varphi}(\xi_t, \varphi)\:r= 
2(J(\xi_t)+ 1-E)\sin \varphi \cos \varphi\: r\;.
\end{eqnarray}
\indent
Let $\varphi_t^E(\xi,\varphi)$ be the solution of the first equation with
initial conditions $(\xi, \varphi)$. The map $\Phi_t^E(\xi,\varphi)=
(\xi_t,\varphi_t^E(\xi, \varphi))$ defines a continuous flow on $K= \Omega
\times S^1$ and also on the projective bundle $\Sigma = \Omega \times P^1$. 
Once their trajectories are known the evolution of $r$ is obtained 
by integration of a simple linear equation.\par
The solutions of (\ref{fluj}) on $R$ define a lift of $\Phi$ on $L= \Omega \times R$
which we denote by the same symbol. The following formula represents the
 rotation number:
\begin{equation}
\alpha(E) =\lim_{t \rightarrow \infty} \,\,\,\frac{\varphi \,\,(t, \xi, 
\varphi)}{t} \,\, =\,\,\, \int_ \Sigma f_E( \xi, \varphi) d \mu
\end{equation}
for every normalized invariant measure. The convergence of the limit above is uniform in
$ \Sigma$.
$\alpha(E)$ is a continuous increasing function and hence it admits a 
derivative almost
everywhere. The properties of the flow at the points of differentiability have yet
 to be explained.\\
\indent Is also known that $\alpha(E)$ is locally 
log-Holder continuous; that means that there is a constant $c$ such that
\begin{equation}
\mid \alpha(E_1)-\alpha(E_2)\mid\leq \frac{-c}{\log \mid 
E_1-E_2\mid}\:\:,\:\:
\mid E_1-E_2\mid <1 \;.
\end{equation}
\indent Kotani and Deift-Simon prove that for almost every $(\theta,E)\in
\Omega \times A$ there exist linearly independent solutions which are $L^2$
almost periodic with the same frequency module as $g_0$. Moreover Deift-Simon
extend Moser's inequality $2\alpha(E) \alpha'(E)\geq 1$ to these points.\par
If $A$ contains an interval $I$, then $\alpha(E)$ is analitic on $I$. This
is a property of the Herglotz functions.
However many examples of almost periodic Schr\"{o}dinger operators with Cantor 
spectrum are known. See \cite{besi} and \cite{mose}. 
References \cite{joh2}, \cite{sina}  contain recent 
results on this problem.\par
 Our theory is based on the existence of a positive invariant measure 
which is absolutely continuous  with respect to $m$.
 We make use of a map of Furstenberg type
which transform the flow on $\Sigma$ into a skew-traslation and preserves the 
rotation number. The idea is similar  to the one used by Brunowski and Hermann
 for
families of diffeomorphisms of the circle. \\
\indent This paper is arranged as follows. In Sections 2 and 3 we present 
neccesary and sufficient conditions which guarantee the existence of the 
type of	invariant measures we need. The cases of square  
integrable density function and continuous density function will be common in
applications. Here we consider a skew-product flow defined by an abstract 
differential equation of the type (\ref{fluj}). The reason is that our argument does not
depend on the linearity of the equation. \\
\indent
In Sections 3 and 4 we apply the previous conclusions
to the Schr\"{o}dinger linear equation. Let $A_{0}$ be the  set of
 energies at which $L_{E}$ has $L^2$ almost periodic solutions. Let us fix 
$E_0\in A_0$.
First we show that $(\Sigma ,\Phi ^{E_0})$ admits an invariant
measure $\mu =1/q\,dm$ such that $1/q \in L^2(\Sigma,m)$
and $q\in L^1(\Sigma,m)$.
 In particular $q$ will be continuous when the solutions of $L_{E_0}$ are bounded.\\
\indent Let $\Omega$ be the hull of an adequate almost periodic $\xi$ and
$\Gamma_0 \in C(\Omega)$ such that $\Gamma_0(\xi_t)=(g_0-E_0)(t)$. We consider
 $\alpha$ as a continuous function from $C(\Omega)$ to $R$. We shall verify 
the Lipschitz variation of $\alpha$  throught $\Gamma_0$.\\
\indent In Sections 5 and 6 we treat the differentiability of $\alpha$ at 
$\Gamma_0$. We introduce the set $C_{ad}(\Omega)$ of admissible 
directions of differentiability by requiring the invariance of a certain integral 
expression. Thus we avoid considering the weak variation of a family of  
invariant measures which appear in the representation of $\alpha(\Gamma_0+
\eta\Gamma)$. We prove that if $\Gamma \in C_{ad}(\Omega)$ the rotation number 
has a derivative at $\Gamma_0$ in the direction $\Gamma$. In particular 
we obtain the classical derivative of $\alpha$ with respect to $E$ at the 
points of $A_0$ and we find an explicit formula easier to calculate .
(This is one  of the advantages of our method).
	\section{The invariant absolutely continuous measure}
 We set $L=\Omega \times R $ and $K=\Omega
\times S^{1}$ where $S^{1}=R/2\pi Z$. The symbols $\rho$ and $\rho_{0}$ stand
 respectively for the Lebesgue measure on $R$ and its normalized 
restriction on $S^{1}$. We represent the normalized invariant measure  on
 $\Omega$ by $m_{0}$ and the completion of the $\sigma$-algebra of its Borel
 sets by \({\cal A}_{0}\). A new measure can be introduced on $K$ (resp. on $L$) in an
 obvious way by taking the product measure $m=m_{0}\otimes \rho_{0}$ 
 (resp. $m_{L}=m_{0}\otimes \rho$) defined on the completion of the Borel sets
 $\cal A$ (resp. $ {\cal A}_{L}$). From now on we always refer to these measures
when no other indication is made.\\
\indent Let $T:R \times \Omega \rightarrow \Omega$  be a continuous
unique ergodic flow on $\Omega$.
Let us assume that both maps $f,\:\:\partial{f}/\partial{\varphi}:K
 \rightarrow R$ are  continuous. In this conditions the flow map
 $\Phi:R\times K \rightarrow K$ ($\Phi:R\times L \rightarrow L$), defined
by the solutions of the differential equation
\begin{equation} \label{dife}
 \dot {\varphi} = f(\xi_t,\varphi) \:, \:  (\xi,\varphi)\in K 
\end{equation}
 is continuous and measurable. In this Section we are going to discuss
the existence and means of constructing a positive invariant measure
 under $\Phi$ which is absolutely continuous with respect to $m$.
We start by identifying the density function of such measures with the positive
measurable solutions of the functional equation
\begin{equation} \label{ecua}
 p(\Phi_{t}(\xi,\varphi))=p(\xi,\varphi)e^{-\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}
\end{equation}
defined along the trajectories of (\ref{dife}). First we have to
remember the following well kown fact of ergodic theory :
\begin{prop}\label{ergo}
Assume that $p_{1}:K\rightarrow R$ is a measurable function such that
at every $t\in R$
\[ p_{1}(\Phi_{t}(\xi,\varphi))=p_{1}(\xi,\varphi)e^{-\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds} \]
for almost every $(\xi,\varphi)\in K$. Then there exists a measurable function\\
$p:K\rightarrow R$ such that $p_{1}=p$ almost everywhere and
\[ p(\Phi_{t}(\xi,\varphi))=p(\xi,\varphi)e^{-\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds} \]
for every $(t,\xi,\varphi)\in R\times K$.
\end{prop}
\begin{prop}
Let $p_{1}\in L^{1}(K,m)$ be a positive function. The following statements
are equivalent:\\
	i) The measure $\mu=p_{1}dm$ is invariant under the flow $\Phi$.\\
	ii)  There exists a measurable solution of (\ref{ecua}) $p$, such
		$p=p_{1}$ almost everywhere.\\
\end{prop}
{\em Proof:} Let us notice that if $q\in L^{1}(S^{1},\rho_{0})$ and
$(t,\xi_{0},\varphi_{o}) \in R\times K$ then 
\[\int_{\varphi_{t}(\xi_{0},0)}^{\varphi_{t}(\xi_{0},\varphi_{0})}q(\varphi)
d\varphi=\int_{0}^{\varphi_{0}}q(\varphi_{t}(\xi_{0},\varphi))
e^{ \int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}d\varphi \;.\]
$i\Rightarrow ii)$ Let $A\subset K$ be a measurable set and $A_{t}=\Phi_
{t}(A)$. According to the above remark the following identities hold:
\[\int_{A_{t}}p_{1}dm=\int_{\Omega}(\int_{S^{1}}p_{1}(\xi,\varphi)\chi_{A_{t}}
(\xi,\varphi)d\rho_{0})dm_{0} =\]
\[\int_{\Omega}(\int_{S^{1}}p_{1}(\xi,\varphi_{t}(\xi_{-t},\varphi))e^{\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi_{-t},\varphi))ds}
 \chi_{A_{t}}(\xi_{-t},\varphi)d\rho_{0})dm_{0} =\]
\[\int_{\Omega}(\int_{S^{1}}p_{1}(\xi_{t},\varphi_{t}(\xi,\varphi))e^{\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi_,\varphi))ds}
 \chi_{A}(\xi,\varphi)d\rho_{0})dm_{0} =\]
\[\int_{A} p_{1}(\xi_{t},\varphi_{t}(\xi,\varphi))e^{\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi_,\varphi))ds}dm =  
\int_{A} p_{1}(\xi,\varphi)dm\;.\]
Therefore
\[p_{1}(\Phi_{t}(\xi,\varphi))e^{\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}
=p_{1}(\xi,\varphi)\]
almost everywhere. Proposition (\ref{ergo}) allows us to obtain a solution of
 (\ref{ecua}).\\
$ii\Rightarrow i)$ If p satisfies (\ref{ecua}) the above identities 
show that  $\int_{A_{t}}p_{1}dm=\int_{A}p_{1}dm$. This means that $\mu=p_{1}dm$
is invariant under $\Phi$.\par
\vspace{.4cm}
\indent The solutions of (\ref{ecua}) satisfy a linear differential equation along the
 trajectories defined by $\Phi$. Unfortunately the local theory cannot be applied
 and all the solutions must be constructed globally.\par
Let us represent the set of continuous functions on $K$ by $C(K)$ and the set 
of Borel measures by $M(K)$. By the Riesz representation theorem  we can identify
 $M(K)$ with the dual space of $C(K)$. Its unit ball $B$ is weakly compact and
 metrizable . Let us consider the average  measures $(\mu_{T})_{T>0}$ defined by
 \[\mu_{T}(A)=\frac{1}{2T} \int_{-T}^{T}m(\Phi_{t}(A))dt =
\frac{1}{2T}\int_{A}\int_{-T}^{T}e^{\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}dtdm \]
for each measurable Borel subset $A\subset K$. If $A$ is an invariant set
we have $\mu_{T}(A)=m(A)$. In particular $\mu_{T}(K)=m(K)$. Hence the family
of measures $(\mu_{T})_{T>0}$ possesses some weakly convergent subsequence.
Moreover $\mu_{T}$ is absolutely  continuous with respect to $m$ and
\begin{equation}\label{fam1}
p_{T}(\xi,\varphi)=\frac{1}{2T}\int_{-T}^{T}e^{\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}dt 
\end{equation}
is its density function. We are going to check that if $\{p_{T}\}_{T>0}$
converges almost everywhere then its limit $p$ satisfies the 
equation (\ref{ecua}).\par
For every $0<l<T$ we have
\[p_{T}(\Phi_{l}(\xi,\varphi))=\frac{1}{2T}\int_{-T}^{T}e^{\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s+l}(\xi,\varphi))ds}dt =\]
\[\frac{1}{2T}\int_{-T}^{T}e^{\int_{l}^{t+l} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}dt =
\frac{1}{2T}\int_{-T}^{T}e^{\int_{0}^{t+l} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds} 
e^{-\int_{0}^{l} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}dt \;.\]
Therefore
\[\frac{T-l}{T}p_{T-l}(\xi,\varphi) \leq
p_{T}(\Phi_{l}(\xi,\varphi))e^{\int_{0}^{l} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds} 
\leq \frac{T+l}{T}p_{T+l}(\xi,\varphi)\;.\]
Taking limits as $T\rightarrow \infty$ we obtain
\[ p(\Phi_{l}(\xi,\varphi))=p(\xi,\varphi)e^{-\int_{0}^{l} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}\] 
almost everywhere. This means that there is an invariant set 
$K_{0}\subset K$ with $m(K_{0})=1$ such that there is convergence 
for all $(\xi,\varphi) \in K_{0}$. Defining  $p$ as zero in $K-K_{0}$ we get 
a solution of (\ref{ecua}).\\
\indent On the other hand, if we define
\begin{equation}\label{fam2}
q_{T}(\xi,\varphi)=\frac{1}{2T}\int_{-T}^{T}e^{-\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}dt 
\end{equation}
then
\begin{equation}\label{desi} 
\frac{1}{q_{T}(\xi,\varphi)}\leq \frac{2T}{\int_{-T}^{T}
e^{-\int_{0}^{t}{\frac{\partial{f}}{\partial{\varphi}}(\Phi_
{s}(\xi,\varphi))ds}}dt}\leq \frac{1}{2T}\int_{-T}^{T}
e^{\int_{0}^{t} \frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))
ds}dt \leq p_{T}(\xi,\varphi)\;.
\end{equation} 

