Content-Type: multipart/mixed; boundary="-------------0908071014531"

This is a multi-part message in MIME format.

---------------0908071014531
Content-Type: text/plain; name="09-137.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="09-137.keywords"

Reducibility, skew-product flows, cocycles, renormalization 
---------------0908071014531
Content-Type: application/x-tex; name="kocic_rspf_mparc_version.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="kocic_rspf_mparc_version.tex"

\documentclass[12pt,twoside,a4paper]{article}

\usepackage[reqno,sumlimits]{amsmath}
\usepackage{latexsym}
\usepackage{graphics}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{theorem}
\usepackage{epic,eepic}
\usepackage{epsfig}
\usepackage{amscd}
\usepackage[T1]{fontenc}
\font\teneufm=eufm10

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\setlength{\oddsidemargin}{-0.04cm}
\setlength{\evensidemargin}{-0.04cm}
\setlength{\textwidth}{16.0cm}
\setlength{\topmargin}{-1cm}
\setlength{\headheight}{1.0cm}
\setlength{\headsep}{0.5cm}
\setlength{\textheight}{21.0cm}
\setlength{\footskip}{1.5cm}
\setlength{\parskip}{2pt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\renewcommand{\theequation}{\thesection.\arabic{equation}}
\numberwithin{equation}{section}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{estimate}[theorem]{Estimate}
{\theoremstyle{plain} {\theorembodyfont{\normalfont\rmfamily}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}{Exercise}[section]
}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\cqd}{\hfill $\Box$\medskip}
\newcommand{\proof}{\noindent{\bf Proof. }}
\newcommand{\proofoftheorem}{\noindent{\bf Proof of Theorem 1.4. }}
\newcommand{\abs}[1]{\left| #1\right|}
\newcommand{\norm}[1]{\left\| #1\right\|}
\newcommand{\pde}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\spec}[1]{{\rm spec}\left( #1\right)}
\newcommand{\id} {\mbox{\rm Id}}
\newcommand{\im} {\mbox{\rm Im\,}}
\newcommand{\re} {\mbox{\rm Re\,}}
\newcommand{\tr} {\mbox{\rm tr\,}}
\newcommand{\lie}{\mbox{\rm Lie}}
\newcommand{\up}{\uparrow}
\newcommand{\down}{\downarrow}
\newcommand{\sgn}{\text{sgn}}

\newcommand{\Aa}{{\mathbb{A}}}
\newcommand{\Bb}{{\mathbb{B}}}
\newcommand{\Cc}{{\mathbb{C}}}
\newcommand{\Dd}{{\mathbb{D}}}
\newcommand{\Ee}{{\mathbb{E}}}
\newcommand{\Ff}{{\mathbb{F}}}
\newcommand{\Gg}{{\mathbb{G}}}
\newcommand{\Hh}{{\mathbb{H}}}
\newcommand{\Ii}{{\mathbb{I}}}
\newcommand{\Jj}{{\mathbb{J}}}
\newcommand{\Kk}{{\mathbb{K}}}
\newcommand{\Ll}{{\mathbb{L}}}
\newcommand{\Mm}{{\mathbb{M}}}
\newcommand{\Nn}{{\mathbb{N}}}
\newcommand{\Oo}{{\mathbb{O}}}
\newcommand{\Pp}{{\mathbb{P}}}
\newcommand{\Qq}{{\mathbb{Q}}}
\newcommand{\Rr}{{\mathbb{R}}}
\newcommand{\Ss}{{\mathbb{S}}}
\newcommand{\Tt}{{\mathbb{T}}}
\newcommand{\Uu}{{\mathbb{U}}}
\newcommand{\Vv}{{\mathbb{V}}}
\newcommand{\Ww}{{\mathbb{W}}}
\newcommand{\Xx}{{\mathbb{X}}}
\newcommand{\Yy}{{\mathbb{Y}}}
\newcommand{\Zz}{{\mathbb{Z}}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\AA{{\mathcal A}}
\def\BB{{\mathcal B}}
\def\CC{{\mathcal C}}
\def\DD{{\mathcal D}}
\def\EE{{\mathcal E}}
\def\FF{{\mathcal F}}
\def\GG{{\mathcal G}}
\def\HH{{\mathcal H}}
\def\II{{\mathcal I}}
\def\JJ{{\mathcal J}}
\def\KK{{\mathcal K}}
\def\LL{{\mathcal L}}
\def\MM{{\mathcal M}}
\def\NN{{\mathcal N}}
\def\OO{{\mathcal O}}
\def\PP{{\mathcal P}}
\def\pp{{\mathfrak p}}
\def\QQ{{\mathcal Q}}
\def\RR{{\mathcal R}}
\def\SS{{\mathcal S}}
\def\ss{{\mathfrak S}}
\def\TT{{\mathcal T}}
\def\UU{{\mathcal U}}
\def\VV{{\mathcal V}}
\def\WW{{\mathcal W}}
\def\XX{{\mathcal X}}
\def\YY{{\mathcal Y}}
\def\ZZ{{\mathcal Z}}
\def\zz{{\mathfrak z}}
\def\hh{{\mathcal h}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\aa{{\mathfrak a}}
\def\bb{{\mathfrak b}}
\def\cc{{\mathfrak c}}
\def\dd{{\mathfrak d}}
\def\eee{{\mathfrak e}}
\def\ff{{\mathfrak f}}
\def\gg{{\mathfrak g}}
\def\hh{{\mathfrak h}}
\def\ii{{\mathfrak i}}
\def\jj{{\mathfrak j}}
\def\kk{{\mathfrak k}}
\def\ll{{\mathfrak l}}
\def\mm{{\mathfrak m}}
\def\nn{{\mathfrak n}}
\def\oo{{\mathfrak o}}
\def\pp{{\mathfrak p}}
\def\qq{{\mathfrak q}}
\def\rr{{\mathfrak r}}
\def\ss{{\mathfrak S}}
\def\tt{{\mathfrak t}}
\def\uu{{\mathfrak u}}
\def\vv{{\mathfrak v}}
\def\ww{{\mathfrak w}}
\def\xx{{\mathfrak x}}
\def\yy{{\mathfrak y}}
\def\zz{{\mathfrak z}}
\def\sl{\mathfrak{sl}}
%\def\ad{\mathfrak{ad}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\fD}{{\mathfrak D}}
\newcommand{\fF}{{\mathfrak F}}
\newcommand{\fN}{{\mathfrak N}}
\newcommand{\fR}{{\mathfrak R}}
\newcommand{\qed}{\hfill\mbox{\bf\scriptsize QED}\par\medskip}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\bfR{{\hbox{\teneufm R}}}
\def\RG{\Re}
\def\mean{{\Bbb E}}
\def\Id{{\Bbb I}}
\def\proj{{\Bbb P}}
\def\natural{{\Bbb N}}
\def\integer{{\Bbb Z}}
\def\rational{{\Bbb Q}}
\def\real{{\Bbb R}}
\def\complex{{\Bbb C}}
\def\torus{{\Bbb T}}
\def\iso{{\Bbb J}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\def\ident{{\mathrm{Id}}}
\def\Id{{\Bbb I}}
\def\id{\mathrm{I}}
\def\inminus{I_n^{^-}}
\def\inplus{I_n^{^+}}
\def\iminus{I^{^-}}
\def\iplus{I^{^+}}
\def\iplusminus{I^{^\pm}}
\def\Lminus{L^{^-}}
\def\Lplus{L^{^+}}
\def\Inminus{{\Bbb I_n^{^-}}}
\def\Innminus{{\Bbb I_{n+1}^{^-}}}
\def\Inplus{{\Bbb I_n^{^+}}}
\def\In1plus{{\Bbb I_{n+1}^{^+}}}
\def\I0plus{{\Bbb I_{0}^{^+}}}
\def\Inplusminus{{\Bbb I_n^{^\pm}}}
\def\Lplusminus{L^{^\pm}}
\def\Iminus{{\Bbb I}^{^-}}
\def\Iplus{{\Bbb I}^{^+}}
\def\Iplusminus{{\Bbb I}^{^\pm}}
\newcommand {\sss}{\scriptscriptstyle}
\newcommand {\wt}{\widetilde}
\newcommand {\wh}{\widehat}
\def\eps{{\varepsilon}}
\def\ssH{{\scriptscriptstyle H}}
\def\ssHp{{\scriptscriptstyle H'}}
\def\ssHpp{{\scriptscriptstyle H''}}
\def\qqqqqqquad{\quad\quad\quad\quad\quad\quad\quad}
\def\bfR{{\hbox{\teneufm R}}}

\def\ssf{{\sss f}}
\def\ssg{{\sss g}}
\def\ssF{{\!\sss F}}
\def\ssH{{\!\sss H}}
\def\ssHp{{\!\sss H'}}
\def\ssK{{\!\sss K}}
\def\ssO{{\sss 0}}
\def\ssR{{\sss R}}
\def\ssX{{\sss X}}
\def\ssXp{{\sss X'}}
\def\ssY{{\sss Y}}
\def\ssZ{{\sss Z}}

\def\DC{{\rm DC}}
\def\BC{{\rm BC}}
\def\GL{{\rm GL}}
\def\SL{{\rm SL}}
\def\SO{{\rm SO}}
\def\SU{{\rm SU}}
\def\G{{\rm G}}
\def\F{{\rm F}}
\def\ad{{\rm ad}}

\def\spa{\parallel}
\def\spe{\perp}

\newcommand\defeq{\stackrel{\mathrm{def}}{=}}
\def\equ(#1){~\eqref{#1}}
\def\clm(#1){\csname c#1\endcsname}
\def\tag(#1){(\number\count5.
          \number\count6)
    \expandafter\xdef\csname e#1\endcsname{
    (\number\count5.\number\count6)}
    \global\advance\count6 by 1}
%\def\equation(#1){\eqno\tag(#1)}
\def\claim #1(#2) #3\par{
    \vskip.1in\medbreak\noindent
    {\bf #1\ \number\count5.\number\count7.\ }{\sl #3}\par
    \expandafter\xdef\csname c#2\endcsname{#1~\number\count5.\number\count7}
    \global\advance\count7 by 1
    \ifdim\lastskip<\medskipamount
    \removelastskip\penalty55\medskip\fi}
\def\claim #1(#2) #3\par{
    \vskip.1in\medbreak\noindent
    {\bf #1\ [section].\number\count7.\ }{\sl #3}\par
    \expandafter\xdef\csname c#2\endcsname{#1~\number\count5.\number\count7}
    \global\advance\count7 by 1
    \ifdim\lastskip<\medskipamount
    \removelastskip\penalty55\medskip\fi}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand {\be}{\begin{equation}}
\newcommand {\bel}[1]{\begin{equation} \label{#1}}
\newcommand {\ee}{\end{equation}}
\newcommand {\bea}{\begin{eqnarray}}
\newcommand {\beal}[1]{\begin{eqnarray} \label{#1}}
\newcommand {\eea}{\end{eqnarray}}
\newcommand {\ch}{\bf}
\newcommand{\subjclassname}{\textup{2000} Mathematics Subject Classification}
\def\clmt(#1){Theorem~\ref{#1}}
\def\clml(#1){Lemma~\ref{#1}}
\def\clmp(#1){Proposition~\ref{#1}}
\def\clmd(#1){Definition~\ref{#1}}
\def\clmr(#1){Remark~\ref{#1}}
\def\clma(#1){Assumption~\ref{#1}}
\def\clme(#1){Example~\ref{#1}}
\def\clmc(#1){Corollary~\ref{#1}}

\renewcommand{\labelenumi}{(\roman{enumi})}
\newcommand{\mnorm}[1]{%
  \left\vert\kern-0.9pt\left\vert\kern-0.9pt\left\vert #1
    \right\vert\kern-0.9pt\right\vert\kern-0.9pt\right\vert}
\newcommand{\mat}[2]{%
\begin{pmatrix}#1 \\
          #2
\end{pmatrix}}

\begin{document}

\title{Reducibility of skew-product systems with multidimensional Brjuno base flows}
\author{Sa\v sa \ Koci\' c \footnote{Current Affiliation: Department of Mathematics, University of Toronto, 40 St.~George Street,
Toronto, Ontario, Canada M5S 2E4} \footnote{Email: kocic@math.utexas.edu}
\\{\small IMPA, Estrada Dona Castorina 110,
Rio de Janeiro, Brasil 22460-320}}

\pagestyle{myheadings}\markboth{Sa\v sa Koci\' c}{Reducibility of
Brjuno skew-product flows}

\date{August 16, 2007; revised May 18, 2009.}
\maketitle


\begin{abstract}
We develop a renormalization method that applies to the problem of
the local reducibility of analytic skew-product flows on $\torus^d\times
\SL(2,\real)$. We apply the method to give a proof of a reducibility
theorem for these flows with Brjuno base frequency vectors.\end{abstract}


\section{Introduction}

We consider real analytic vector fields on $\MM=\Tt^d\times {\G}$,
where $\Tt=\Rr/(2\pi\Zz)$, and ${\G}\subset \GL(n,\Rr)$ is a Lie group with the Lie algebra $\gg$. Vector fields of
the form,
    \bel{skewproduct}
    X(x,y)=(\omega,f(x)y)
    \ee
where $\omega\in\Rr^d$ and $f:\Tt^d\to\gg$, generate the
following system of differential equations
    \bel{diffeq}
\frac{dx}{dt}=\omega\,,\qquad \frac{dy}{dt}=f(x)y,
    \ee
where $(x,y)\in\MM$.

We can assume, without restriction, that $\omega$ is incommensurate
with respect to integers, i.e.  $\omega\cdot\nu\ne 0$ for all
$\nu\in\Zz^d\backslash\{0\}$. The linear flow on the base $\Tt^d$,
given by $x\mapsto x+\omega t$, for $t\ge 0$, is in this case quasiperiodic with
frequency vector $\omega$.

The flow of~\eqref{diffeq}, referred to as the skew-product flow, for every $t\in\Rr$, is the mapping on $\MM$,
    \bel{cosflow}
(x,y)\mapsto (x+\omega t,\Phi^t(x)y),
    \ee
where $\Phi^t(x)$ is the matrix solution of~\eqref{diffeq} that satisfies $\Phi^0(x)=\id$, with $\id$ the identity in $\G$.  $\Phi^t(x)$ satisfy the {\em cocycle property}
    \bel{cocycleidentity}
    \Phi^{t+s}(x)=\Phi^t(x+s\omega)\Phi^s(x),\qquad \forall t,s\in\Rr\,, \quad \forall x\in\Tt^d.
    \ee

On $\MM$, one can perform coordinate changes of the
form
    \bel{conjugacypsi}
    \psi(x,y)=(x,\Psi(x)^{-1}y),\qquad x\in\Tt^d\,, \quad y\in\G,
    \ee
with $\Psi:\Tt^d\to\G$, that preserve the class of vector
fields that generate skew-product flows. Under such a diffeomorphism $\psi$, a vector field $X$ of the
form~\eqref{skewproduct}, transforms as
$X\mapsto\psi^*X:=(D\psi)^{-1}X\circ\psi$, where the pullback of the
vector field $X$ under a coordinate change $\psi$ is given by
    \be
\psi^*X(x,y)=(\omega,\Psi^\star f(x)y)\,,\qquad
\Psi^\star  f=\Psi^{-1}[f-D_\omega]\Psi.
    \ee
Here $D_\omega$ is the directional derivative on $\Tt^d$ in the direction of $\omega$.

The problem of reducibility of the system~\eqref{diffeq} concerns the existence of a coordinate change $z=\Psi(x)^{-1}y$ that transforms the system into
    \bel{diffeq2}
\frac{dx}{dt}=\omega\,,\qquad \frac{dz}{dt}=uz,
    \ee
with a constant function $x\mapsto u\in\gg$.

\begin{definition} We say that $X$ is ($C^\omega$) {\em conjugated} to $Y$ if there exists a
($C^\omega$) coordinate change $\psi$ on $\MM$, such that $\psi^*
X=Y$. We call $X$ of the form~\eqref{skewproduct} {\em reducible}
if it is conjugated to $K$, where $K(x,y)=(\omega,uy)$, with some
constant element $u\in\gg$.
\end{definition}

The system of equations~\eqref{diffeq} is equivalent to a linear
system of ODE's with time dependent coefficients. In the case $d=1$,  the coefficients are periodic in time and
the results of Floquet and Lyapunov show that the system is always reducible.