\indent In a similar way, if the family $\{q_{T}\}_{T>0}$ converges almost everywhere to a
 function $q$, not identically zero, then 
\begin{equation}\label{ecuA}
 q(\Phi_{t}(\xi,\varphi))=q(\xi,\varphi)e^{\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}\;.
\end{equation}
The set $C=\{(\xi,\varphi)\in K  / q(\xi,\varphi)>0\}$ is measurable, invariant
and moreover $m(C)>0$. Defining
\[q_{1}(\xi,\varphi)=\left\{ \begin{array}{ll}
		0 & \mbox{if $(\xi,\varphi)\in K-C$}\\[.2cm]
		{\displaystyle \frac{1}{q(\xi,\varphi)}} & \mbox{if $(\xi,\varphi)\in K$}
	\end{array}  \right. \]
$q_{1}$ satisfies
\[ q_{1}(\Phi_{t}(\xi,\varphi))=q_{1}(\xi,\varphi)e^{-\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}\]
and we obtain a measurable solution of (\ref{ecua}).\\
\indent Retaining the previous notation we will prove the following
\begin{prop}\label{cond}
i) If $p=\lim_{T\rightarrow\infty}p_{T}$ exists in an  invariant set $C$
with $m(C)>0$, where p does not vanish, then $p\chi_{C} \in L^{1}(K,m)$ and
$p\chi_{C}dm$ is an invariant measure under the flow.\\
ii) If $q=\lim_{T\rightarrow\infty}q_{T}$ exists in an  invariant set $C$
with $m(C)>0$, where  q does not vanish, then $q_{1}\chi_{C} \in L^{2}(K,m)$ and
$q_{1}\chi_{C}dm$ is an invariant measure under the flow.
\end{prop}
{\em Proof:} $i)$ Actually we only have to check that $p\chi_{C} \in L^{1}(K,m)$.
This is a consequence of Fatou's lemma: one has
\[0< \frac{1}{\lambda}=\int p\chi_{C}dm =\int \lim_{T\rightarrow\infty} \inf
 p_{T}\chi_{C}dm\leq  \lim_{T\rightarrow\infty} \inf \int p_{T}\chi_{C}dm=m(C)
\;.\]
In these conditions $\mu=\lambda p\chi_{C}dm$ is an invariant measure
 absolutely continuous with respect to $m$ and moreover it is equivalent 
to $m$ in $C$. It follows from this fact that the family $\{p_{T}\}_{T>0}$
is uniformly integrable.\par
As a matter of fact, for every $\epsilon >0$ there exists $\delta_{1}>0$
 such that if $A\subset C$ is a measurable set  with $\mu(A)<\delta_{1}$
 then $m(A)<\epsilon$.\par
In an analogous way there exists $\delta_{2}>0$ such that if $A\subset C$ is
 a measurable set  with $m(A)<\delta_{2}$ then $\mu(A)<\delta_{1}$.
Hence if $A\subset C$ and $m(A)<\delta_{2}$ then $\mu(A)=\mu(\Phi_{t}(A))<
\delta_{1}$ and finally $m(\Phi_{t}(A))<\epsilon$ for every $t\in R$.\par
This proves
that $\int_{A}p_{T}dm <\epsilon$ for every $T\in R$. We may apply now the theorem
 of Vitali to get
\[0< \frac{1}{\lambda}=\int p\chi_{C}dm = \lim_{T\rightarrow\infty}
\int p_{T}\chi_{C}dm =m(C)\;.\]
This also shows that $\mu$ preserves the original measure of each invariant
subset of $C$. \\
$ii)$ Using  the inequality (\ref{desi}) and repeating the above argument we
get that $p_{1}\chi_{C}\in L^{1}(K,m)$. Hence, if $1/\lambda=
\int_{C}
q_{1}dm$ then  $\mu=\lambda q_{1}\chi_{C}dm$ is a normalized invariant measure.
Besides, according  to Birkhoff's ergodic theorem we have
\[\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}q_{1}(\Phi_{T}
(\xi,\varphi))\chi_{C}(\xi,\varphi)dt=\]
\[\lim_{T\rightarrow \infty}\frac{1}{2T}
\int_{-T}^{T}q_{1}(\xi,\varphi)\chi_{C}(\xi,\varphi)e^{-\int_{0}^{t} 
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds}dt =
\chi_{C}(\xi,\varphi)\]
for almost every $(\xi,\varphi)\in K$. Since $\chi_{C}\in L^{1}(K,\mu)$  
we finally conclude that $q_{1}\chi_{C}\in L^{2}(K,\mu)$ and $\int_{C}q_{1}^{2}dm=
\int_{C}q_{1}dm\leq1$\\
Notice that the last identity holds only if $C=K$, $q_{1}=1$ and therefore
$m$ is an invariant measure.\par
\vspace{.4cm}
\indent The converse statements are also true:
\begin{prop}\label{coni}
$i)$ Let us assume that $\mu_{1}=p_{1}dm$ is a normalized invariant 
measure with $p_{1} \in L^{1}(K,m)$. Then there exists  $p=\lim_{T\rightarrow\infty}
p_{T}$ in an invariant set $C$ with $m(C)>0$ where $p$ does not vanish.\\
$ii)$ Let us assume that $\mu_{2}=p_{2}dm$ is a normalized invariant 
measure with $p_{2} \in L^{2}(K,m)$. Then there exists  $q=\lim_{T\rightarrow\infty}
q_{T}$ in an invariant set $C$ with $m(C)>0$ where $q$ does not vanish.
\end{prop} 
{\em Proof:} $i)$ Let $D=\{(\xi,\varphi) \in K/p_{1}(\xi,\varphi)>0\}$. In our 
conditions is obvious that $1/p_{1}\:\chi_{D} \in L^{1}(K,\mu_{1})$.
 Birkhoff's ergodic theorem 
guarantees the existence of a measurable function $p_{1}^{\star}\in 
L^{1}(K,\mu_{1})$ such that
\[p_{1}^{\star}(\xi,\varphi)=\lim_{T\rightarrow\infty}\frac{1}{2T} \int_{-T}^{T}
(\frac{1}{p_1}\chi_D)(\Phi_t(\xi,\varphi))dt=
\lim_{T\rightarrow\infty}
\frac{p_{T}(\xi,\varphi)}{p_{1}(\xi,\varphi)}\chi_{D}(\xi,\varphi)\;.\]
\indent  This implies that for almost every $(\xi,\varphi)\in K$
\[\lim_{T\rightarrow\infty}
p_{T}(\xi,\varphi)\chi_{D}(\xi,\varphi)=
p_{1}^{\star}(\xi,\varphi)p_{1}(\xi,\varphi)\]
and moreover
\[\int_{D}p_{1}^{\star}p_{1}dm=\int_{D}pdm=m(D)\;.\]
In fact the last equality holds in every invariant subset $D'\subset D$.
If the set $C$ contains all the points of convergence where $p$ does not vanish
then
\[m(D)=\int_{D}p_{1}^{\star}p_{1}dm=\int_{C}p_{1}^{\star}p_{1}dm=m(C)\]
which proves the claim.\\
$ii)$ Let $D=\{(\xi,\varphi) \in K/p_{2}(\xi,\varphi)>0\}$. Repeating the above 
 argument we guarantee the existence of a measurable function  
$p_{2}^{\star}\in L^{1}(K,\mu_{2})$ such that
\[p_{2}^{\star}(\xi,\varphi)=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}
(p_{2}\chi_D)(\Phi_{t}(\xi,\varphi))dt=p_2(\xi,\varphi)
\lim_{T \rightarrow \infty}q_T(\xi,\varphi)\chi_D(\xi,\varphi)\]
This  implies that for almost  every  $(\xi,\varphi)\in K$
\[\lim_{T\rightarrow \infty}(q_{T}\chi_{D})(\xi,\varphi)=
(\frac{p_{2}^{\star}}{p_{2}}\chi_{D})(\xi,\varphi)\]
and moreover
\[\int_{D}p_{2}^{\star}p_{2}dm=\int_{D}p_{2}^{2}dm\]
As before, if $C$ contains the points  of convergence where $p_{2}^{\star}$
 does not vanish then $m(C)=m(D)$. Thus $(ii)$ is established.\par
\vspace{.4cm}
\indent These facts point out  some ways to reach invariant  absolutely continuous 
measures. Actually the existence  of such measures  depends on the behaviour 
of the families of functions $\{p_{T}\}_{T>0}$, $\{q_{T}\}_{T>0}$. Of 
course if $\{q_{T}\}_{T>0}$ converges to a non identically zero limit, 
so does $\{p_{T}\}_{T>0}$. In this case the invariant measure $\mu=
\lambda q_{1}dm$ has a density function within $L^{2}(K,m)$. We will show
that if $\mu$ is not ergodic there are  many other invariant measures whose 
density functions do not belong to  $L^{2}(K,m)$.\\
\indent Indeed, using a result by Fustenberg \cite{furs} , we know that there are invariant
sets of arbitrary $\mu-$measure. In fact we can choose a sequence $(D_{n})_
{n\in N}$ of dijoints sets such that $\mu(D_{n})=1/2^{n}$ for every
$n\in N$. Thus we have
\[\int_{D_{n}}q_{1}dm=\int_{D_{n}}q_{1}^{2}dm=\frac{1}{\lambda 2^{n}}\;.\]
Taking $\lambda_{n}=2^{\frac{n}{2}}$ for each $n\in N$ and $q_{2}=
\sum_{n=1}^{\infty}\lambda_{n}q_{1}\chi_{D_{n}}$ then
\[\int q_{2}dm=\sum_{n=1}^{\infty}\int_{D_{n}} \lambda_{n}q_{1}dm= 
\sum_{n=1}^{\infty}\frac{1}{\lambda 2^{\frac{n}{2}}}<\infty\]
but
\[\int q_{2}^{2}dm=\sum_{n=1}^{\infty}\int_{D_{n}} \lambda_{n}^{2}q_{1}dm= 
\infty\;.\]
This proves that $q_{2}dm$ is an invariant measure but $q_{2}$ is not 
square integrable.



	\section{The continuous density function}
Let $(X,d)$ be a compact metric space and $\Phi$ a continuous flow on
$X$. Let us remember the following concepts
\begin{defi}
We say that the flow $(X,\Phi)$ is distal when for each pair $x,y$ of different
 elements of $X$ there is a $\delta>0$ such that $d(\Phi_{t}(x),\Phi_{t}
(y))>\delta $ for every $t\in R$.
\end{defi}
\begin{defi}
We say that the flow $(X,\Phi)$ is almost periodic when  for every $\epsilon >0$
there is a $\delta >0$ such that if $x,y$ are elements of $X$ with $d(x,y)<
\delta $ then  $d(\Phi_{t}(x),\Phi_{t}(y))<\epsilon $ for every $t\in R$.
\end{defi}
If $(X,\Phi)$ is almost-periodic it is distal. The converse is not true, 
even if $(X,\Phi)$ is minimal and distal it need not to be almost-periodic.
\\[0.4 cm]
\indent In this Section we assume that $(\Omega,T)$ represent a distal and unique
ergodic flow. We are going to deal with the  case where  continuous solutions 
of the equation (\ref{ecua}) exist and hence invariant measures with continuous
density functions  can be constructed. In fact we are interested in positive
 solutions, so if we take logarithms in (\ref{ecua}) and write $k=\log p$ 
we obtain the new relation  
\begin{equation}\label{ecul}
k(\Phi_{t}(\xi,\varphi))-k(\xi,\varphi)=\int_{0}^{t}\frac{\partial{f}}
{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds \;.
\end{equation}
\indent Several conditions assuring the existance of continuous solutions of (\ref{ecul})
 appear in the literature. Let us note the following result which comes from
Gottschalk-Hendlund \cite{gohe} and is also proved in \cite{fuks} and 
\cite{joh0}.
\begin{prop}\label{solc}
Let assume that $(X,\Phi)$ is a minimal flow. Let $h \in C(X),$ $x_0\in X$
and suppose that $\int_{0}^{t}h(\Phi_{s}(x_{0}))ds$ is bounded. Then there 
exists  $k\in C(X)$ such that
\[k(\Phi_{t}(x))-k(x)=\int_{0}^{t}h(\Phi_{s}(x))ds \]
 for every $ x\in X, t\in R\;. $
\end{prop}

\indent We derive the next version of this proposition which works for non  
minimal fibred systems defined by (\ref{dife}). The basic results on 
topological dynamics we use can be found in Ellis \cite{elli}. However all of them
are direct consequences of our argument, so we think the following one is a 
self-contained proof. We make use of the classical properties of the 
diffeomorfims of the circle.
\begin{theo}\label{solC}
The following statements are equivalent:\\
$i)$ There exists a continuous function $k:K\rightarrow R$, satisfying
\[k(\Phi_{t}(\xi,\varphi))-k(\xi,\varphi)=\int_{0}^{t}\frac{\partial{f}}
{\partial{\varphi}}(\Phi_{s}(\xi,\varphi))ds\;.\]
$ii)$ There exists  $\xi_{0}\in \Omega $, such that the family of functions
\[
\begin{array}{rcl}
 		F_{t}:S^{1}&\longrightarrow &R\\
		\varphi \: &\rightarrow    &\int_{0}^{t}\frac{\partial{f}}
{\partial{\varphi}}(\Phi_{s}(\xi_{0},\varphi))ds
\end{array}
\]
is a conditionally compact subset of $C(S^{1})$.
\end{theo}
{\em Proof:} If the flow $(K,\Phi)$ is minimal,  proposition (\ref{solc})
applies directly, so we do not consider this option. 
 The implication $ii)\Rightarrow i)$ is trivial. As for the other one, 
we first show that the functions $(F_{t})_{t\in R}$ are uniformly bounded
 and equicontinuous. Notice that
\[ \frac{\partial{\varphi}}{\partial{\varphi_{0}}}(t,\xi_{0},\varphi_{0})=
e^{\int_{0}^{t}\frac{\partial{f}}
{\partial{\varphi}}(\Phi_{s}(\xi_{0},\varphi_{0}))ds}\;.\]
This assures the existence of constants $m, M$ such that
\begin{equation}\label{desl}
m\mid \varphi_{1}-\varphi_{2}\mid \leq \mid \Phi_{t}(\xi_{0},\varphi_{1})-
\Phi_{t}(\xi_{0},\varphi_{2}) \mid \leq M\mid \varphi_{1}-\varphi_{2}\mid \;.
\end{equation}
\indent Let $(t_{n})_{n=1}^{\infty}$ be a sequence of real numbers. We define
\[
\begin{array}{rcl}
\Psi_{n}:S^{1}&\rightarrow & S^{1} \\
 \varphi &\rightarrow &\varphi_{t_{n}}(\xi_{0},\varphi)
\end{array}
\]
By the Arzela-Ascoli theorem the family of diffeomorphisms $(\Psi_{n})_{n=1}^
{\infty}$ possesses a convergent subsequence in the $C^{1}({S^{1}})$ topology.
 Its limit $\Psi$ is a diffeomorphism which also preserves the orientation.\\
\indent Let us fix $(\xi_{1},\varphi_{1})\in K$. Since $(\Omega,T)$ is minimal 
there is a sequence $(t_{n})_{n=1}^{\infty}$ 
such that $\xi_{1}=\lim_{n\rightarrow \infty}T_{t_{n}}(\xi_{0})$. Moreover we can
 assume that  $\varphi_{1}=\lim_{n\rightarrow \infty}\varphi_{t_{n}}(\xi_{0},
\varphi_{0})$ for an adequate $\varphi_{0}\in S^{1}$.\par
Let $\eta$ be a positive constant with $\mid \int_{0}^{t}\partial{f}/
\partial{\varphi}\,(\Phi_{s}(\xi_{0},\varphi))ds\mid \leq \eta$ for every 
$\varphi \in S^{1} , t\in R$. Then
\[\int_{0}^{t}\frac{\partial{f}}
{\partial{\varphi}}(\Phi_{t_{n}+s}(\xi_{0},\varphi))ds=\int_{t_{n}}^{t+t_{n}}
\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi_{0},\varphi))ds=\]
\[\int_{0}^{t+t_{n}}\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi_{0},
\varphi))ds-\int_{0}^{t_{n}}\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi_{0},
\varphi))ds \]
hence
\[\mid \int_{0}^{t}\frac{\partial{f}}{\partial{\varphi}}(\Phi_{s}(\xi_{1},
\varphi_{1}))ds
\mid =\lim_{n\rightarrow \infty}\mid \int_{0}^{t}\frac{\partial{f}}
{\partial{\varphi}}(\Phi_{s+t_{n}}(\xi_{0},\varphi))ds \mid \leq 2\eta\;.\]
This extends the inequality (\ref{desl}) and proves that there are 
global constants $m, M$ such that
\begin{equation}\label{desg}
m\mid \varphi_{1}-\varphi_{2}\mid \leq \mid \Phi_{t}(\xi,\varphi_{1})-
\Phi_{t}(\xi,\varphi_{2}) \mid \leq M\mid \varphi_{1}-\varphi_{2}\mid
\end{equation}
for every pair of points $(\xi,\varphi_{1}), (\xi,\varphi_{2})$ from $K$.\par
It follows that the flow is distal and semisimple. This means that the compact 
set K is a disjoint union of an uncountable collection $(K_{j})_{j\in J}$ of minimal 
compact subsets. According to the proposition (\ref{solc}) a continuous solution 
of (\ref{ecul}) can be constructed in each of these sets $K_{j}$. The main 
problem  consists of making a good choice which allows us to define a continuous 
function on K. In order to solve this problem we analize now how these subsets 
cover the entire space $K$.\\[.3 cm]
\indent Let us consider $j_{0}\in J$ such that $(\xi_{0},0)\in K_{j_{0}}$ and let us identify
$S^{1}$ with $\{\xi_{0}\} \times S^{1}\subset K$. Let $A=K_{j_{0}}\cap S^{1}=
\{\varphi_{i}/i\in I_{j_{0}}\}$. For every $i\in I_{j_{0}}$ there exists a 
sequence of real numbers $(t^i_n)_{n=1}^{\infty}$ such that the diffeomorphisms $\Psi_{n}^{i}:S^{1}
\rightarrow S^{1} ; \varphi \rightarrow \varphi_{t_{n}^{i}}(\xi_{0},\varphi)$
 converge to a diffeomorphism $\Psi^{i}$ with $\varphi_{i}=\Psi^{i}(0)=
\lim_{n\rightarrow \infty}\varphi_{t_{n}^{i}}(\xi_{0},0)$. The families
${\cal F}=\{\Psi^{i}\}, {\cal G}=\{(\Psi^{i})^{-1}\}$ are equicontinuous and 
$\Psi^{i}(A)=A$ for each $i\in I_{j_{0}}$.\\
\indent This  shows that if $A'$ does not vanish then $A'=A$. In these conditions
 for all $\epsilon >0$ there is $\delta (\epsilon)>0$ such that for every
$\varphi \in A$ can be found $\varphi' \in A$  with $\delta <\varphi-\varphi'
<\epsilon$. 
We set $\delta_{1}=m\delta $ and $\epsilon_{1}=M\epsilon $, there exists a finite 
chain $\varphi_{0},\varphi_{1},\cdots \varphi_{N}$ of elements of $A$ such 
that $\varphi_{0}=0, \varphi_{N}>2\pi$ and $\delta_{1}<\varphi_{i+1}-\varphi_{i}<
\epsilon_{1},\: i=0\cdots N-1$. Indeed, if $\varphi_{0},\cdots \varphi_{k}$ are the 
first $k+1$ elements of the chain we choose $\varphi_{k-1}^{\star}\in A$ with
$\delta<\varphi_{k-1}^{\star}-\varphi_{k-1}<\epsilon $ and we take a diffeomorphism
$\Psi$ with $\Psi(\varphi_{k-1})=\varphi_{k}$ and $\delta_{1}<
\Psi(\varphi_{k-1}^{\star})-\varphi_{k}<\epsilon_{1}$. Defining $\varphi_{k+1}=
\Psi(\varphi_{k-1}^{\star})$ we obtain the next element. This proves that
 $A'=S^{1}$ and this situation only arises when the flow is minimal.\\[.3 cm]
\indent We conclude that $A'=\emptyset$ and $A=\{ \varphi_{0}=0,\varphi_{1},\cdots \varphi
_{N}=2\pi\}$ is a finite set. If $C_{i}=[\varphi_{i},\varphi_{i+1})\subset S^
{1}$, and $\Psi ^{i}$ denote the diffeomorphism of ${\cal F}$ with $\Psi ^{i}
(0)=\varphi_{i}$, then $\Psi ^{i}(C_{0})=C_{i}$ for $i=0\cdots N-1$. This shows
that  $C_{i}$ only contains one element of each subset $K_{j}$. Hence $K_{j}$
 is a finite cover of $\Omega$.\\
Let $\varphi_{j}\in C_{0} \cap K_{j}$, for each $j\in J$. We denote by $k_{j_{
0}}$ the solution of (\ref{ecul}) on $K_{j_{0}}$ satisfying $k_{j_{0}}(0)=0$
 and assume that $k_{j_{0}}(\varphi_{1})=\eta$. We denote by $k_{j}$
 the solution of (\ref{ecul}) on $K_{j}$ satisfying $k_{j}(\varphi_{j})=
\varphi_{j}/\varphi_{1}\,\eta $. All the $(k_{j})_{j\in J}$ together 
define a function $k$ which is solution of (\ref{ecul}) on $K$. Our last step consist 
in proving that $k$ is continuous.\\[.3 cm]
\indent If $\{(\xi_{n},\varphi_{n})\}_{n=1}^{\infty}$ is a sequence of elements of $K$ 
 with limit $(\xi ',\varphi ')$ there exist indices $(j_{n})_{n=1}^{\infty }$
 and real numbers $(t_{n})_{n=1}^{\infty }$ such that $(\xi_{n},
\varphi_{n})\in M_{j_{n}}$, $d(\Phi_{t_{n}}(\xi_{0},\varphi_{j_{n}}),(\xi_{n}
,\varphi_{n}))<1/n$ and $\mid k(\Phi_{t_{n}}(\xi_{0},\varphi_
{j_{n}}))-k(\xi_{n},\varphi_{n})\mid <1/n$.\\
We may suppose that $(\varphi_{j_{n}})_{n=1}^{\infty }$ is a convergent 
sequence 
(otherwise we would choose an adequate subsequence) with limit $\varphi _{0}'$.
 The points $(\xi ',\varphi ')$ and $(\xi_{0},\varphi_{0}')$ belong to the same
minimal subset where $k$ is continuous. Moreover $(\xi ',\varphi ')=\lim_
{n\rightarrow \infty}\Phi_{t_{n}}(\xi_{0},\varphi_{0}')$. So we immediately deduce that
$k((\xi ',\varphi '))=\lim_{n\rightarrow \infty}k(\Phi_{t_{n}}(\xi_{0},
\varphi_{0}'))$. Now we can write
\[\mid k(\xi ',\varphi ')-k(\xi_{n},\varphi_{n})\mid \leq\frac{1}{n}+
\mid k(\xi ',\varphi ')-k(\Phi_{t_{n}}(\xi_{0},\varphi_{j_{n}}))\mid \leq\]
\[\frac{1}{n}+
\mid k(\xi ',\varphi ')-k(\Phi_{t_{n}}(\xi_{0},\varphi_{0}'))\mid +
\mid k(\Phi_{t_{n}}(\xi_{0},\varphi_{0}'))-k(\Phi_{t_{n}}(\xi_{0},\varphi_
{j_{n}}))\mid\;.\] 
On the other hand
\[\mid k(\Phi_{t_{n}}(\xi_{0},\varphi_{0}'))-k(\Phi_{t_{n}}(\xi_{0},\varphi_
{j_{n}}))\mid \leq\]
\[\mid k(\xi_{0},\varphi_{0}')-k(\xi_{0},\varphi_{j_{n}})\mid +
\mid \int_{0}^{t_{n}}\frac{\partial{f}}
{\partial{\varphi}}(\Phi_{s}(\xi,\varphi_{j_{n}}))ds-\int_{0}^{t_{n}}\frac{\partial{f}}
{\partial{\varphi}}(\Phi_{s}(\xi_{0},\varphi_{0}))ds\mid=\]
\[\mid k(\xi_{0},\varphi_{0}')-k(\xi_{0},\varphi_{j_{n}})\mid +
\mid F_{t_{n}}(\varphi_{j_{n}})- F_{t_{n}}(\varphi_{j_{0}}) \mid\;.\]
Taking limits as $n\rightarrow \infty $, we obtain that $k(\xi ',\varphi ')=
\lim_{n\rightarrow \infty}k(\xi_{n},\varphi_{n})$. This guarantees the continuity 
of k and completes the proof.\\[.3cm]
\indent Obviously the solution of (\ref{ecul}) is not unique. Two different continuous 
solutions of this equation differ in a constant on each subset
 $K_{j}$. Conversely every continuous function $\kappa :
[0,\varphi_{1}]\rightarrow R$ with $\kappa(\varphi_{1})-\kappa(0)=\eta$
allows us to find a solution of (\ref{ecul}).\\[.5cm]
\indent When the methods of this  and the previous Sections are applied they give invariant 
 measures under $\Phi$ which can be different. We now discuss when the families
 $\{p_{T}\}$, $\{q_{T}\}$ converge to a continuous limit. This fact depends on 
the set of points where there is convergence.
\begin{prop}\label{conj}
$i)$ For every $\epsilon >0$ there is a measurable subset $\Omega_{\epsilon}
\subset \Omega$ with $m(\Omega_{\epsilon })>1-\epsilon$ such that  $\{p_{T}\}$ and 
$\{q_{T}\}$  converge uniformly in $\Omega_{\epsilon }\times S^{1}$.\\
$ii)$ The functions $\{p_{T}\}$, $\{q_{T}\}$ converge almost everywhere for every
invariant measure under the flow.
\end{prop}
{\em Proof:} Let $h$ be a continuous function defined on $K$ and
\begin{equation}\label{cpun}
h_{T}(\xi,\varphi)=\frac{1}{2T}\int_{-T}^{T}h(\Phi_{t}(\xi,\varphi))dt\;.
\end{equation}
The family $\{h_{T}\}$ converges  almost everywhere to a measurable 
limit $h^{\star} $. Let $C $ be the set of points of convergence and  $C_
{\xi}$ its section at $\xi $. We are going to show  that  $C_{\xi}$ is 
closed for every $\xi \in \Omega$.\\
Let $(\varphi_{n})_{n=1}^{\infty}$ be a sequence of  points of $C_{\xi}$
with limit $\varphi$. We derive from (\ref{desg}) and from the continuity of
$h$  that for every $\epsilon >0$ there is $n_{0}$ such that if $n\geq n_{0}$
then
\[d(\Phi_{t_{n}}(\xi,\varphi_{n}),\Phi_{t_{n}}(\xi,\varphi)) \leq\epsilon\]
and 
\[\mid h(\Phi_{t}(\xi,\varphi_{n}))-h(\Phi_{t_{n}}(\xi,\varphi))\mid 
\leq\epsilon\;.\]
Moreover there exists $T$ such that if $T_{1}>T_{2}>T$ then
\[\mid \frac{1}{2T_{1}}\int_{-T_{1}}^{T_{1}}h(\Phi_{t}(\xi,\varphi_{n_{0}}))dt-
\frac{1}{2T_{2}}\int_{-T_{2}}^{T_{2}}h(\Phi_{t}(\xi,\varphi_{n_{0}}))dt\mid \leq
\frac{\epsilon}{3}\;.\]
\indent Combining the last two inequalities we get
\[\mid \frac{1}{2T_{1}}\int_{-T_{1}}^{T_{1}}h(\Phi_{t}(\xi,\varphi))dt-\frac{1}
{2T_{2}}\int_{-T_{2}}^{T_{2}}h(\Phi_{t}(\xi,\varphi))dt\mid \leq\epsilon\;.\]
This means that $h_{T}$ converges at $(\xi,\varphi)\;.$\par
If $\xi \in \Omega$ and $\rho_{0}(C_{\xi })=1$ then $C_{\xi }=S^{1}$. 
Since $m(\Omega )=1$ there exists $\Omega_{0}$ with $m(\Omega_{0})= 1$ such that 
$C_{\xi }=S^{1}$ for  every $\xi \in \Omega_{0}$.\par
Let $\xi \in \Omega_{0}$. The functions $h_{T}(\xi,\varphi)$ are uniformly 
bounded and equicontinuous on $S^{1}$ where they converge  uniformly. Hence 
$h^{\star}$ is continuous on $\{\xi \} \times S^{1}$. The functions $H_{T}:
\Omega \rightarrow R$ defined by 
\[H_{T}(\xi)=\left\{ \begin{array}{ll}
		{\displaystyle \sup_{\varphi \in S^{1}}}{\mid h_{T}
		(\xi,\varphi)- h^{\star}
		(\xi,\varphi)\mid} &\;\; \mbox{if $\xi \in \Omega_{0}$}\\[.2cm]
		0 & \;\;\mbox{otherwise}
		\end{array}
\right. 	\]
are measurable and converge to $0$. By Egoroff's theorem for every $\epsilon >0$
 there is a measurable subset $\Omega_{\epsilon}\subset \Omega$ with
 $m(\Omega_{\epsilon })>1-\epsilon$ where $\{H_{T}\}$
converge uniformly. It follows that $\{h_{T}\}$ converge uniformly in $\Omega_
{\epsilon}\times S^{1}$.\par
Let $k$ be a continuous solution of (\ref{ecul}) . Then
\[p_{T}(\xi,\varphi)=e^{k(\xi,\varphi)} \frac{1}{2T}\int^{T}_{-T}e^{-k(\Phi_{t}
(\xi,\varphi))dt} \]
\[q_{T}(\xi,\varphi)=e^{-k(\xi,\varphi)} \frac{1}{2T}\int^{T}_{-T}e^{k(\Phi_{t}
(\xi,\varphi))dt}\]
and all the above conclusions about convergence hold.\par
Let $\mu $ be a normalized invariant measure on $K$. Note  that $\mu$ is 
mapped into the unique normalized invariant measure on the base under the natural
 projection $\Pi:K\rightarrow \Omega;(\xi,\varphi)\rightarrow \xi$. Hence 
$\mu(\Omega_{\epsilon }\times S^{1})>1-\epsilon$ and
 the functions $\{p_{T}\}$, $\{q_{T}\}$ converge almost everywhere for every
invariant measure under the flow.\\[.5cm]
\indent
Now we suppose that $(\Omega,T)$ is an almost-periodic flow. Theorem 
(\ref{solC}) permits an alternative proof of several results contained 
in \cite{joh0} and \cite{joh1}. If $(K,\Phi)$ 
is not minimal then each subset $K_{j}$ is a finite cover of $\Omega$. See 
\cite{sase}. The functions $\{p_{T}\}$, $\{q_{T}\}$ converge everywhere to a continuous 
function on each subset $K_{j}$. These limits are also continuous with respect 
to $\varphi$ and which as shown in theorem (\ref{solC}) implies the continuity
on $K$. \par
Similar conclusions can be derived when $(K,\Phi)$  is a unique ergodic flow.
 From proposition (\ref{conj}) we deduce that there are constants $\eta_{1}$, 
$\eta_{2}$ such that
\[\eta_{1}=\lim_{T\rightarrow \infty}\frac{1}{2T}\int^{T}_{-T}e^{-k(\Phi_{t}
(\xi,\varphi))dt} \]
and
\[\eta_{2}=\lim_{T\rightarrow \infty}\frac{1}{2T}\int^{T}_{-T}e^{k(\Phi_{t}
(\xi,\varphi))dt} \]
almost everywhere. In fact the convergence holds everywhere and moreover is 
uniform in $K$.\par
However a different situation occurs when the flow on $K$ is minimal but many
invariant measures exist. A slight modification of an argument of Furstenberg
\cite{furs} allows to prove that if
 $\{p_{T}\}$ or $\{q_{T}\}$ converge everywhere  their limits
 are continuous functions and the convergence is
 uniform. Otherwise  $\stackrel{\circ}{\Omega}_{1/n}=\emptyset$ for 
every $n\in N$ and hence $\bigcup_{n\in N}({\Omega}_{1/n}\times S^{1})$ 
is a set of first cathegory in $K$ whose elements are points of convergence.