We are interested in the case $d>1$ and the following reducibility problem: given a constant matrix $u\in\gg$ and the associated system in constant coefficients~\eqref{diffeq2}, we are interested in the reducibility of systems ``nearby'' to the one with constant coefficients. This is an old and well-studied problem in dynamics. The problem typically involves small divisors, and further conditions are needed in order to achieve reducibility.

Classical results~\cite{Bogoljubov,DS} and the later extensions~\cite{Eliasson,Katok,Krikorian1,Krikorian2,PS} have already shown that this is possible in the case of  Diophantine frequency vectors $\omega$, i.e. those for which there exist a constant $\kappa>0$ and $\beta\ge 0$, such that
    \bel{DiophantineCond}
 |\omega\cdot\nu|\ge \kappa\|\nu\|^{-(d-1+\beta)}\;,\qquad\forall \nu\in\Zz^d\backslash\{0\}.
    \ee

In the case of ${\G}=\SL(2,\Rr)$, motivated by the study of one-dimensional Schr\" odinger equation with
a quasi-periodic potential, reducibility results have been obtained in the perturbative regime by KAM methods by Dinaburg and Sinai and Eliasson~\cite{DS,Eliasson}, following the seminal ideas of Moser~\cite{Moser}. Eliasson has obtained, in the continuous-time case, the reducibility of analytic skew-product systems, close to a constant coefficient system of the form~\eqref{diffeq2}, in the case of Diophantine $\omega$ and fibered rotation number (associated to eigenvalues of $u$) either Diophantine or rational with respect to $\omega$. The closeness condition does not depend on the fibered rotation number. A discrete-time version of this result was obtained in~\cite{Amor}.  In the case of compact groups ${\G}$, local reducibility results have been obtained in~\cite{Krikorian1,Krikorian2}.

A non-perturbative version of the Eliasson result~\cite{Eliasson} (the closeness to a constant does not depend on the frequency) in the single frequency case can be found in~\cite{AJ2}. In the multifrequency case, it was shown by Bourgain that the results fail to extend to the non-perturbative regime~\cite{Bourgain}.

Another approach to the reducibility problems is based on renormalization methods. For discrete-time cocycles on ${\G}$, i.e. pairs $(\alpha,A):(x,y)\mapsto (x+\alpha, A(x)y)$ on $\Tt^d\times {\G}$, over rotations by an irrational $\alpha\in\Tt^1$, and for ${\G}=\SU(2)$, a renormalization method was introduced by Rychlik~\cite{Rychlik}. The method was later improved by Krikorian who obtained several global results on reducibility of cocycles~\cite{Krikorian3,Krikorian4}. Similar methods have been applied in~\cite{HouYou} to prove that, for almost all $\alpha$, a measurable conjugacy of an analytic quasi-periodic cocycle on $\SO(n,\Rr)$ to a constant, implies analytical reducibility. Avila and Krikorian have proved a global reducibility result for analytic cocycles on $\SL(2,\Rr)$~\cite{AK}: for almost every $\alpha$, if such a cocycle is $L^2$-conjugated to some constant cocycle in $\SO(2,\Rr)$, then it is analytically reducible to it.

Different renormalization methods have been applied to the problem of reducibility of skew-product flows on $\Tt^d\times\SL(2,\Rr)$ in~\cite{KLD,LopesDias06}. These methods are similar to the ones previously used in the context of invariant tori for Hamiltonian flows~\cite{Kocic} and more general vector fields on $\Tt^d\times\Rr^m$~\cite{KK1}. For skew-product flows, the $d=2$, analytic, Brjuno case was considered by Lopes Dias~\cite{LopesDias06}, while Koch and Lopes Dias considered the Diophantine multi-frequency case~\cite{KLD}, in the $C^k$ setting, with $k$ finite and sufficiently large.  The former approach relies on the estimates coming from the one-dimensional continued fraction algorithm and is consequently restricted to $d=2$. In the latter, the results have been extended to $d>2$, in the Diophantine case, by using estimates from a multidimensional continued fraction algorithm~\cite{KLM}.

In this paper, we develop a renormalization method that applies to the problem of reducibility of analytic skew-product flows with multidimensional Brjuno base rotation. It extends the results of~\cite{LopesDias06} to dimensions $d>2$, and the results of~\cite{KLD} (in the analytic case) to Brjuno frequency vectors.

\begin{definition} A vector $\omega\in\Rr^d$ is {\em Brjuno} if the
following condition is satisfied~\cite{Brjuno1,Brjuno2}
    \bel{BrjunoCond}
\sum_{n=1}^\infty 2^{-n}\ln(1/\Omega_n)<\infty\,,\qquad
\Omega_n:=\min_{\nu\in\Zz^d,0<|\nu|\le 2^n}|\omega\cdot\nu|\,.
    \ee
\end{definition}
The set of Brjuno vectors is of full Lebesgue measure. In particular, it contains all Diophantine vectors, for all $\kappa>0$ and all $\beta\ge 0$. Conversely, there are vectors that are Brjuno and are not Diophantine for any $\kappa$ and $\beta$.

A condition less restrictive than Diophantine (that agrees with Brjuno in the $d=2$ case) has also been recently considered in~\cite{LopesDias08}, in the contexts of linearizability of torus flows. The multidimensional Brjuno $\omega$ case has also been recently considered in the skew-product case by Gentile~\cite{Gentile},
 using a different renormalization method based on resummation of
perturbation series as in quantum field theory (see e.g.
~\cite{Gallavotti,GG}). In fact, Gentile's results are valid under a Brjuno condition on the fibered rotation number with respect to $\omega$.

We will consider vector fields close to an integrable vector field
$K(x,y)=(\omega,uy)$, with $u\in \sl(2,\Rr)$ Diophantine with
respect to $\omega$. We will assume that $u$ has purely
imaginary eigenvalues.

\begin{definition}
A matrix $u\in \sl(2,\Rr)$ with eigenvalues $\pm i\rho$ is {\em
Diophantine with respect to} $\omega$ if there exists constants
$\CC,\tau>0$ such that
    \bel{Diophantinewrtomega}
|\omega\cdot\nu-2\rho|>\frac{\CC}{|\nu|^\tau}\,,\qquad\forall \nu\in\Zz^d\backslash\{0\}.
    \ee
\end{definition}

As a corollary of our main result (\clmt(main1) below), we will obtain the following.

\begin{theorem}\label{main2}
Let $\omega\in\Rr^d$ be a Brjuno vector and $K$ a vector field of
the form $K(x,y)=(\omega,uy)$, with $u\in \sl(2,\Rr)$ Diophantine
with respect to $\omega$. If $X$ is a real analytic vector field on
$\Tt^d\times \SL(2,\Rr)$ of the form~\eqref{skewproduct} with the base frequency vector $\omega$, $C^\omega$-sufficiently
close to $K$, then there exists $v\in \sl(2,\Rr)$ such that $X_v$, where
$X_v(x,y)=X(x,y)+(0,vy)$, for $(x,y)\in\Tt^d\times \SL(2,\Rr)$, is $C^\omega$-reducible to $K$. The map $X\mapsto v$ is analytic.
\end{theorem}

This theorem is often referred to as a normal form theorem~\cite{Krikorian2}. We will not use this terminology here, in order to avoid the confusion with a different theorem that is an essential part of the methods used here.
We note that this result can be used to improve the arithmetic condition on $\omega$ in the main result of~\cite{DS}. Following Chapter $4$ of~\cite{Krikorian2}, by looking at the dependence of $v$ on $u$, and using Whitney extension, one can show the existence of a positive measure set of parameters (energy) for which the one-dimensional quasi-periodic Schr\"{o}dinger equation is reducible.