	\section{The case of the Schr\"odinger equation} 
Let us consider  in this Section the one dimensional Schr\"{o}dinger equation
\begin{equation}\label{Scho}
L_E x=-x''+(\xi (t)-E)x=0  \qquad \xi \in \Omega, 
\end{equation}
 with almost-periodic potential where we are going to apply 
the conclusions obtained in the previous Sections. 
The phase of the solutions of (\ref{Scho}) satisfy  the differential equation
\begin{equation}\label{fibr}
\dot{\varphi}= \cos^{2}\varphi-(J(\xi_{t})-E)\sin^{2}\varphi \:.
\end{equation}
The map $\Phi^E(\xi, \varphi)=(\xi_t,\varphi_t^E(\xi,\varphi))$ defines 
a continuos flow on $\Sigma = \Omega \times P^1$. This flow allows a complete 
description of the solutions of (\ref{Scho}).\par
 We shall identify $P^{1}$ with $S^{1}/\pi Z$ and  maintain on $P^1$ the notation 
used on $S^1$ in the previous Sections.\par
 Let $x(t,\xi,\varphi)$ stand for the solution of (\ref{Scho}) verifying
$x'(0)+ix(0)= e^{i\varphi}$ and $r(t, \xi, \varphi)= x^{2}
(t, \xi, \varphi)+(x')^{2}(t,\xi,\varphi)$. Then one has 
\[ r(t, \xi, \varphi)= e^{-\int^t_{0}\frac{\partial f}{\partial \varphi}
(\Phi_s(\xi,\varphi))ds}\]
and
\[\frac{\partial \varphi}{\partial \varphi_0}(t,\xi,\varphi_{0})=
 \frac{1}{r(t,\xi,\varphi_{0})} =
e^{\int^t_{0}\frac{\partial f}{\partial \varphi}
(\Phi_s(\xi,\varphi_{0}))ds}\;.\]
\indent Let us fix $E\in R$. If all the solutions of the equation (\ref{Scho}) with potential
$\xi_{0} \in \Omega$ are bounded, then all of them are bounded for every $\xi
 \in \Omega$. In this conditions we can state
\begin{prop}
If the solutions of (\ref{Scho}) are bounded
 then $(\Sigma, \Phi^{E})$ admits an invariant absolutely continuous 
measure with continuous density function. 
\end{prop}
{\em Proof:} We recall the following inequality for the first derivative
\[{\displaystyle \max_{t\in R}} \mid x'\mid \leq 
{\displaystyle \max_{t\in R}} \mid x \mid+{\displaystyle \max_{t\in R}}
 \mid x''\mid \;.\]
This means that the family $(F_t)_{t\in R}$
introduced in theorem (\ref{solC}) is bounded. Moreover for every $\epsilon >0$
there is a $\delta >0$ such that for any two elements $\varphi_1, \varphi_2$ of 
$S^1$ with $\mid \varphi_1-\varphi_2 \mid < \delta$ we have
\[\mid \int^t_0 \frac{\partial f}{\partial \varphi}(\Phi_s(\xi,\varphi_1))
-\int^t_0 \frac{\partial f}{\partial \varphi}
(\Phi_s(\xi,\varphi_2)) ds \mid =
\mid \ln r(t,\xi,\varphi_1) -\ln r(t,\xi,\varphi_2) \mid < \epsilon\]
for every $t\in R$. This shows that $(F_t)$ is conditionally compact. Hence
our claim is a consequence of theorem (\ref{solC}).\par
\vspace{0.4cm}
This is not the only case where invariant absolutely continuous measures with
$L^{2}$-density function occur. Let $\gamma(E)$ be the Lyapunov exponent of
 $L_E$. 
 Then the set $A=\{ E\in R/ \gamma(E)=
0 \}$ is known to be the essential support with respect to the Lebesgue measure
of the absolutely continuous part of the spectral measure . This is a consequence
of the work by Pastur, Ishii and Kotani. The following theorem was demonstrated
 by Kotani \cite{kota} and Deift-Simon \cite{desi} and 
show the kind of
 almost-periodic solutions we find for these values of the energy.
\begin{theo}\label{cape}
There exist two linearly independent solutions $u_{\pm} (t,\xi, E)$ of
$x''+\xi(t) x=Ex$ for almost every $(\xi, E) \in \Omega \times A$ and two 
positive measurable functions $P_1, P_2: \Omega \longrightarrow R$ for almost
every $E \in A$ such that\\
$i) u_{+}=\bar{u}_{-} $.\\
$ii) \mid u_{\pm} (t, \xi, E)\mid = P_1(\xi_t,E), \quad \mid u'_{\pm} (t, \xi, E)
\mid = P_2(\xi_t,E) $.\\
$iii) \int_\Omega [P^{2}_2(\xi, E)+ P^{2}_2(\xi, E)]dm_0 =
 \eta_{E} \in (0,\infty)$.
\end{theo}
\indent
 We take $E\in A$ where the claim of the theorem holds. Our objetive is to prove
 that  an invariant square integrable measure under $\Phi^E$ can be 
constructed.\\[.4 cm]
 It follows from Birkhoff's ergodic theorem that
\[\lim_{T \rightarrow \infty} \frac{1}{2T} \int^T_{-T}[\mid u_{\pm}^{2}(t, \xi, E)\mid
+\mid u'^{2}_{\pm} (t, \xi, E)\mid]dt= \int_\Omega [P_1^{2}(\xi, E)+ P_2^{2}(\xi, E)]
dm_0\]
for almost every $\xi \in \Omega$. Now it is easy to check that

\begin{equation}\label{cres}
\lim_{\mid t \mid \rightarrow \infty } \frac{\mid u_{\pm}^{2}(t, \xi, E) \mid +
\mid u'^{2}_{\pm}(t, \xi, E) \mid}{t}=0
\end{equation}
for almost every $\xi \in \Omega$. In consequence

\begin{equation}\label{creS}
\lim_{\mid t \mid \rightarrow \infty} \frac{x^{2}(t,\xi, E)+ x'{^{2}}(t,\xi, E)}{t}=0
\end{equation}
for every solution of these linear equations.\par
If $u_{+}(0, \xi, \varphi)=\alpha_1(\xi)$, $u'_{+}(0, \xi, E)=\alpha_2(\xi)$, it follows
 from the proof of the theorem that $\alpha_1,\alpha_2 \in L^{2}(\Omega, m_0)$ 
and moreover
$\mid \alpha_1^{2}(\xi)\mid+\mid \alpha_2^{2}(\xi)\mid \geq 1$. Then we have
\[x(t, \xi,\varphi) = \alpha (\xi,\varphi) u_{+}(t,\xi,E)+\overline{\alpha 
(\xi, \varphi)}\overline{u_{+}(t,\xi,E)}\]
 where \[\alpha (\xi, \varphi) = \frac{1}{2i}
(\overline{\alpha_2 (\xi)} \sin \varphi - \alpha_1 (\xi)\cos \varphi)\;.\]
Taking 
$\beta(\xi) = 1/2\,( \mid \alpha_1^{2} (\xi) \mid+\mid \alpha_2^{2} 
(\xi)\mid)$ we see that $\beta \in L^1(\Omega, m_0)$. Moreover
  $1/2 \leq \beta(\xi)$ and $\mid \alpha^{2} (\xi,\varphi) 
\mid \leq \beta(\xi)$ for every  $(\xi,\varphi)\in \Sigma$  .\par
In order to achieve an invariant absolutely continuous measure whose density
function belongs to $L^{2}(\Sigma,m)$, we  consider the family
\[q_T(\xi,\varphi)=
\frac{1}{2T}\int^T_{-T}[x^{2}(t,\xi,\varphi)+(x')^{2}(t,\xi,\varphi)]
dt =\]
\[\frac{1}{2T}\int^T_{-T}2\Re\{\alpha^{2}(\xi,\varphi)(u_{+}^{2}(t,\xi, E)+u'^{2}_{+}
(t,\xi,E))\}dt+\]
\[ \frac{1}{2T}\int^T_{-T}2\mid\alpha^{2}(\xi,\varphi)\mid(\mid u^{2}_{+}(t,\xi,E)
\mid+
\mid u'^{2}_{+}(t,\xi,E)\mid)\}dt\;.\]
We cannot prove the pointwise convergence of $\{q_T\}$ directly from theorem
 (\ref{cape}). In
consequence we will establish the convergence of $\sqrt{q_T}$ in a weak-topology
whose limit allows us to obtain a positive solution  of the functional equation 
(\ref{ecuA}). Of course this finally implies the pointwise convergence of $\{q_T\}$ to a 
finite limit.\par
 We will always refer through the proof to the measures
$ \nu_0 = 1/\beta (\xi)\, dm_0$ on $\Omega$ and $\nu = \nu_0 \times
 d\rho$ on $\Sigma$. Notice that
\[ \frac{q_T (\xi,\varphi)}{\beta(\xi)} \leq \frac{1}{2T} \int^T_{-T} 
4(\mid u_{+}^{2}
(t,\xi, E)\mid + \mid u'^{2}_{+}(t,\xi, E)\mid)dt \]
 and 
\[ \int_{\Sigma} q_T(\xi,\varphi) d \nu \leq \frac{1}{2T}\int^T_{-T} 
\int_{\Omega} 4(P_1^{2}(\xi_t,E)+P_2^{2}(\xi_t,E))dm_0dt= 4\eta_E < 
\infty \;.\]   
Then $\{ \sqrt{q_T} \}$ is a bounded subset of $L^{2}_{\nu}=L^{2}(\Sigma,\nu)$ 
which possesses a
subsequence  $\{ \sqrt{q_{T_{j}}} \}$, with $\lim_{j \rightarrow \infty} T_j=
\infty$, satisfying:
\[ \left\{
 \begin{array}{ll}
i) \sqrt{q_{T_j}} &\mbox{converges in the weak topology $\sigma(L^{2}_{\nu}
,L^{2}_{\nu})$ to a limit $\sqrt{q}$.}\\[.4cm]
ii) {\displaystyle \frac{q_{T_{j}}}{T_j}} &\mbox{converges to zero 
almost everywhere.}
\end{array} 
\right.\]
\indent The next lemmas show certain important properties of this sequence.
\begin{lemm}\label{lem1} 
Let $l\in Z$ be fixed. Then \\
i) $\{q_{T_{j}+l} -q_{T_{j}}\}$ converges to zero almost everywhere. \\
ii) $\{\sqrt{q_{T_{j}+l}}\}$ converges to $\sqrt{q}$ in the $\sigma(L^{2}_{\nu}
,L^{2}_{\nu})$-topology.
\end{lemm} 
{\em Proof:} $i)$ We suppose that $l$ is positive. The other case is completely analogous
\[  q_{T_{j}+l}(\xi,\varphi)-q_{T_j}(\xi,\varphi)=\]
\[\frac{1}{2(T_{j}+l)}\int^{T_{j}+l}_{-T_{j}-l}e^{-\int^t_0 \frac{\partial f}
{\partial \varphi}(\Phi_s(\xi, \varphi))ds} dt-\frac{1}{2T_j}\int^{T_j} _
{-T_j} e^{-\int^t_0 \frac{\partial f}{\partial \varphi}(\Phi_s(\xi, \varphi)ds}dt
= \]
\[ S_{1,j}(\xi, \varphi)+S_{2,j}(\xi, \varphi)+S_{3,j}(\xi, \varphi). \]
where
\[S_{1,j}(\xi,\varphi)= \frac{-l}{2T_j(T_{j}+l)} \int^{T_j}_{-T_j}
e^{ -\int^t_0 \frac{\partial f}
{\partial \varphi}(\Phi_s(\xi, \varphi))ds} dt\]
\[S_{2,j}(\xi,\varphi)= \frac{1}{2(T_{j}+l)}
\int^{T_{j}+l}_{T_j}e^{ -\int^t_0\frac{\partial f}
{\partial \varphi}(\Phi_s(\xi, \varphi))ds} dt\]
\[S_{3,j}(\xi,\varphi)= \frac{1}{2(T_{j}+l)}
\int^{-T_j}_{-T_{j}-l}e^{-\int^t_0\frac{\partial f}
{\partial \varphi}(\Phi_s(\xi, \varphi))ds} dt \;.\]
\vspace{0.1cm}
\indent First we know that 
\[\lim_{j\rightarrow \infty} S_{1,j}(\xi,\varphi)=\lim_{j \rightarrow \infty} \frac{-l}{T_{j}+l} q_{T_j}
(\xi,\varphi)=0 \]
almost everywhere. Besides
\[ S_{2,j}(\xi, \varphi)=\frac{1}{2(T_{j}+l)}e^{-\int^{T_j}_{0} \frac{\partial f}
{\partial \varphi} (\Phi_s(\xi, \varphi) ds}\int^{T_{j}+l}_{T_j}
e^{-\int^t_{T_j} \frac{\partial f}{\partial \varphi} 
(\Phi_s(\xi, \varphi) ds}dt\;. \]
If $M=max \left\{ \mid \frac{\partial f}{\partial \varphi}(\xi,\varphi) \mid/
(\xi, \varphi)  \in \Sigma \right\} $ we obtain using (\ref{creS}) that
\[ \lim_{j \rightarrow \infty} S_{2,j}(\xi, \varphi)=l e^{M.l}\lim_
{j \rightarrow \infty} \frac{2r(T_j,\xi,\varphi)}{2(T_{j}+l)}= 0 \:. \]
\indent Similarly $\lim_{j \rightarrow \infty}S_{3,j} (\xi,\varphi)=0 $. This
completes the proof.