The renormalization approach that we pursue here provides an alternative method to KAM theory to deal with
small divisors. By scaling the torus,
one turns some of them into ``large divisors'', that can be
eliminated by a coordinate change. Construction of such a coordinate change (see Section~\ref{NF}) is an important part of our renormalization method that differs it from renormalization methods used for discrete time cocycles (see e.g.~\cite{AK}). These are the basic steps of a
one step renormalization operator $\RR$, defined between spaces of vector
fields of the form~\eqref{skewproduct}. We will construct a sequence
of renormalization operators, with an integrable fixed set, and show
that there is a ``stable manifold'' associated to them. We will show
that the vector fields on the stable manifold are reducible. The
result of \clmt(main2) is obtained simply by showing that the family
of vector fields parameterized by $v$ intersects the stable
manifold. The proof is given at the end of Section~\ref{reducibility}.

The scaling of $\Tt^d$ is achieved by a sequence of non-integer matrices $T_n\in\GL(d,\Rr)$. This is in contrast to the earlier approaches where a sequence of scaling matrices in $\SL(d,\Zz)$ is generated from a continued fraction algorithm. The flow on a homogeneous space $\SL(d,\Zz)\backslash\SL(d,\Rr)$ can provide a way to construct a sequence of matrices $P_n\in\SL(d,\Zz)$ and the corresponding matrices $T_n=P_{n-1}^{-1}P_{n}$~\cite{KLM,Lagarias}. The flow is generated by the right action on a matrix $M_\omega$ associated to $\omega\in\Rr^d$, of a one-parameter subgroup $E^t$, with $t\ge 0$, of $\SL(d,\Rr)$, that has the property to expand the direction of $\omega$ and contract perpendicular directions. Given a sequence of stopping times $n\mapsto t_n$, $n\in\Nn$, one can generate a sequence of matrices $P_n\in\SL(d,\Zz)$, whose left action brings the resulting matrix  back to the fundamental domain for the left action of $\SL(d,\Zz)$ on $\SL(d,\Rr)$. In order for these matrices to be useful for renormalization, one needs to have a good control of the matrices $P_n$. In~\cite{KLM}, the appropriate bounds have been obtained for the case of Diophantine frequency vectors.
In the approach pursued here, applicable to the larger set of Brjuno frequency vectors, we perform the scaling of $\Tt^d$ using non-integer matrices, that expand the direction of $\omega$ and contract all perpendicular directions. The scaling produces a deformation of the Fourier lattice $\VV$ and we control the deformation produced by repeated scaling.

Let $\Id$ be the identity operator and $\Ee$ be the torus averaging projection operator on spaces of analytic functions from $\Tt^d$ into $\gg$ (see Section~\ref{spaces}). We will use the same symbols for the corresponding operators on spaces of vector fields on $\Tt^d\times \G$ of the form~\eqref{skewproduct}. In particular, $\Ee X(x,y)=(\omega,\Ee f (x) y)$.

\begin{theorem}\label{main1}
Let $\omega\in\Rr^d$ be a Brjuno vector and $K$ a vector
field of the form $K(x,y)=(\omega,uy)$, with $u\in \sl(2,\Rr)$
Diophantine with respect to $\omega$. There exists an open neighborhood $\DD_0$ of $K$ of real analytic vector fields on $\Tt^d\times\SL(2,\Rr)$, of the form~\eqref{skewproduct}, and an analytic function $W:(\Ii-\Ee)\DD_0\to\Ee \DD_0$, satisfying $W(0)=K$ and $DW(0)=0$.
Every vector field $X$ on the graph $\WW$ of $W$ is $C^\omega$ reducible to $K$ via
a conjugacy $\psi_X$ of the form~\eqref{conjugacypsi}. Moreover, the map $X\mapsto \psi_X$ is analytic.
\end{theorem}


This theorem follows from the results of Section \ref{sequence}
(\clmt(BRRnComposeT)) and Section \ref{reducibility}
(\clmt(AnalyticConjugacy)). It immediately implies \clmt(main2).

We remark that the frequency vector $\omega$ is fixed, and an open neighborhood of $K$ means $\{\omega\}\times B$, where $B$ is an open neighborhood of $u$ in the space of analytic functions from $\Tt^d$ into $\sl(2,\Rr)$. Our second remark is that our estimates are uniform within classes of Brjuno vectors $\omega$ and $u\in\sl(2,\Rr)$ Diophantine with respect to $\omega$, described at the end of Section~\ref{sequence1}.

The paper is organized as follows. In Section~\ref{spaces} we
introduce the spaces of analytic vector fields and maps that we
consider. In Section~\ref{single} we construct a one step
renormalization operator. In Section~\ref{sequence} we construct a
sequence of renormalization operators with a ``trivial'' limit set and
show that there exists a stable manifold $\WW$ of vector fields
associated to this set. Section~\ref{reducibility} deals with the
reducibility of vector fields on the stable manifold.
Section~\ref{NF} is independent, and contains a proof of a normal
form theorem for skew-product flows on $\Tt^d\times\G$, that has been used in the
construction of a single renormalization step.

\section{Preliminaries} \label{spaces}

\subsection{Spaces of vector fields with skew-product
flows}

On $\real^n$ and $\complex^n$ we use the following norms: $|v|=\sum_{j=1}^n|v_j|$ and $\|v\|=\max_j|v_j|$. Let $\varrho>0$. Denote by $D_\varrho$ the set of all points
$x\in\complex^d$ characterized by $\|\im x\|<\varrho$.

Let $\{e_1,\dots,e_d\}$ be a basis and $\ZZ=\{\sum_{i=1}^dz_ie_i|z_i\in\Zz\}$ be a lattice in $\Rr^d$. Let also $\VV$ be its dual
lattice, i.e. the set of points $v\in\Rr^d$ satisfying $\exp(iv\cdot
z)=1$ for all $z\in\ZZ$.

Let $\F^{n\times n}$ be the space of $n\times n$ matrices with entries in $\F$, where $\F$ is either $\Rr$ or $\Cc$.
We will consider functions defined from the $d$-dimensional torus $\Tt^d=\Rr^d/\ZZ$ into $\Rr^{n\times n}$, that can be regarded as maps from $\Rr^d$ into $\Rr^{n\times n}$, by lifting to the universal cover. We will make use of analyticity to extend them to the complex domain $D_\varrho$.  Let
$\FF_\varrho(\VV)$ be the Banach space of all
$\ZZ$-periodic analytic maps $f:D_\varrho\to\Cc^{n\times n}$, that can
be expanded in Fourier series as
    \bel{fExpansion}
f(x)=\sum_{v\in\VV}f_{v} e^{iv\cdot x} \,,
    \ee
where $v\cdot x=\sum_jv_jx_j$, and for which the following norm is
finite
    \bel{fnorm}
    \|f\|_\varrho=\sum_{v\in\VV}\|f_{v}\| e^{\varrho|v|}.
    \ee
Here and in what follows, for matrices $M$ with matrix elements $M_{ij}$, we use the operator norm corresponding to the $|\cdot|$ norm of vectors, that is the matrix norm $\|M\|=\max_{j}\sum_{i}|M_{ij}|$. The space $\FF_\varrho(\VV)$, with the norm $\|\cdot\|_\varrho$ is a Banach algebra.

The torus averaging projection operator $\Ee$ is defined on functions $f\in\FF_\varrho(\VV)$, with the expansion~\eqref{fExpansion}, as $\Ee f=f_0$. The identity operator acts as $\Ii f=f$.

We define $\FF_\varrho'(\VV)$ to be the subspace of all maps in $f\in\FF_\varrho(\VV)$ for which the norm
    \be
    \|f\|_\varrho'=\|D_\omega f\|_\varrho+\|f\|_\varrho
    \ee
is finite. $D_\omega f=\sum_{i=1}^d\omega_i\cdot\frac{\partial}{\partial x_i} f$ is the directional derivative of $f$ in the direction of $\omega\in\Rr^d$.

Let ${\G}\subset \GL(n,\Cc)$ be a Lie group and $\gg$ its corresponding Lie algebra.
The subspace of maps in $\FF_\varrho(\VV)$ that take values in $\gg$ is denoted by $\FF_\varrho(\gg,\VV)$.
Similarly, the subset of maps in $\FF_\varrho(\VV)$ that take values in $\G$ is denoted by $\FF_\varrho(\G,\VV)$.
Analogously, we can define $\FF_\varrho'(\gg,\VV)$ and $\FF_\varrho'(\G,\VV)$.

The space of vector fields on $D_\varrho\times\G$ that we consider, $\AA_\varrho(\VV)$, is an affine space of vector fields $X$, of the form $X(x,y)=(\omega,f(x)y)$, for $(x,y)\in D_\varrho\times\G$, where $f\in\FF_\varrho(\gg,\VV)$, with the norm $\|X\|_\varrho=\|\omega\|+\|f\|_\varrho$. The space $\AA_\varrho(\VV)$ corresponding to $\omega=0$ is a Banach spaces that will be denoted by $\AA_\varrho^0(\VV)$. The norm of $Z\in\AA_\varrho^0(\VV)$ with $Z(x,y)=(0,f(x)y)$ is thus $\|Z\|_\varrho=\|f\|_\varrho$. For these spaces, we will also use the notation $\AA_\varrho(\gg,\VV)$ and $\AA_\varrho^0(\gg,\VV)$, respectively, when we would like to emphasis which group $\G$ we consider. The subspace of vector fields $Z\in\AA_\varrho^0(\VV)$ for which the norm $\|Z\|_\varrho'=\|f\|_\varrho'$ is finite will be denoted by ${\AA_\varrho^0}'(\VV)$.

We define $\BB_\varrho(\VV)$ as the space of maps $\psi$ on
$D_\varrho\times {\G}$, of the form  $\psi(x,y)=(x,\Psi^{-1}(x)y)$, where
$\Psi\in\FF_\varrho(\G,\VV)$. The distance from the identity map $\ident\in\BB_\varrho(\VV)$ is measured by the norm $\|\psi-\ident\|_\varrho=\|\Psi^{-1}-\id\|_\varrho$. The subspace of maps $\psi\in\BB_\varrho(\VV)$ for which the norm $\|\psi-\ident\|_\varrho'=\|\Psi^{-1}-\id\|_\varrho'$ is finite, will be denoted by $\BB_\varrho'(\VV)$.

For the maps between Banach spaces, we will use the following notion of analyticity~\cite{HP}. Such a map will be
called analytic if it is Fr\' echet differentiable. Thus, sums, products and compositions of analytic maps are analytic. We also have that the limit
of a uniformly convergent sequence of such analytic maps is analytic.

A map $X\mapsto \Psi$, from $\{\omega\}\times B\subset \AA_\varrho(\VV)$ into $\FF_\delta(\VV)$, will be called analytic if the map $f\mapsto \Psi$, from $B\subset\FF_\varrho(\gg,\VV)$ into $\FF_\delta(\VV)$, is analytic. The map $X\mapsto\psi$ of the form $\psi(x,y)=(x,\Psi^{-1}(x)y)$, from $\{\omega\}\times B\subset \AA_\varrho(\VV)$ into $\BB_\delta(\VV)$, will be called analytic if the map $X\mapsto \Psi$ is analytic.

In the following, when the lattice $\VV$ is fixed and no ambiguity
can arise, we will simplify the notation by writing $\AA_\varrho$
instead of $\AA_\varrho(\VV)$, and analogously for other spaces.

\subsection{Lie algebra $\sl(2,\Rr)$}

The focus of this paper will be on the Lie group ${\G}=\SL(2,\Rr)$ and its
Lie algebra $\gg=\sl(2,\Rr)$. The group ${\G}=\SL(2,\Rr)$ is a
non-compact Lie group of real $2\times 2$ matrices with determinant
$1$. The corresponding Lie algebra $\gg=\sl(2,\Rr)$ consists of
$2\times 2$ real traceless matrices. The Lie bracket is given by the usual matrix commutator. A basis in $\sl(2,\Rr)$ is
given by $\{H,E_+,E_-\}$, where
    \be
H=\left[ \begin{array}{cc}1&0\\
0&-1\\\end{array}\right],\qquad E_+=\left[ \begin{array}{cc}0&1\\
0&0\\\end{array}\right],\qquad E_-=\left[ \begin{array}{cc}0&0\\
1&0\\\end{array}\right].
    \ee

These matrices satisfy the following commutation relations
 \be \ad_HE_+=2E_+,\qquad \ad_H E_-=-2E_-,\qquad \ad_{E_+} E_-=H. \ee

The adjoint operator $\ad_u$ on $\gg$, for a given $u\in\gg$, is defined by
the action $\ad_uv=uv-vu$, on an arbitrary $v\in\gg$.

Given $u\in \sl(2,\Rr)$ with imaginary eigenvalues, there exists a
unique $\rho$, such that \be
u=\rho M\left[ \begin{array}{cc}0&1\\
-1&0\\\end{array}\right]M^{-1}, \ee for some $M\in \SL(2,\Rr)$.

The spectrum of the adjoint operator $\ad_u$, with $u\in\sl(2,\Rr)$, is
$\{0,2i\rho,-2i\rho\}$, where $\rho$ is the unique number from the
above decomposition.


\section{Single renormalization step}\label{single}

Let $\varrho>0$ and let $\omega\in\Rr^d$ be a fixed vector of unit length. Let $K$ be a fixed vector field of the form $K(x,y)=(\omega,uy)$, where $u\in\gg$, with eigenvalues $\pm i\rho$, and $\rho>0$.

\subsection{Renormalization operator}
On a subset of $\AA_\varrho$ the one-step renormalization operator
is defined (formally) as
    \be
    \RR(X)=\eta^{-1}\TT^*\UU_\ssX^*(X),
    \ee
where $\TT^*$ is the pullback of the map $\TT:(x,y)\mapsto (Tx,y)$, with
$Tx=\eta^{-1}x_\spa+\beta x_\spe$. Here $x=x_\spa+x_\spe$ is the
decomposition of $x\in\Rr^d$ into components parallel and
perpendicular to $\omega$. More precisely, if $X(x,y)=(\omega,f(x)y)$, then, using $T\omega=\eta^{-1}\omega$, we find
    \be
    \TT^*(X)(x,y)=(\eta\omega,f\circ T(x)y).
    \ee
We will assume that the renormalization parameters $\eta$ and $\beta$ are chosen such that they are positive numbers smaller than $1$.

The map $\UU_\ssX\in \BB_\varrho$ is a change of coordinates
chosen in such a way that $\UU_\ssX^*X$ is in a normal
form, referred to as {\em resonant}. The precise definition of
resonant is given below. It will follow from the definition that the
vector field $K(x,y)=(\omega,uy)$, with $u\in\gg$, is resonant and $\UU_\ssK$ is the
identity. An explicit calculation shows that $\RR(K)(x,y)=(\omega,\eta^{-1}uy)$.

The renormalization parameters $\eta$ and $\beta$ and the renormalization operator $\RR$ depend on the
renormalization step giving rise to a sequence of renormalization
operators $\RR_n$, $n\in\Nn$, and a sequence of vector fields
$K_n=\RR_n(K_{n-1})$, with $K_0=K$.

\subsection{Resonant versus non-resonant modes}

\begin{definition}
Let $\tilde\tau>0$ be given. We define the resonant index set as $\iplus=\{v\in\VV :\|T
v\|\le\tilde\tau|v|\}$. The non-resonant index set $\iminus$ is defined as
its complement in $\VV$. When necessary, we will explicitly state
the dependence of $\iminus$ or $\iplus$ on $\VV$, as $\iminus(\VV)$
or $\iplus(\VV)$, respectively. The terms in the
expansion~\eqref{fExpansion} of an $f\in\FF_\varrho(\VV)$ will be
called modes; those with $v\in\iplus$ will be called resonant, those
with $v\in\iminus$ will be called non-resonant. The projection
operator onto the sets of resonant and non-resonant modes in
$\FF_\varrho(\VV)$ will be denoted by $\Iplus$ and $\Iminus$,
respectively.