\vspace{0.2cm}
ii) Since $q_{T_{j}+l}-q_{T_j}= \left( \sqrt{q_{T_{j}+l}}-\sqrt{q_T}\right)
\left( \sqrt{q_{T_{j}+l}}+\sqrt{q_T}\right)$ we deduce that 
$\sqrt{q_{T_{j+l}}}-\sqrt{q_T}$ converges to zero almost everywhere.\\
Moreover the family $\{ \sqrt{q_T} \}$ is uniformly integrable . Indeed, if 
$\epsilon$ is a positive constant, $\delta_{\epsilon}= \epsilon^{2}/4\eta_E $
 and $A$
is a measurable set with $\nu(A) \leq \delta_{\epsilon}$ then
\[ \int_A \sqrt{q_T} d\nu= \int_{\Sigma} \sqrt{q_T}. \chi_A d\nu \leq
\left( \int_{\Sigma} q_T d \nu \right)^{1/2} \nu(A)^{1/2} \leq \epsilon \;.\]
\indent Let $B\subset L^{2}(\Sigma,\nu)$ be the unit ball and $B'=4\eta_E^{1/2}B$. 
The topologies $\sigma(L^{2}_{\nu},L^{2}_{\nu})$ and $\sigma(L^{2}_{\nu},C)$
 where $C=C(\Sigma)$,
are identical on $B'$, so we will prove the convergence of $\sqrt{q_{T_{j}+l}}
-\sqrt{q_{T_j}}$ in the last topology.\par
If $h\in C(\Sigma)$ the sequence $\{(\sqrt{q_{T_{j}+l}}-\sqrt{q_{T_j}})h\}$
is uniformly integrable and converges to zero almost evertywhere. According to
Vitali's theorem we find that $\lim_{j \rightarrow \infty} \int_{\Sigma} \mid
(\sqrt{q_{T_{j}+l}}-\sqrt{q_{T_j}})h\mid d\nu=0$ . \par
This confirms that $\{\sqrt{q_{T_{j}+l}}\}$ converges to $\sqrt{q}$ in the weak
 topology.
\begin{lemm}\label{lem2}
Let us fix $l\in Z$. Then \\
i) $\{\sqrt{q_{T_j} (\Phi_l(\xi,\varphi))}\}$ converges to $\sqrt{q
(\Phi_l(\xi,\varphi))}$ in the $\sigma(L^{2}_{\nu},L^{2}_{\nu})$ -topology. \\
ii) $q$ satisfies the functional equation
\[q(\Phi_l(\xi,\varphi))=q(\xi,\varphi).e^{\int^l_0 \frac{\partial f}{\partial \varphi}
(\Phi(\xi,\varphi))ds}\;.\]
\end{lemm}
{\em Proof:} $i)$ Again we suppose that $l$ is positive.
We will prove the convergence in the $\sigma(L^{2}_{\nu},C)$-topology. 
If $h\in C(\Sigma)$, and
\[\upsilon(\xi,\varphi)= \frac{h(\Phi_{-l}(\xi,\varphi))}{\beta(\xi_{-l})}
 e^{\int^{-l}_0 \frac{\partial f}
{\partial \varphi}(\Phi_{s}(\xi,\varphi)) ds} \beta(\xi) \]
then
\[ \int_{\Sigma} \left[ \sqrt{q_{T_j}(\Phi_l(\xi,\varphi))}-\sqrt{q(\Phi_l
(\xi,\varphi))}\right]
h(\xi,\varphi)d\nu =\]
\[ \int_{\Sigma} \left[ \sqrt{q_{T_j}(\xi,\varphi)}-\sqrt{q(\xi,\varphi)}\right]
\upsilon(\xi,\varphi)d\nu \;.\]
\indent We see that there exists a constant $M$ such that $\upsilon(\xi,\varphi)\leq
M \beta(\xi)$ and $\int_{\Sigma} \beta^{2} (\xi) d\nu=\int_\Omega \beta(\xi)
dm_0<\infty $.
Hence $\upsilon \in L^{2}(\Sigma,d\nu)$. Now using lemma (\ref{lem1}), we get 
\[ \lim_{j \rightarrow \infty} \int_{\Sigma} \left[ \sqrt{q_{T_j}(\Phi_l(\xi,\varphi))}-
\sqrt{q(\Phi_l(\xi,\varphi))}\right] h(\xi,\varphi)d\nu=0 \;.\]

$ii)$ The argument used in Section (1) yields
\[ \sqrt{\frac{T_j-l}{T_j}}q_{T_{j}-l}^{\frac{1}{2}}(\xi,\varphi)\leq
 q_{T_j}^{\frac{1}{2}}(\Phi_l(\xi,\varphi))e^{-\frac{1}{2}\int^l_0
\frac{\partial f}{\partial \varphi}(\Phi_s(\xi,\varphi))ds}
 \leq \sqrt{\frac{T_j+l}{T_j}}q_{T_{j}+l}^{\frac{1}{2}}(\xi,\varphi) \:.\]
\indent Taking limits in the $\sigma(L^{2}_{\nu},L^{2}_{\nu})$-topology as $j \rightarrow \infty$ we obtain
\[ \sqrt{q(\Phi_l(\xi,\varphi))}=\sqrt{q(\xi,\varphi)} e^{{\frac{1}{2}} 
\int^l_0 \frac{\partial f}{\partial \varphi}(\Phi (\xi,\varphi))ds} \]
almost everywhere. Taking squares we prove $(ii)$.
We can assume that $q$ satisfies (\ref{ecuA}) for every $t \in R$ and $(\xi,\varphi)
\in \Sigma$.\\[.2cm]
\indent Finally we verify that $q$ is different from zero almost everywhere.
The set $A=\{(\xi,\varphi) \in \Sigma \;/\; q(\xi,\varphi)=0 \} $ is invariant and it 
follows from (\ref{desi}) that 
\[m(A) \leq (\int_A \sqrt{q_{T_j}}dm)^{\frac{1}{2}}(\int_{\Sigma}\frac{1}{ \sqrt{q_{T_j}}} dm)
^{\frac{1}{2}} \leq \]
\[ (\int_A \sqrt{q_{T_j}}dm)^{\frac{1}{2}}(\int_{\Sigma} \frac{1}{q_{T_j}} dm)
^{\frac{1}{4}} \leq (\int_A \sqrt{q_{T_j}}dm)^{\frac{1}{2}}\]
then
 \[ m(A) = \lim_{j \rightarrow \infty} \int_A \sqrt{q_{T_j}} dm =
\int_A \sqrt{q}dm =0 \;.\]
\indent This confirms the pointwise convergence of $\{q_{T}\}$. We may apply proposition
 (\ref{cond}) to get
\begin{theo}
The flow $(\Sigma,\Phi^{E})$ admits an invariant measure, equivalent to
$m$ whose density function belongs to $L^{2}(\Sigma ,m)$ for almost every
 $E\in A$.
\end{theo}


Let $q$ be the limit of the above family of functions $\{q_{T}\}$. Note that
\begin{equation}\label{tipo}
(i)\:q\in L^{1}(\Sigma,m)\:\:\:,\:\:\:(ii)\:\frac{1}{q}\in L^{2}(\Sigma,m)\;.
\end{equation}
These conditions on convergence will be essential in the next Sections.
 From now on we denote
 by $A_{0}\subset A$ the set of energies where they  are satisfied.\\
\indent If $(\Sigma,\Phi^{E})$ is uniquely ergodic then $\gamma(E)=0$.   
Let us assume that  $\gamma(E)>0$. In this conditions the function
$r(t,\xi,\varphi) $ increases exponentially 
for almost every $(\xi,\varphi) \in \Sigma$ when  $ t\rightarrow {\pm}\infty$
 and hence
\[ \lim_{T \rightarrow \infty} p_T(\xi,\varphi)= \lim_{T \rightarrow \infty}
\frac{1}{2T}\int^T_T \frac{1}{r(t,\xi,\varphi)} dt =0  \]
almost everywhere.
>From proposition (\ref{coni}) we deduce that $(\Sigma,\Phi^{E})$ does not admit
an invariant absolutely continuous measure. In fact invariant singular measures
which induce  purely discontinuous measures on the fibres are known in this 
case. See \cite{furs}, \cite{molc}.
 




	\section{The admissible directions of differentiability}
We consider again the subset $A_0$ where the functions $\{q_T\}$ converge 
as shown  in (\ref{tipo}).
Let us fix $E\in A_0$ and $\Gamma \in C(\Omega)$. 
If $f_{E}(\xi,\varphi)$ defines the differential equation
 (\ref{fibr}) then $\partial f/\partial E\,=\sin^{2} \varphi$. The family
\begin{equation}\label{nuev}
{\tilde{q}}_{T,\Gamma}(\xi,\varphi)=\frac{1}{2T}\int_{-T}^{T}e^{-\int_{0}^{t}\frac{\partial
f}{\partial \varphi}(\Phi_{s}(\xi,\varphi))ds}\Gamma(\xi_t)\sin^{2}
\varphi_{t}(\xi,\varphi)dt=
\end{equation}
\[\frac{1}{2T}\int_{-T}^{T}\frac{q(\xi,\varphi)}
{q(\Phi_{t}(\xi,\varphi))}\Gamma(\xi_t)\sin^{2}
\varphi_{t}(\xi,\varphi)dt=
\frac{1}{2T}\int_{-T}^{T}x^{2}(t,\xi,\varphi)\Gamma(\xi_t)dt\]
converges to a limit ${\tilde{q}}_{\Gamma}(\xi,\varphi)$. Moreover
${\tilde{q}}_{\Gamma}(\xi,\varphi)=q(\xi,\varphi)q^{\star}_{\Gamma}(\xi,\varphi)$ where
$q^{\star}_{\Gamma}(\xi,\varphi)$ is an invariant function. This also shows that
$q_{\Gamma}(\xi,\varphi)$ satisfies  the functional equation (\ref{ecuA}).\par
We introduce the following set of directions:
\begin{defi}\label{admi}
We say that $\Gamma$ is an admissible direction of differentiability if 
\begin{equation}\label{TIPO}
(i)\:{\tilde{q}}_{\Gamma}\in L^{1}(\Sigma,m)\:\:\:,\:\:\:(ii)\:{\tilde{q}}_{1,\Gamma}=\frac{1}
{{\tilde{q}}_{\Gamma}}\in L^{2}(\Sigma,m)
\end{equation}
\end{defi}
The set $C_{ad}(\Omega)$ contains the admissible directions of differentiability.
 This Section is transitional. We are going to show the basic properties of the  measure 
$\mu_{\Gamma}={\tilde{q}}_{1,\Gamma}dm$ for $\Gamma \in C_{ad}(\Omega)$. 
Our conclusions will be essential in Section 6. \\