We will use the same symbols for the corresponding
projection operators in $\AA_\varrho$, defined by $\Iminus X(x,y)=(0,\Iminus f (x) y)$, $\Iplus X(x,y)=(\omega,\Iplus f (x) y)$, on an arbitrary $X$ of the form~\eqref{skewproduct}.
\end{definition}

We assume that the components of $\omega$ are rationally independent
with respect to $\VV$, in the sense that the first component
$v_\spa$ of any nonzero vector $v\in\VV$ is nonzero. Then, given any
$L\ge 1$, we can find $\ell>0$ such that
    \bel{emptyRectangle}
|v_\spe|>L \quad {\rm or}\quad |v_\spa|\ge\ell\,,\qquad \forall
v\in\VV\setminus\{0\}\,.
    \ee
In other words, all points in $\VV$, except for the origin, lie
outside the disc $|v_\spe|\le L$ and $|v_\spa|<\ell$.

\subsection{Elimination of non-resonant modes}

In this section, given a vector field $X\in\AA_\varrho(\sl(2,\Rr))$, close to $K(x,y)=(\omega,uy)$, we construct
a change of coordinates $\UU_\ssX\in \BB_\varrho$  such that
$\Iminus\UU_\ssX^*X=0$.  The explicit construction is carried out in
Section~\ref{NF}, where a normal form theorem is proved that applies
to skew-product flows on $\Tt^d\times\G$. A similar theorem was proved in~\cite{KK1} in
the context of vector fields on $\Tt^d\times\Rr^m$, but that theorem
does not apply in the present setting, since the main assumption (Assumption 5.1 of Chapter 5 of~\cite{KK1}) is not satisfied.


Given $X\in\AA_\varrho$, define the operator $\hat X$ as $\hat X Z:=[X,Z]$, for an arbitrary $Z$ in $\AA_\varrho$ or $\AA_\varrho^0$. Here, the right hand side of
the equality denotes the usual commutator of the vector fields, i.e.
$[X,Z]=(DZ)X-(DX)Z$.

The action of the operator $D X$, with $X(x,y)=(\omega,f(x)y)$, on an arbitrary vector field $Y$ of the form $Y(x,y)=(\omega,g(x)y),$ where $f,g\in\FF_\varrho$, is given by
    \be
[(D X) Y](x,y) =(0,D_\omega f(x)+f(x)g(x) y).
    \ee
The action of $DX$ on a vector field $Z$ of the form $Z(x,y)=(0,g(x)y)$ is given by
    \bel{DXZ}
[(D X) Z](x,y) =(0,f(x)g(x) y).
    \ee

Note that for $X\in\AA_\varrho$, the operator $\hat X$ is well defined on vector fields $Z\in\AA_\varrho^0$, of the above form, with $g\in\FF_\varrho'(\gg)$, and acts as
    \be
(\hat X Z)(x,y) =(0,\hat f g(x)y),\qquad \hat f g(x)=D_\omega g(x)-\ad_{f(x)}g(x).
    \ee
In particular, for $K(x,y)=(\omega,uy)$, the operator $\hat K$ takes the form
    \be
\hat K Z(x,y)=(0,\hat u g(x) y),\qquad  \hat u= D_\omega-\ad_u.
    \ee

The following proposition will be used repeatedly.

\begin{proposition}\label{facts1}
If $f\in\FF_\varrho$, and $X\in\AA_\varrho$, we have
\begin{enumerate}
\item[$(i)$] $\|f(x)\|\le\|f\|_\varrho$
\item[$(ii)$] $\|\ad_f\|_\varrho\le 2\|f\|_\varrho$
\item[$(iii)$] $\|DX|_{\AA^0_\varrho}\|_\varrho\le \|X\|_\varrho$
\end{enumerate}
\end{proposition}

\proof The property $(i)$ follows from the definition of the norms. Inequality $(ii)$ follows directly from the
Banach algebra property of $\FF_\varrho$, i.e. the fact that for all $f,g\in\FF_\varrho$, we have  $\|fg\|_\varrho\le
\|f\|_\varrho\|g\|_\varrho$. The same property also implies $(iii)$, using the identity~\eqref{DXZ}.\qed


%\begin{assumption}\label{Parameters}
We assume that the renormalization parameters $\eta,\beta,\ell,L,\tilde\tau,\sigma$ are
positive and satisfy
    \bel{LdeltaCond}
\eta<1/2\,,\qquad 2\sigma L<\ell\,,\qquad
\sigma=\frac{1}{2}(1-\beta)\eta,\qquad \tilde\tau=\frac{1}{2}(1+\beta)\,.
    \ee
%\end{assumption}

In the following, let $\gamma>0$ be such that
    \bel{gamma}
\gamma^{-1}=\min\limits_{v\in\iminus,|v|<8\rho/\sigma}\left\{6^{-1},\frac{|v_\spa-2\rho|}{2+|v_\spa|}\sigma^{-1}\right\}.
    \ee

\begin{proposition}\label{BKXBound} Let $0<\sigma<1/4$, $\varrho>0$, let $\omega\in\Rr^d$ be a unit vector, and let $u\in\sl(2,\Cc)$ with eigenvalues $\pm i\rho$ be given. There exist $\gamma>6$ such that the operator $\hat u: \Iminus\FF_\varrho'(\sl(2,\Cc))\to\Iminus\FF_\varrho(\sl(2,\Cc))$ has an inverse and for every $g\in\Iminus\FF_\varrho(\sl(2,\Cc))$,
    \bel{BKXBounds}
\|\hat u^{-1} g\|'_\varrho\le\frac{\gamma}{\sigma}\|g\|_\varrho\,.
    \ee
\end{proposition}

\proof If $v$ belongs to $\iminus$, then we have $|Tv|>\tilde\tau|v|$, and
thus, $\eta^{-1}|v_\spa|+\beta|v_\spe|>\tilde\tau|v_\spa|+\tilde\tau|v_\spe|$.
This immediately implies that $|v_\spa|>\sigma|v_\spe|$. This
inequality, together with \eqref{emptyRectangle} and
\eqref{LdeltaCond}, also implies that $|v_\spa|>\sigma$.

Let $f_v(x)=c_ve^{ix\cdot v}\in\Iminus\FF_\varrho(\sl(2,\Cc))$ and $g_v=\hat u f_v$. Then, we have
$g_v(x)=(i v_\spa-\ad_u)f_v(x)$,
    \be
\|f_v\|_\varrho\le\max\left\{\frac{1}{|v_\spa|},\frac{1}{|v_\spa\pm
2\rho|}\right\}\|g_v\|_\varrho
    \ee
and
    \bel{BomegadotnuInv}
\left\|D_\omega f_v\right\|_\varrho
\le\max\left\{1,\frac{|v_\spa|}{|v_\spa\pm
2\rho|}\right\}\|g_v\|_\varrho\,.
    \ee

If $v\in\iminus$ and $|v|>8\rho/\sigma$, then
    \be
    \frac{|v_\spa\pm 2\rho|}{2+|v_\spa|}\ge\frac{|v_\spa|-
    2\rho}{2+|v_\spa|}>\sigma/6
    \ee
Here we have also used that $\sigma<1/4$. The number of modes
indexed by $v\in\iminus$ with $|v|<8\rho/\sigma$ is finite. This gives $\|f_v\|_\varrho'\le (\gamma/\sigma)\|g_v\|_\varrho$.

Taking into account that the image of $\Iminus\FF'_\varrho(\sl(2,\Cc))$ under $\hat u$ is dense in $\Iminus\FF_\varrho(\sl(2,\Cc))$,
the above bounds show that if $f\in\Iminus\FF'_\varrho(\sl(2,\Cc))$ and $g=\hat u f$,
then
    \bel{BomegadotnuInv}
\left\|f\right\|'_\varrho \le{\gamma\over\sigma}\|g\|_\varrho\,.
    \ee
This proves the claim.    \qed

\clmp(BKXBound) verifies the main assumption of the
normal form theorem proved in Section~\ref{NF}, with
$\zeta=\sigma/\gamma$. The following lemma follows directly from the
main result of that section (\clmt(ElimFull)).

\begin{lemma}\label{BElim} Let $\varrho>0$, $0<\sigma<1/4$, let $\omega\in\Rr^d$ be a unit vector, and let $u\in\sl(2,\Cc)$ with eigenvalues $\pm i\rho$, and $\rho>0$, be given.  Let $\gamma$ be chosen as in \eqref{gamma}. There exist universal constants $C_1,C_2>0\,$,
such that the following holds.  If $X$ is any vector field in
$\AA_\varrho(\sl(2,\Cc))\,$ of the form~\eqref{skewproduct} satisfying
    \bel{BElimCond}
\|X-K\|_\varrho\le C_1(\sigma/\gamma)\,,\qquad \|\Iminus
X\|_\varrho\le C_1(\sigma/\gamma)^2\,,\qquad
    \ee
where $K(x,y)=(\omega,uy)$, then there exists a vector field $Z\in\Iminus{\AA_\varrho^0}'(\sl(2,\Cc))$ solving
the equation $X+[Z,X]=0$ and a change of coordinates
$\UU_\ssX:D_{\varrho}\times {\SL(2,\Cc)}\to D_\varrho\times {\SL(2,\Cc)}\,$, of the form $\UU_\ssX(x,y)=(x,U_f^{-1}(x)y)$, with $U_f\in\FF_\varrho'(\SL(2,\Cc))$, solving the
equations $\Iminus \UU_\ssX^*X=0$. The vector field $\UU_\ssX^\ast
X$ belongs to $\AA_{\varrho}(\sl(2,\Cc))\,$, and
    \beal{BElimBounds}
    \|Z\|_\varrho'&\le&
C_2(\gamma/\sigma)\|\Iminus X\|_\varrho\,,\nonumber\\
\|\UU_\ssX-\ident\|_{\varrho}' &\le&
C_2(\gamma/\sigma)\|\Iminus X\|_\varrho\,,\nonumber\\
\|\UU_\ssX^\ast X-X\|_{\varrho} &\le&
C_2(\gamma/\sigma)\|\Iminus X\|_\varrho\,,\\
\|\UU_\ssX^\ast X-X-[Z,X]\|_{\varrho} &\le&
C_2(\gamma/\sigma)^3\|\Iminus X\|_\varrho^2\,.\nonumber
    \eea
The maps  $f\mapsto U_f$ and $X\mapsto\UU_\ssX$ are continuous in the region defined by
\eqref{BElimCond}, and analytic in its interior. Moreover, if $X:\Rr^d\times\SL(2,\Rr)\to \Rr^d\times\SL(2,\Rr)$, then $U_f:\Rr^d\to \SL(2,\Rr)$.
\end{lemma}

\subsection{Scaling of resonant modes}

We now consider the action of $\TT^*$ on resonant vector fields,
i.e. those consisting of only resonant modes. In particular, the
following lemma assures contraction of $v\ne 0$ resonant modes.

\begin{lemma}\label{BContractionL} $\TT^\ast$ is a bounded operator from
$\Iplus\AA_{\varrho}(\VV)$ to $\AA_\varrho(T\VV)$, where
$\varrho>0$, with the property that
    \beal{BContraction}
\bigl\|\TT^\ast\mean X\|_\varrho &\le&
\|\mean X\|_{\varrho}\,,\nonumber\\
\bigl\|\TT^\ast\Iplus(\Id-\mean)X\|_\varrho &\le&
e^{-\varrho{1-\beta\over 2}L}\|\Iplus(\Id-\mean)X\|_{\varrho}\,.
   \eea
\end{lemma}

\proof Consider first a vector field $Y_0(x,y)=(\omega,c_0y)$, with $c_0\in\gg$. Then,
$\TT^*(Y)(x,y)=(\eta\omega,c_0y)$, and the first bound in
\eqref{BContraction} follows.

To obtain the second bound, because of our choice of the norm, it
suffices to verify it for a single non-zero $v\in\iplus$. It follows
essentially from the definitions that if $Y_v(x,y)=(0,c_ve^{ix\cdot
v}y)$ then
    \bel{BJAbound}
\|\TT^\ast Y_v\|_\varrho \le e^A\|Y_v\|_{\varrho}\,,\qquad
A=\varrho|T v|-\varrho|v|\,.
    \ee

Assume now that $v$ belongs to
$\iplus$, and that $v\not=0$. Thus $|T v|\le\tilde\tau|v|$. It leads to
$|v_\spa|<2\sigma|v_\spe|$, if we use that $\eta\tilde\tau<1/2$. This
inequality excludes frequencies $v$ that satisfy $|v_\spe|\le L$ and
$|v_\spa|\ge\ell$, due to the condition \eqref{LdeltaCond}. Thus, we
must have $|v_\spe|>L$ by condition \eqref{emptyRectangle}.
Consequently,
    \bel{BAboundOne}
A\le-\varrho(1-\tilde\tau)|v| \le-\varrho{1-\beta\over 2}L\,,
    \ee
and the second bound in \eqref{BContraction} follows.
 \qed

\subsection{Estimates for a single renormalization step}

Combining the two preceding lemmas, we obtain the following theorem.
Notice that the subspace $\Ee\AA_\varrho$ is invariant under $\RR$,
and the restriction of $\RR$ to that subspace defines a linear operator
that will be denoted by $\LL$, by the action $\RR(Y)(x,y)=(\omega,\LL(c)y)$, on an arbitrary $Y\in\Ee\AA_\varrho$ with $Y(x,y)=(\omega,cy)$, for all $(x,y)\in D_\varrho\times\G$.

\begin{theorem}\label{BRRBoundsT} Let $\G=\SL(2,\Cc)$ and $\gg=\sl(2,\Cc)$. There exist universal constants $C,R>0$, such
that the following holds, under the previous assumptions on
$L,\ell,\eta,\beta$, and $\gamma$. Let $B$ be the open ball in
$\AA_\varrho(\VV)$ of radius $R(\sigma/\gamma)^2$, centered at $K$.
Then $\RR$ is a bounded analytic map from $B$ to
$\AA_\varrho(T\VV)$, satisfying $\|\LL^{-1}\|\le 1$ and
    \beal{BRRBounds}
\|(\Id-\mean)\RR(X)\|_\varrho
&\le&C\eta^{-1}(\gamma/\sigma)e^{-\varrho{1-\beta\over 2}L}
\|(\Id-\mean)X\|_\varrho\,,\nonumber\\ \|\mean\RR(X)-\RR(\mean
X)\|_\varrho &\le& C\eta^{-1}(\gamma/\sigma)^3
\|(\Id-\mean)X\|_\varrho^2\,.
    \eea
Moreover, if $X$ takes values in $\{\omega\}\times\SL(2,\Rr)$ when restricted to $\Rr^d\times\SL(2,\Rr)$, so does $\RR(X)$.
\end{theorem}

\proof There exists a universal constant $R>0$, such that the
conditions \eqref{BElimCond} in \clml(BElim) hold, whenever $X$
belongs to the domain $B$, defined by
$\|X-K\|_\varrho<R(\sigma/\gamma)^2$.

By \clml(BContractionL), we have
    \beal{BRRBOne}
\|(\Id-\mean)\RR(X)\|_\varrho
&=&\eta^{-1}\|\TT^\ast(\Id-\mean)\UU_\ssX^\ast X\|_\varrho\nonumber\\
&\le& \eta^{-1}e^{-\varrho{1-\beta\over 2}L}
\bigl[\|(\Id-\mean)X\|_{\varrho} +\|\UU_\ssX^\ast
X-X\|_{\varrho}\bigr]\,.
    \eea
Using the bound in \eqref{BElimBounds} on the norm of $\UU_\ssX^\ast
X-X$, together with the fact that $\Iminus\mean=0$, we obtain the
first inequality in \eqref{BRRBounds}.