The condition $(i)$ of the definition always holds, so only $(ii)$ has
 to be verified. We represent
\[{\tilde{q}}_{T}(\xi,\varphi)=\frac{1}{2T}\int_{-T}^{T}e^{-\int_{0}^{t}\frac{\partial
f}{\partial \varphi}(\Phi_{s}(\xi,\varphi))ds}\sin^{2}\varphi_{t}(\xi,\varphi)dt=
\frac{1}{2T}\int_{-T}^{T}x^{2}(t,\xi,\varphi) \;.dt\]
Note that for almost $\xi \in \Omega$ one has
\[ \lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}x'^{2}(t,\xi,\varphi)dt=
\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}x^{2}(t,\xi,\varphi)
J(\xi_t)dt\;.\]
Thus, there is a constant $M$ such that 
${\tilde{q}}\leq q\leq M{\tilde{q}}$. Hence ${\tilde{q}}_{1}\in 
L^{2}(\Sigma,m)$ and ${\tilde{q}}_{1}dm$ is an invariant measure. This shows
that $\Gamma \equiv 1 \in C_{ad}(\Omega)$.\\
\begin{prop}\label{pro1}
There exists an invariant set $\Omega_{0}\subset \Omega$ with 
$m_{0}(\Omega_{0})=1$ such that if $\xi \in \Omega_0$ then \\
$i)$ $\{{\tilde{q}}_{T,\Gamma}(\xi,\varphi)\}$ converges to 
${\tilde{q}}_{\Gamma}(\xi,\varphi)$ for  every $\varphi\in P^1$.\\
$ii)$ ${\tilde{q}}_{1,\Gamma}(\xi,\varphi) \in C^{\infty}(P^{1})$ and $\int_{P^1}
{\tilde{q}}_{1,\Gamma}(\xi,\varphi)d\rho_0$=$1/\lambda_{\Gamma}$ (constant).\\ 
$iii)$  $\int_{\Sigma}{\tilde{q}}_{1,\Gamma}(\xi,\varphi)\Gamma(\xi)\sin^{2}\varphi d\mu=1$
for every invariant measure under the flow.
\end{prop}
{\em Proof:} 
If $\varphi_{1},\varphi_{2}$ are different elements of $\Sigma$ we represent 
\[x(t,\xi, \varphi)=c_1(\varphi)x(t,\xi,\varphi_1)+c_2(\varphi)x(t,\xi,\varphi_2)\]
where $c_1,c_2:P^1\rightarrow R$ are smooth functions. We  have
\begin{equation}\label{suma}
{\tilde{q}}_{\Gamma}(\xi,\varphi)=
2c_1(\varphi)c_2(\varphi)\lim_{T\rightarrow \infty}\frac{1}{2T}
\int_{-T}^{T}x(t,\xi,\varphi_1)x(t,\xi,\varphi_2)\Gamma(\xi_t)dt\:\:+
\end{equation}
\[c_1^2(\varphi)\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}x^{2}
(t,\xi,\varphi_1)
\Gamma(\xi_t)dt+c_2^2(\varphi)\lim_{T\rightarrow \infty}\frac{1}{2T}
\int_{-T}^{T}x^{2}(t,\xi,\varphi_2)\Gamma(\xi_t)dt\]
when these limits exist. If we choose $\xi$ of $\Omega$ and $\varphi_1,\varphi_2
,\varphi=\varphi_3$ of  $\Sigma$ such that $\{{\tilde{q}}_T(\xi,\varphi_i)\}$ 
converges for
 every $i=1,2,3$ then it is obvious that there exists
$\lim_{T\rightarrow \infty}
1/2T\:\int_{-T}^{T}x(t,\xi,\varphi_1)x(t,\xi,\varphi_2)\Gamma(\xi_t)dt$ 
and in consequence $\{{\tilde{q}}_{T,\Gamma}(\xi,\varphi)\}$ converges for every 
$\varphi \in P^1$.\par
 The map $I:\Omega\rightarrow R\:,\:\xi \rightarrow 
\int_{P^1}{\tilde{q}}_{1,\Gamma}(\xi,\varphi)d\rho_0$ is invariant. We
 can find an invariant subset $\Omega_0 \subset \Omega$ with $m_0(\Omega_0)=1$ 
and a constant $\lambda_{\Gamma}$ such that
 $\{{\tilde{q}}_{T,\Gamma}(\xi,\varphi)\}$ converge,
${\tilde{q}}_{1,\Gamma}(\xi,\varphi) \in L^2(P^1,\rho_0)$  and
$\int_{P^1}{\tilde{q}}_{1,\Gamma}
(\xi,\varphi)d\rho_0=1/\lambda_{\Gamma}$ for every 
$\xi \in \Omega_0$.\par
Let us fix $\varphi_0 \in P^1$ . Choosing $\varphi_1=\varphi_0$,
$\varphi_2=\varphi_0+\pi/2$ we obtain
\[x(t,\xi, \varphi+\varphi_0)=\cos\varphi\: x(t,\xi,\varphi_0)+
\sin\varphi\: x(t,\xi,\varphi_0+\frac{\pi}{2})\:.\]
Denoting $\kappa_1(\xi)={\tilde{q}}_{\Gamma}(\xi,\varphi_0)$, $\kappa_2(\xi)=
{\tilde{q}}_{\Gamma}(\xi,\varphi_0+\frac{\pi}{2})$ and
\[\kappa_3(\xi)=\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}
x(t,\xi,\varphi_0)x(t,\xi,\varphi_0+\frac{\pi}{2})\Gamma(\xi_t)dt\]
we find
\begin{equation}\label{repr}
{\tilde{q}}_{\Gamma}(\xi,\varphi+\varphi_0)=\cos^2 \varphi\: \kappa_{1}(\xi)+
\sin^2 \varphi\: \kappa_{2}(\xi)+
2 \sin \varphi\: \cos \varphi\: \kappa_{3}(\xi)
\end{equation}
and
\begin{equation}\label{reps}
{\tilde{q}}_{1,\Gamma}(\xi,\varphi+\varphi_0)=\frac{1}{\cos^2\varphi\:\kappa_{1}(\xi)+
\sin^2\varphi\:\kappa_{2}(\xi)+
2 \sin\varphi\:\cos\varphi\:\kappa_{3}(\xi)}\;.
\end{equation}
\indent If $\xi \in \Omega_0$ it is obvious that ${\tilde{q}}_{\Gamma}(\xi,\varphi)$ is a 
smooth function in $\varphi$. Since ${\tilde{q}}_{1,\Gamma}\in L^{2}
(P^1,\rho_0)$ then ${\tilde{q}}_{\Gamma}(\xi, \varphi) \neq 0$  for every $\varphi 
\in P^{1}$. This shows that ${\tilde{q}}_{1,\Gamma}\in C^{\infty}(P^1)$. We 
assume that ${\tilde{q}}_{\Gamma}$ is always positive on $\Omega_0 \times P^1$.
Then  we have 
\[\frac{1}{\lambda_{\Gamma}}=\frac{1}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{1}
{\cos^2\varphi\:\kappa_{1}(\xi)+\sin^2\varphi\:\kappa_{2}(\xi)+
2 \sin(\varphi)\:\cos(\varphi)\:\kappa_{3}(\xi)}d\varphi\:.\]
Taking $s=\tan\varphi$ and $\kappa_0=\sqrt{\kappa_1\kappa_2-\kappa_{3}^{2}}\:$ we
obtain
\begin{equation}\label{form}
\frac{1}{\lambda_{\Gamma}}=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{1}
{\kappa_{1}(\xi)+s^2\kappa_{2}(\xi)+2s\kappa_{3}(\xi)}ds=
\end{equation}
\[\frac{2}{\pi \kappa_0(\xi)}\left[\tan^{-1}\left(\frac{2\kappa_3(\xi)+2\kappa_{2}(\xi)s}
{\kappa_0(\xi)}\right)\right]_{-\infty}^{\infty} =\frac{2}{\kappa_{0}(\xi)}\;.\]
\indent Moreover since
\[\lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}{\tilde{q}}_{1,\Gamma}
(\Phi_t(\xi,\varphi)\Gamma(\xi_t)\sin^{2}
\varphi_{t}(\xi,\varphi)dt=1\]
at every $(\xi,\varphi)\in \Omega_0\times P^1$ we conclude that
$\int_{\Sigma}{\tilde{q}}_{1,\Gamma}(\xi,\varphi)\Gamma(\xi)\sin^{2}\varphi 
d\mu=1$ for every 
normalized invariant measure under the flow. This completes the proof.\\[.4cm]

\indent The sequence $\{ \tilde{q}_{T,\Gamma}\}$ converges to a limit 
$ \tilde{q}_{\Gamma}\in L^1(\Sigma,m)$. We can assume  that 
${\tilde{q}}_{\Gamma}>0$ on 
$\Omega_0 \times P^1$, otherwise we would change $\Gamma$ to $-\Gamma$.
Using the relation (\ref{repr}) we define
\begin{equation}\label{beta}
\beta(\xi)=\frac{1}{2}({\tilde{q}}_{\Gamma}(\xi,\varphi)+{\tilde{q}}_{\Gamma}(\xi,\varphi+
\frac{\pi}{2}))=\frac{1}{2}(\kappa_{1}(\xi)+\kappa_{2}(\xi))\;.
\end{equation}
Then it is obvious that $\beta \in L^1(\Omega,m_0)$ and ${\tilde{q}}_{1}\beta
 \in L^1(\Sigma,m_0)$.\par
 We take $\varphi_0 \in P^1$ such that ${\tilde{q}}_{1,\Gamma}(\xi,\varphi_0),
{\tilde{q}}_{1,\Gamma}(\xi,\varphi_0+\pi/2) \in L^{2}(\Omega_0,m_0)$ and
 $\beta(\xi){\tilde{q}}_{1,\Gamma}(\xi,\varphi_0),\:\: \beta(\xi)
{\tilde{q}}_{1,\Gamma}(\xi,\varphi_0+\frac{\pi}{2}) \in L^{1}(\Omega,m_0)$.
 We assume that  
\[\int_{\Omega}(\beta(\xi)+{\tilde{q}}_{1,\Gamma}(\xi,\varphi_0)
){\tilde{q}}_{1,\Gamma}(\xi,\varphi_0)dm_0\leq \parallel{\tilde{q}}_{1,\Gamma} 
\parallel_2^2 +\frac{1}{\lambda_{\Gamma}}\parallel
\beta \parallel_1\;.\]
\begin{lemm}\label{lem3}
The functions $\kappa_{3} /\kappa_{1}^{2}$ and 
$\kappa_{3}/\kappa_{2}^{2}$ belong to $L^1(\Omega,m_0)$.
\end{lemm}
{\em Proof:} We start by showing that  $\kappa_3
/\kappa_1^2 \in L^1(\Omega,m_0)$. We have proved that
$ \mid\kappa_{3}(\xi)\mid \leq \sqrt{\kappa_1(\xi)\kappa_2(\xi)}$. 
>From (\ref{beta}) we deduce
\[1+\frac{{\tilde{q}}_{\Gamma}(\xi,\varphi+\frac{\pi}{2})}{{\tilde{q}}_
{\Gamma}(\xi,\varphi)} \leq 2\beta(\xi){\tilde{q}}_{1,\Gamma}(\xi,\varphi)\]
and hence
\[\frac{\kappa_2(\xi)}{\kappa_1(\xi)}=\frac{{\tilde{q}}_{\Gamma}
(\xi,\varphi_0+
\frac{\pi}{2})}{{\tilde{q}}_{\Gamma}(\xi,\varphi_0)}\leq 2\beta(\xi)
{\tilde{q}}_{1,\Gamma}(\xi,\varphi_0)\in L^1(\Omega,m_0)\;.\]
\indent Combining these inequalities, we obtain
\[\int_{\Omega}\frac{\mid \kappa_3 \mid }{\kappa_1^2}dm_0\leq
\int_{\Omega}\sqrt{\frac{\kappa_2}{\kappa_1}}\frac{1}{\kappa_1}dm_0\leq
\left[\int_{\Omega}{\frac{\kappa_2}{\kappa_1}}dm_0\right]^{\frac{1}{2}}
\left[\int_{\Omega}\frac{1}{\kappa_1^2}dm_0\right]^{\frac{1}{2}} \leq\infty\;.\]
The other case is completely analogous.
Moreover
\begin{equation}\label{nor2}
\| \frac{\kappa_3}{\kappa_1^2}\| _1\leq \parallel{\tilde{q}}_{1,\Gamma} 
\parallel_2^2 +\parallel{\tilde{q}}_{\Gamma}\parallel_1 \parallel
{\tilde{q}}_{1,\Gamma}\parallel_1\;.
\end{equation}
\\[.3 cm]
\indent Let  $\mu$ be an invariant measure on $\Sigma$ which is mapped into  $m_0$ 
under the natural projection on $\Omega$. Then $\tilde{q}_{1,\Gamma}\in 
L^1(\Sigma,\mu)$. This is the main consequence of the next 
\begin{prop}\label{elep}
There exist $\kappa\in L^1(\Omega,m_0)$,  $\kappa'\in L^2(\Omega,m_0)$ such that 
$0<\kappa'(\xi)\leq {\tilde{q}}_{1,\Gamma}(\xi,\varphi)\leq\kappa(\xi)$ for every 
$(\xi,\varphi) \in \Omega_0 \times P^1$. Moreover there exists a constant $M$ 
such that $\| \kappa \|_1\leq M( \parallel{\tilde{q}}_{1,\Gamma} 
\parallel_2^2 +\parallel{\tilde{q}}_{\Gamma}\parallel_1 \parallel
{\tilde{q}}_{1,\Gamma}\parallel_1)\;.$
\end{prop}
{\em Proof:} It is known that ${\displaystyle \frac{1}{2\beta}}\leq {\tilde{q}}_{1,\Gamma}
(\xi,\varphi)$ for every $(\xi,\varphi) \in \Omega_0 \times P^1$. Therefore the
 only problem is to obtain the function $\kappa$ of our claim.\par
Let us identify $\Omega \times \{\varphi_0\}$ with $\Omega$. We are going to 
prove that ${\tilde{q}}_{1,\Gamma}$ possesses continuous derivative along the 
trajectories lying on $\Omega_0$.\par
Let us fix $\xi \in \Omega_0$. We know that
\[{\tilde{q}}_{1,\Gamma}(\xi_t,\varphi_0)={\tilde{q}}_{1,\Gamma}(\xi,\varphi_{-t}
(\xi_t,\varphi_0))e^{-\int_0^t \frac{\partial{f}}{\partial{\varphi}}(\Phi_s(\xi,\varphi_{-t}
(\xi_t,\varphi_0)))ds}\]
and ${\tilde{q}}_{1,\Gamma}$ is smooth with respect to $\varphi$. Thus taking limits 
as $t\rightarrow 0$ it follows that $\lim_{t\rightarrow 0}
{\tilde{q}}_{1,\Gamma}(\xi_t,\varphi_0)={\tilde{q}}_{1,\Gamma}(\xi,\varphi_0)$.
The same conclusion can be derived  for any  $\varphi$ of $S^1$.\par
In this way we conclude that  $ \lim_{t\rightarrow 0}\kappa_i
(\xi_t)=\kappa_i(\xi)$ for $i=1,2,3$. We decompose
\[{\tilde{q}}_{1,\Gamma}(\xi_t,\varphi_0)-{\tilde{q}}_{1,\Gamma}(\xi,\varphi_0)=\]
\[\left[{\tilde{q}}_{1,\Gamma}(\xi_t,\varphi_0)-{\tilde{q}}_{1,\Gamma}
(\xi_t,\varphi_{t}(\xi,\varphi_0))\right]+
\left[{\tilde{q}}_{1,\Gamma}(\xi_t,\varphi_{t}(\xi,\varphi_0))-{\tilde{q}}_{1,\Gamma}
(\xi,\varphi_0)\right]\;.\]
\indent First we have
\[{\tilde{q}}_{1,\Gamma}(\xi_t,\varphi_0)-{\tilde{q}}_{1,\Gamma}
(\xi_t,\varphi_{t}(\xi,\varphi_0))=-\frac{\partial{{\tilde{q}}_{1,\Gamma}}}
{\partial{\varphi}}(\xi_t,\varphi'_t)(\varphi_t(\xi,\varphi_0)-\varphi_0)\]
where $\varphi'_t$ is an intermediate point between $\varphi_0$ and $\varphi_t
(\xi,\varphi_0)$ and 
\[\frac{\partial{{\tilde{q}}_{1,\Gamma}}}{\partial{\varphi}}(\xi_t,\varphi'_t)=
\frac{2\sin(\varphi'_t-\varphi_0)\cos(\varphi'_t-\varphi_0)\left[\kappa_1(\xi_t)-
\kappa_2(\xi_t)\right]}{{{\tilde{q}}_{1,\Gamma}^2}(\xi_t,\varphi'_t)}\:+\]
\[\frac{2\left[\sin^2(\varphi'_t-\varphi_0)-\cos^2(\varphi'_t-\varphi_0)
\right] \kappa_3(\xi_t)}{{{\tilde{q}}_{1,\Gamma}^2}(\xi_t,\varphi'_t)}\]
thus
\[\lim_{t\rightarrow 0}\frac{{\tilde{q}}_{1,\Gamma}(\xi_t,\varphi_0)-{\tilde{q}}_{1,\Gamma}
(\xi_t,\varphi_{t}(\xi,\varphi_0))}{t}=\frac{2\kappa_3(\xi)}{\kappa_1^2(\xi)}
f_E(\xi,\varphi_0)\;.\]
\indent We also see that
\[\lim_{t\rightarrow 0}\frac{{\tilde{q}}_{1,\Gamma}(\xi_t,\varphi_{t}(\xi,\varphi_0))
-{\tilde{q}}_{1,\Gamma}(\xi,\varphi_0)}{t}=-{\tilde{q}}_{1,\Gamma}(\xi,\varphi_0)
\frac{\partial{f_{E}}}{\partial{\varphi}}(\xi,\varphi_0)\;.\]
 \indent Therefore, from lemma (\ref{lem3}) we obtain 
\[\frac{d{\tilde{q}}_{1,\Gamma}(\xi_t,\varphi_0)}{dt}|_{t=0}
=\frac{2\kappa_3(\xi)}{\kappa_1^2(\xi)}f_E(\xi,\varphi_0)
-{\tilde{q}}_{1,\Gamma}(\xi,\varphi_0)
\frac{\partial{f_{E}}}{\partial{\varphi}}(\xi,\varphi_0)\in L^1(\Omega,m_0)\;.\]
\indent On the other hand there exists $T>0$ such that for every $(\xi,\varphi) \in \Sigma$ 
we can find $t=t(\xi,\varphi)\in \left[-T,T\right]$ with $\varphi_
{t}(\xi,\varphi)=\varphi_0$.\par
If $M_0=max\{\mid \partial{f}/\partial{\varphi}(\xi,\varphi)\mid /
(\xi,\varphi) \in \Sigma\}$ and we define
\[\kappa(\xi)=e^{M_0 T}\left[{\tilde{q}}_{1,\Gamma}(\xi,\varphi_0)+\int_{-T}^{T}
\mid\frac{d{\tilde{q}}_{1,\Gamma}}{ds}(\xi_s,\varphi_0)\mid dt\right]\]
then $\kappa \in L^1(\Omega,m_0)$ and finally
\[{\tilde{q}}_{1,\Gamma}(\xi,\varphi)={\tilde{q}}_{1,\Gamma}(\xi_t,\varphi_0)
e^{-\int_0^{-t} \frac{\partial{f}}{\partial{\varphi}}
(\Phi_s(\xi_t,\varphi_0)ds}=\]
\[\left[{\tilde{q}}_{1,\Gamma}(\xi,\varphi_0)+\int_{0}^{t}
\frac{d{\tilde{q}}_{1,\Gamma}}{ds}(\xi_s,\varphi_0)ds\right]
e^{-\int_0^{-t} \frac{\partial{f}}{\partial{\varphi}}(\Phi_s(\xi_t,\varphi_0)
dt}\leq \kappa(\xi)\;.\]
\indent This proves the first part of the statement. The bound of $\|\kappa\|_1$ comes 
from inequality (\ref{nor2}). Thus the proposition is established.


	\section{The differentiability of the rotation number}
We shall describe the behaviour of the rotation number $\alpha$ as function of 
the potential $g$. We recall the equation (\ref{prim})
\begin{equation}\label{segu}
-x''+(g_0(t)-E)x=0\::.
\end{equation}
If $E_0 \in A_0$ then $(\Sigma, \Phi^{E_0})$ admits an invariant measure equivalent
to $m$ whose density function belongs to $L^2(\Sigma, m)$, we may 
define a fibered map which transforms $\Phi^{E_0}$ into a skew-translation
and preserves the rotation number. This is a variant of an old idea used by Brunowski \cite{brun} 
and Hermann \cite{herm} for families of diffeomorphims of the circle which 
give us  
 an effective tool to analyze  the
variation of the rotation number across $g=g_0-E_0$. In this paragraph we
 will verify the Lipschitz character  of the rotation number on $g_0-E_0$
and will deduce its differentiability properties.\\[.2 cm]
\indent In order to formulate properly the problem we are going to deal we remember 
some known facts.
If ${\cal M}(g_0)$ stands for the frecuency module of $g_0$ and
 $A({\cal M})$  for
the set of all almost-periodic functions with frequency module contained in 
${\cal M}$, then it is possible to identify
 $A{\cal(M)}$ with $C(\Omega)$. In fact they are isomorphic as Banach
 algebras.\par
Now if $g'$ is a second almost-periodic function with ${\cal M}(g_0)\subset
{\cal M}(g')$ then for any sequence $(t_n)_{n=1}^{\infty}$ for which
$\{g'(t+t_n)\}_{n\in N}$ converges uniformly to a limit $\xi'$ , also
the family $\{g_0(t+t_n)\}_{n\in N}$ converges uniformly ; we denote  its limit by $\xi$ .
The map $\Pi:\Omega'\rightarrow \Omega, \xi'
\rightarrow \xi$ is a continuous and surjective homomorphism. The Haar
measure on $\Omega'$ is the image under $\Pi$ of the Haar measure on 
$ \tilde{\Omega}$. Thus if $p\in L^1 (\Omega,m_0)$ then 
$p\circ \Pi \in L^1(\Omega', m'_0)$ and $\int_{\Omega}
pdm_0 =\int_{\Omega'} p\circ \Pi dm'_0$.\\[.2 cm]
\indent The solutions of (\ref{segu}) can be described by the trajectories of the
flow defined  on $\Omega' \times P^1$ by
\begin{equation}\label{tras}
\dot{\varphi}= \cos^2 \varphi-(J\circ \Pi(\xi'_t)-E)\sin^2 \varphi 
\end{equation}
Let $pdm$ be an square integrable invariant measure under $\Phi$ on 
$\Sigma$. If we define $p'(\xi',\varphi) =p(\Pi(\xi'),\varphi)$ then 
$p'dm'$ is an invariant measure under $\Phi'$ on $\Sigma'$. Moreover 
$p' \in L^2(\Sigma',dm')$.\\[.2 cm]
\indent Let us consider a second almost periodic potential  $g$. We can find  a constant 
$\epsilon$ such that if $g'=g_0+
\epsilon g$ then ${\cal M}(g_0)\cup {\cal M}(g) \subset {\cal M}(g')$.
The solutions of the Schr\"{o}dinger equations with potentials $g$ and $g_0-E_0$
can be described by flows of type (\ref{tras}) defined on a common compact 
metric space.\\[.4 cm]
\indent We state the variation of the rotation number under the following conditions:\\
\indent {\em i) $\Omega$  stands by the hull of an adequate almost periodic
 function.\\
$\Gamma_0,\Gamma $ are continuous functions on  $C(\Omega)$}\\
\indent {\em ii)  $\Phi^{\eta}_t(\xi,\varphi)=
(\xi_t,\varphi_t^{\eta}(\xi,\varphi))$ 
denote  the  flow defined on $\Sigma$ by the differential equation}
\begin{equation}\label{dedi}
\dot{\varphi}= \cos^2 \varphi-[(1-\eta)\Gamma_0(\xi_t)+\eta\Gamma(\xi_t)]
\sin^2 \varphi =f_{\eta}(\xi,\varphi)\;.
\end{equation}
\indent {\em We assume that for $\eta=0$ the flow $(\Sigma ,\Phi)$ admits invariant 
measures with square integrable density function}.\\
We will consider $\alpha$ as a continuous function from $C(\Omega)$ to $R$ and
we  analyze the variation of the rotation number on $\Gamma_0$ in the
 direction  $\Gamma$.\\[.4 cm]
\indent Let $B'$ the set of the functions $p \in L^2(\Sigma,m)$ such that\\
i) $\mu=pdm$ is a normalized invariant measure under $\Phi$. \\
ii) $p(\xi,\varphi) \in C^{\infty}(P^1)$ for almost every $\xi \in \Omega$\\
iii) There exists $\kappa_{p} \in L^1(\Omega,m_0)$ such that $p(\xi,\varphi)\leq
\kappa_{p}(\xi)$ for almost every  $\xi \in \Omega$.\\[.2 cm]
\indent We take $p\in B'$.  Arguing as in the previous Section we know that there exists $\Omega _0 \subset
\Omega$ with $m_0(\Omega_0)=1$ such that $\int_{P1} p(\xi,\varphi) dm_0=1$
for every $\xi \in \Omega_0$. Actually $p(\xi, \varphi)dm_0$ is nothing but the
conditional measure induced by $\mu$ on $\{ \xi \} \times P^1$. \par
We introduce the transformation given by
\[
\begin{array}{rcl} 
H: \Sigma& \longrightarrow& \Sigma \\
 (\xi,\varphi)& \longrightarrow &\left\{ \begin{array}{ll}
   (\xi, \int^{\varphi}_0 p(\xi,s)ds\:  & \mbox{if}\:\: \xi \in \Omega_0 \\
   (\xi,\varphi) & \mbox{if}\:\: \xi \in \Omega-\Omega_0\:\:\:.
   \end{array}  \right. 
\end{array}
 \]
\indent The same formula extends the definition of $H$ to $L=\Omega \times R$. 
If $\xi \in \Omega_0$ and $\varphi_2 = n \pi + \varphi_1$ then
\[ \int^{\varphi_2} _0 p(\xi, s)ds= n\pi + \int^{\varphi_1}_0 [p(\xi,s)-1]ds\;. \]
This shows that if $(\xi,\varphi) \in L$ and $(\xi,\psi)= H(\xi,\varphi)$
then  $\psi$ $= \varphi+h(\xi,\varphi)$ where the function $h$ is 
measurable, bounded and periodic in $\varphi$ .\par
 If $\Psi ^{\eta}_t= H.\Phi^{\eta}_t. H^{-1}$ and
$\psi^{\eta}_t = \varphi^{\eta}_t+h(\xi_t, \varphi_t ^{\eta})$ then
\begin{equation}\label{nrot}
 \lim_{t \rightarrow \infty} \frac{\psi ^{\eta}_t}{t}=
\lim_{ t \rightarrow \infty} \frac{\varphi^{\eta}_t+h(\xi_t,\varphi^{\eta}_t)}{t}=
\lim_{t \rightarrow \infty} \frac{ \varphi ^{\eta}_t}{t} 
\end{equation}
when both limits exist. Hence $H$ preserves the rotation number.\par
 Note that $H$ is a bijective map and setting ${\displaystyle r(\xi, \psi)
=\frac{1}{p(\xi,\varphi(\xi,\psi))}}$ then
\[
\begin{array}{rcl}
H^{-1}: \Sigma& \longrightarrow & \Sigma \\
 (\xi,\psi)& \longrightarrow &\left\{ \begin{array}{ll}
(\xi, \int^{\psi}_0 r(\xi, s)ds)\: & \mbox{if} \xi \in \Omega_0 \\
(\xi, \psi)  & \mbox{if} \xi \in \Omega-\Omega_0 
\end{array}
\right.  
\end{array}
\]
defines its inverse.
We search  for the transformation of the flow under $H$.
\begin{prop}\label{nvar}
If $(\xi,\varphi) \in \Omega_0 \times P^1$ and $(\xi_t,\psi_{t}^{\eta}
(\xi, \psi))= H(\xi_t,\varphi_t^{\eta} (\xi,\varphi))$ then
\begin{equation}\label{fvar}
 \dot{\psi}_t= p(\xi_t,0)+[f_{\eta}(\xi_t,\varphi_t^{\eta})-
f_{0}(\xi_t,\varphi_t^{\eta})]\;p(\xi_t,\varphi_t^{\eta})=f'_{\eta}(\xi_t,\psi_t^{\eta}) \:.
\end{equation}
\end{prop}
{\em Proof:} First we consider the case $\eta=0$. 
The symbol $\varphi^{\star}_t(\xi)$ stands by the unique solution on $L$ of the implicit 
equation $\varphi_t(\xi,\varphi^{\star}_t(\xi))=0$. 
Taking a derivative with respect to $t$, we obtain
\[ \frac{d}{dt}\varphi_t(\xi,\varphi^{\star}_t(\xi))= f_{0}(
\xi_t,0)+ \frac{\partial \varphi_t}{\partial \varphi}(\xi,\varphi^{\star}_t(\xi))
\frac{d\varphi^{\star}_t}{dt}=0  \]
which yields
\[ \frac{d \varphi^{\star}_t(\xi)}{dt}=-f_{0}(\xi_t,0)e^{-\int
^t_0 \frac{\partial \varphi}{\partial \varphi}(\Phi_s(\xi,\varphi^{\star}_t(\xi))
ds}\:. \]
\indent On the other hand
\[ \psi_t(\xi, \psi)= \int^{\varphi_t(\xi,\varphi)}_0
p(\xi_t,s)ds=\int^{\varphi}_{\varphi_t^{\star}(\xi)} p(\xi,s)ds\:. \]
\indent Since $p(\xi,\varphi) \in C^{\infty}(P^1)$ then the above transformation
admits the derivative with respect to $t$ and
\[ \frac{d \psi_t(\xi,{\bf \varphi)}}{dt}=f_{0}(\xi_t,0)
p(\xi_t,\varphi^{\star}_t(\xi)) e^{-\int^t_0 \frac{\partial \varphi}
{\partial \varphi} (\Phi_s(\xi,\varphi^{\star}_t(\xi))ds} =p(\xi_t,0)\:. \]
\indent The case $\eta \neq 0$ can be verified in an analogous way. 
Let $\varphi^{\star}_t(\xi,\varphi)$ be the unique solution on $L$ of the
equation $\varphi_t(\xi, \varphi^{\star}_t(\xi,\varphi))=
\varphi^{\eta}_t(\xi,\varphi)$. Differentiating with respect to $t$ we obtain
\[ \frac{d\varphi_t}{dt}(\xi,\varphi^{\star}_t(\xi,\varphi))=
f_{0}(\xi,\varphi^{\eta}_t(\xi,\varphi))+\frac{\partial{\varphi_t}}
{\partial{\varphi}}(\xi,\varphi^{\star}_t(\xi))\frac{d \varphi^{\star}_t}
{dt}= f_{\eta}(\xi_t,\varphi^{\eta}_t(\xi,\varphi))  \]
which leads us to
\[ \frac{d\varphi^{\star}_t}{dt}=\left[ f_{\eta}(\xi_t,\varphi^{\eta}_t(\xi,\varphi))
-f_{0}(\xi_t,\varphi^{\eta}_t(\xi,\varphi)) \right] e^{-\int^t_0 
\frac{\partial \varphi}{\partial \varphi} (\Phi_s(\xi,\varphi^{\star}_t
(\xi))ds}\:. \]
\indent Now we can write
\[ \psi^{\eta}_t(\xi,\psi)=\psi^{\eta}_t(\xi,\psi)
-\psi_t(\xi,\psi)+\psi_t(\xi,\psi) \]
where the last term has already been considered. Then
\[ \psi^{\eta}_t(\xi,\psi)-\psi_t(\xi,\psi)=
\int^{ \varphi^{\eta}_t(\xi,\varphi)}_{ \varphi_t(\xi,\varphi)}
p(\xi_t,s)ds=\int^{\varphi^{\star}_t(\xi,\varphi)}_{\varphi} p(\xi,s)ds\:. \]
\indent Taking the derivative with respect to $t$, we get
\[ \frac{d}{dt} \left[ \psi^{\eta}_t(\xi,\psi)-\psi_t
(\xi,\psi) \right]= p(\xi_t,\varphi) \frac{d \varphi^{\star}_t}{dt}= \]
\[ \left[  f_{\eta}(\xi_t,\varphi^{\eta}_t)-f_{0}(\xi_t,\varphi^{\eta}_t
(\xi,\varphi)) \right] p(\xi_t,\varphi^{\eta}_t(\xi,\varphi)) \]
which yields
\[ \frac{d}{dt}\psi^{\eta}_t(\xi, \psi)=p(\xi_t,0)+
\left[ f_{\eta}(\xi_t,\varphi^{\eta}_t)-f_{0}(\xi_t,\varphi^{\eta}_t
(\xi,\varphi)) \right] p(\xi_t,\varphi^{\eta}_t)\:. \]
This completes the proof.\\[.4 cm]
\indent The following statement shows the Lipschitz variation of $\alpha$ 
throught $\Gamma_0$.
\begin{prop} 
Let  $C_{\Gamma_0}=\min\{ \|\kappa_{p}\|_1 /\:p\in B'\}$. Then
\begin{equation}\label{lips}
 \mid \alpha(\Gamma)-\alpha(\Gamma_0) \mid \leq C_{\Gamma_0}\| \Gamma-\Gamma_0 \|_{\infty} 
\end{equation}
for every $\Gamma \in C(\Omega)$.
\end{prop}
{\em Proof:} Let $p\in B'$. Using the proposition (\ref{nvar}) we obtain
\begin{equation}\label{roin}
 \alpha(\Gamma_0)= \lim_{t \rightarrow \infty}\frac{\psi_t
(\xi, \psi)}{t}=\lim_{t \rightarrow \infty}
\frac{1}{t}\int^t_0 p(\xi_s,0)ds=\int_{\Omega}p(\xi,0)dm_0  \:.
\end{equation}
\indent Similarly, we find that
\[ \psi^{\eta}_t(\xi,\psi)-
\psi= \int^t_0 \{p(\xi_s,0)+ [f_{\eta}(\xi_s,
\varphi^{\eta}_s)-f_{0}(\xi_s,\varphi^{\eta}_s)]p(\xi_s,\varphi^{\eta}_s) \} ds \]
and in consequence we get
\begin{equation}\label{vain}
 \alpha(\Gamma)-\alpha(\Gamma_0)=  \int_{\Sigma}[f_{1}(\xi,\varphi)-
f_{0}(\xi,\varphi)] p(\xi,\varphi) d\mu_{1} 
\end{equation}
for every normalized invariant measure under $\Phi^{1}$. Then one has
\[ \mid \alpha(\Gamma)-\alpha(\Gamma_0)\mid \leq \int_{\Sigma}
\mid \Gamma(\xi)-\Gamma_0(\xi) \mid
p(\xi,\varphi)d\mu \leq \| \Gamma-\Gamma_0 \|
\int_{\Sigma}\kappa_{p}(\xi)dm_0 \]
and hence 
\[ \mid \alpha(\Gamma)-\alpha(\Gamma_0)\mid \leq C_{\Gamma_0}\| \Gamma-
\Gamma_0\|_{\infty}\:. \]
\\[.4 cm]
\indent Let  $\Gamma \in C(\Omega)$ and assume that it exists $p\in B'$ 
such that $\int_{\Sigma} p(\xi,\varphi)\Gamma(\xi)d\mu$ takes the same value for 
every normalized invariant measure. This condition avoids considering  in
(\ref{vain}) the variation of the invariant measures $(\mu_{\eta})$ with respect 
to $\eta$. The theorem (\ref{fina}) assures the differentiability of $\alpha$
at $\Gamma_0$ in the direction  $\Gamma$.\par
That above property suggests the definition of  $C_{ad}(\Omega)$. There are functions
 which satisfy it but do not belong to  $C_{ad}(\Omega)$. For instance
 if $(\Sigma, \Phi)$ 
is a unique ergodic flow every continuous function satisfy that property but  
 $C_{ad}(\Omega)$ is a dense subset of  $C(\Omega)$. However the definition
(\ref{admi}) propose a practical verification of that condition. 
\begin{theo}\label{fina}
Let $\Gamma \in C_{ad}(\Omega)$. The rotation number has a derivative
at $\Gamma_0$ in the direction $\Gamma$ and
\begin{equation}\label{deri}
 d\alpha(\Gamma_0,\Gamma)=\lambda^2_{\Gamma}\int_{\Sigma}{\tilde{q}}^2
_{1,\Gamma} (\xi,\varphi) \Gamma(\xi) \sin^2 \varphi dm= \lambda_{\Gamma} \:.
\end{equation}
\end{theo}
{\em Proof:} We assume that $\mid \Gamma \mid_{\infty}\leq 1$.
 The variation of $\alpha$ in the direction $\Gamma$ can be
analyzed in the differential equation
\[\dot{\varphi}=\cos^2\varphi-(\Gamma_0(\xi_t)+\eta \Gamma (\xi_t))
 \sin^2\varphi\:. \]
We denote $I_{\eta}=\int_{\Sigma} {\tilde{q}}
_{1,\Gamma} (\xi,\varphi) \Gamma(\xi) \sin^2 \varphi d \mu_{\eta}$. Then
\[ \frac{\alpha (\Gamma_0 +\eta \Gamma)-\alpha(\Gamma_0)}{\eta}
=\lambda_{\Gamma}I_{\eta}\:.\]
\indent  There exists a function $\kappa \in L^1(\Omega,m_0)$ satisfying 
${\tilde{q}}_{1,\Gamma}(\xi,\varphi) \leq \kappa (\xi)$ for every $(\xi,
\varphi) \in \Omega_0 \times P^1$. 
For all $\epsilon \in [0,1]$ there is $\delta_{\epsilon} >0$ such that
if $D$ is a measurable subset of $\Omega$ with $m_0(D)<\delta_{\epsilon}$
then  $\int_{P^1}\kappa(\xi)dm_0<\epsilon/4\:.$\par
We set $D_T=\{ \xi \in \Omega \:/\: \kappa (\xi) < T-1\}$  and we select
a constant $T>0$ with $m_0(D_T)>1-\delta_{\epsilon}$.
By virtue of Egoroff theorem we can aproximate $\kappa_i(\xi)$ by continuous
functions in a measurable compact set $C_0 \subset \Omega$ with
 $C_0\subset D_T$ and $m_0(C_0)>1-\delta_{\epsilon}$ for $i=1,2,3$. Let 
$C=C_0\times P^1$. Repeating the combination
 (\ref{repr}) we find a function $h(\xi,\varphi)$ continuous on $\Sigma$ such
that $\mid h(\xi,\varphi)-{\tilde{q}}_{1,\Gamma}(\xi,\varphi) \mid <
\epsilon/4$ at every $(\xi,\varphi) \in C$. Thus we have 
\[I_{\eta}=\int_{\Sigma-C}{\tilde{q}}_{1,\Gamma}(\xi,\varphi)\Gamma(\xi)
\sin^2 \varphi d \mu_{\eta}+\int_C{\tilde{q}}_{1,\Gamma}(\xi,\varphi) \Gamma(\xi)
\sin^2 \varphi d \mu_{\eta} \leq \]
\[ \frac{\epsilon}{2} +\int_C h(\xi,\varphi) \Gamma(\xi)
\sin^2 \varphi d \mu_{\eta}\:. \]
\indent Moreover $h(\xi,\varphi)<T$ for every $(\xi,\varphi)\in C$. Now we consider
an open set $U_0$ such that $C_0 \subset U_0$, $h(\xi,\varphi)<T$ for every 
$(\xi,\varphi) \in U_0\times P^1$ 
and $m_0(U_0-C_0)<\epsilon/4T$. Let $U=U_0\times P^1$. If
$\varrho$ is a continuous function
on $\Sigma$ with $\chi_{C}\leq \varrho\leq \chi_{U}$ then
\begin{equation}\label{inpa}
I_{\mu} \leq \frac{\epsilon}{2}+\int_{\Sigma}h(\xi,\varphi) \Gamma(\xi)
\varrho(\xi,\varphi) \sin^2 \varphi d \mu_{\eta} \:.
\end{equation}
\indent We take a sequence $(\eta_n)^{\infty}_{n=1}$ which tends to $0$. We assume that 
$(\mu_{\eta_n})$ converges in the weak topology to $\mu_0$. This limit is a normalized
invariant measure under $\Phi$. In this way we derive
\[ \lim_{n \rightarrow \infty} \sup I_{\mu_n} \leq \frac{\epsilon}{2}\:+
\int_{\Sigma} h(\xi,\varphi) \Gamma(\xi)\varrho(\xi,\varphi)
\sin^2 \varphi d \mu_0 \leq  \]
\[ \frac{\epsilon}{2}+\int_C h(\xi,\varphi) \Gamma(\xi)\varrho(\xi,\varphi)
\sin^2 \varphi d \mu_0 +\int_{U-C}h(\xi,\varphi) \Gamma(\xi)\varrho(\xi,\varphi)
\sin^2 \varphi d \mu_0 \leq\]
\[ \frac{3 \epsilon}{4}+\int_C h(\xi,\varphi) \Gamma(\xi)
\sin^2 \varphi d \mu_0 \leq \epsilon+ \int_{\Sigma}{\tilde{q}}_{1,\Gamma}
(\xi,\varphi) \Gamma(\xi) \sin^2 \varphi d \mu_0 \:. \]
\indent Similarly we can show that
\[ \lim_{n \rightarrow \infty} \inf I_{\mu_n} \geq -\epsilon +
\int_{\Sigma}{\tilde{q}}_{1,\Gamma}
(\xi,\varphi) \Gamma(\xi) \sin^2 \varphi d \mu_0 \:.\]
\indent But $\epsilon$ is an arbitrary constant and $\int_{\Sigma} {\tilde{q}}_{1,
\Gamma}(\xi,\varphi) \Gamma(\xi) \sin^2 \varphi d \mu_0$ is independent on
 the invariant measure $\mu_0$ we use. We  conclude that
\[ \lim_{\mu \rightarrow 0} I_{\mu}=\lambda^2_{\Gamma}
\int_{\Sigma}{\tilde{q}}_{1,\Gamma}
(\xi,\varphi) \Gamma(\xi) \sin^2 \varphi d \mu =\lambda_{\Gamma}\:.\]
\indent Note that (\ref{form}) gives an alternative representation of 
$\lambda_{\Gamma}$ involving  a basis   of solutions of the Schr\"{o}dinger 
equation.

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\end{thebibliography}

\vspace{.4cm}
\indent OBAYA: DEPARTAMENTO DE MATEMATICA APLICADA A LA INGENIERIA. 
E.T.S. DE INGENIEROS INDUSTRIALES. UNIVERSIDAD DE VALLADOLID.
47011 VALLADOLID. SPAIN\\

\vspace{.2cm}
\indent PARAMIO: DEPARTAMENTO DE MATEMATICAS. FACULTAD DE VETERINARIA.
UNIVERSIDAD DE LEON. 24071 LEON. SPAIN\\ 

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