By \clml(BContractionL), we also have a bound
    \beal{BRRBThreeA}
\|\mean\RR(X)-\RR(\mean X)\|_\varrho
=\eta^{-1}\|\TT^\ast\mean(\UU_\ssX^\ast X-X)\|_\varrho\le
\eta^{-1} \|\mean(\UU_\ssX^\ast X-X)\|_{\varrho}\,.
    \eea
Using \clml(BElim), the norm on the right hand side of
\eqref{BRRBThreeA} can be estimated as follows:
    \bel{BRRBThreeB}
\|\mean(\UU_\ssX^\ast X-X)\|_{\varrho} \le
C_2(\gamma/\sigma)^3\|(\Id-\mean)X\|_\varrho^2
+\|\mean[Z,X]\|_{\varrho}\,,
    \ee
where $Z=\Iminus Z$ is the vector field described in \clml(BElim). Since
$\mean Z=0$, we have $\mean[Z,\mean X]=0$. As a result,
    \beal{BRRBThreeC}
\|\mean[Z,X]\|_{\varrho} =\|\mean[Z,(\Id-\mean)X]\|_{\varrho} \le
C_3\|Z\|_\varrho\|(\Id-\mean)X\|_\varrho \le C_4(\gamma/\sigma)
\|(\Id-\mean)X\|_\varrho^2\,.
    \eea
Here, we have used \clmp(facts1) and the bound on $\|Z\|_\varrho$
from \clml(BElim). Combining the last three equations yields the
second inequality in \eqref{BRRBounds}.

In order to bound the inverse of $\LL$, let $Y$ be a vector field in
$\Ee\AA_\varrho\,$. Then $Y$ can be written as $Y(x,y)=(\omega,cy)$, for some $c\in\gg$,
and the bound on $\|\LL^{-1}\|$ now follows from the fact that
    \bel{BLLinverse}
\RR (Y) (x,y)=\eta^{-1}(T^{-1}\omega,cy)=(\omega,\eta^{-1} c y)\,,
    \ee
and thus $\LL(c)=\eta^{-1}c$. This completes the proof. \qed


\section{A sequence of renormalization operators}\label{sequence}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\omega\in\Rr^d$ be a given vector that satisfies the Brjuno condition~\eqref{BrjunoCond}. For the sake of convenience, we will perform a linear transformation of our coordinate system (a rotation and a scaling) such that
$\omega$ takes the form $\omega=(1,0,\dots,0)$. The lattice obtained
from $(2\pi\Zz)^d$ under this transformation will be denoted by
$\ZZ_0$, and its dual lattice by $\VV_0$.
Notice that, after that transformation, the Brjuno condition~\eqref{BrjunoCond} on
$\omega$ and the Diophantine with respect to $\omega$ condition~\eqref{Diophantinewrtomega} on $u$ can
be restated in the same form with $\omega=(1,0,\dots,0)$ and
$\nu\in\VV_0$ (with not necessarily the same constant $\CC>0$). This
$\omega$ that satisfies the Brjuno condition with frequencies
$\nu\in\VV_0$ will be called {\em $\VV_0$-Brjuno}, and $u$ that is
Diophantine with respect to this $\omega$ and frequencies
$\nu\in\VV_0$ will be called {\em $\VV_0$-Diophantine} with respect
to $\omega$.

Starting with a  lattice $\VV_0$ in $\Rr^d$ and a
$\VV_0$-Brjuno frequency vector $\omega$, the objective of this
section is to construct a sequence of matrices $T_n$, a sequence of
lattices $\VV_n=T_n\VV_{n-1}$, and a sequence of renormalization
operators $\RR_n:\AA_\varrho(\VV_{n-1})\to\AA_\varrho(\VV_{n})$ with a fixed set $K_n\in\Ee\AA_\varrho$,
$n\in\Nn$, and a stable manifold $\WW$ associated to this set.

\subsection{Construction of the renormalization transformations}\label{sequence1}

We begin by restating the Brjuno condition~\eqref{BrjunoCond} in
another form. Let now $\VV_0$ be a  lattice in $\real^d$, and
let $\omega$ be $\VV_0$-Brjuno frequency vector. Using the same set
of frequencies, define
    \bel{BanDef}
a_n=\sum_{k=n}^\infty 2^{n-k}\Bigl[
2^{-k-\kappa}\ln(1/\Omega'_{k+\kappa})+(k+\kappa')^{-2}\Bigr]\,,\qquad
\Omega_n'=\min_{0<|\nu_\spe|<2^n}|\nu_\spa|\,,
    \ee
for all positive integers $n$. Here $\kappa,\kappa'>2$ are two
integer constants that will be determined later on. Then the Brjuno
condition \eqref{BrjunoCond} is equivalent to the condition that the
resulting sequence $a_n$ is summable. The terms involving
$(k+\kappa')^{-2}$ are clearly summable, and do not influence the
summability of other terms. Parameters $\kappa$ and $\kappa'$
produce ``shifts'' of the terms in the resulting series.

Define $\lambda_0=1$ and
    \bel{BlambdaEtc}
\lambda_n=2^{-n-\kappa}e^{-2^{n+\kappa}a_n}\,,\quad
\eta_n={\lambda_n\over\lambda_{n-1}}\,,\quad A_n=\sum_{k=n}^\infty
a_k\,,\quad \beta_n={A_{n+1}\over A_n}\,,
    \ee
for all positive integers $n$. Consider the corresponding scaling
transformations
    \be
P_n(x)=\lambda_n^{-1}x_\spa+A_1^{-1}A_{n+1}x_\spe \,,\qquad
T_n(x)=\eta_n^{-1}x_\spa+\beta_n x_\spe\,.
    \ee
Notice that $P_n=T_1 T_2\dots T_n$ by equation \eqref{BlambdaEtc}.
These quantities will now be used to define the $n$-th step
renormalization transformation $\RR_n\,$. We need to verify that the
renormalization parameters satisfy the assumptions made in the
previous section at each renormalization step. Clearly, $\beta_n$ is
positive and less than one, since $n\mapsto A_n$ is a decreasing
sequence. The condition $\eta_n<1/2$  follows from the fact that
$a_n>a_{n-1}/2$ for $n>1$, and that $\lambda_1<1/2$.

The geometric data $\VV$, $L$ and $\ell$ used in step $n$ are
    \bel{BVVLdeltaDef}
\VV_{n-1}=P_{n-1}\VV_0\,,\qquad
L_{n-1}=A_1^{-1}A_n2^{n+\kappa}\,,\qquad
\ell_{n-1}=2^{n+\kappa}\eta_n\,.
    \ee
These definitions immediately imply bounds \eqref{LdeltaCond}. The following proposition shows that the
condition \eqref{emptyRectangle} holds for all $v\in\VV_{n-1}$.

\begin{proposition}\label{Bindeed} If $v\in\VV_{n-1}$ is nonzero, then
either $|v_\spa|\ge\ell_{n-1}$ or $|v_\spe|> L_{n-1}\,$.
\end{proposition}

\proof Assume that $v\in\VV_{n-1}$ satisfies $0<|v_\spe|\le
L_{n-1}$. Then the corresponding lattice point $\nu=P_{n-1}^{-1}v$
in $\VV_0$ satisfies $|\nu_\spe|\le A_1A_n^{-1}L_{n-1}=
2^{n+\kappa}$, and thus $|\nu_\spa|\ge\Omega'_{n+\kappa}$ by
\eqref{BanDef}. Since we have
$\lambda_n<2^{-n-\kappa}\Omega'_{n+\kappa}$, this yields
    \bel{Bindeed}
|v_\spa|=\lambda_{n-1}^{-1}|\nu_\spa|
\ge\eta_n\lambda_n^{-1}\Omega'_{n+\kappa}
>\eta_n 2^{n+\kappa}=\ell_{n-1}\,,
    \ee
as claimed. \qed

\begin{definition}
Let us define $\rho_{n-1}=\lambda_{n-1}^{-1} \rho$ and $\sigma_n=\frac{1}{2}(1-\beta_n)\eta_n$ for $n\in\Nn$.
We also define
    \be
\gamma_n=\max\limits_{v\in\iminus(\VV_{n-1}),\;|v|<8\rho_{n-1}/\sigma_n}\left\{6,\frac{2+|v_\spa|}{|v_\spa-2\rho_{n-1}|}\sigma_n\right\}.
    \ee
\end{definition}

\begin{proposition}\label{Gamman}
If $u\in \sl(2,\Rr)$ with eigenvalues $\pm i\rho$ is
$\VV_0$-Diophantine with respect to $\omega$, then there exists a
constant $\xi$ that may depend only on $\rho$, $\CC$ and $\tau$,
such that for all $n\in\Nn$,
    \be
\gamma_n<\xi(\lambda_{n-1}(\prod_{i=1}^{n-1}\beta_i)\sigma_n)^{-\tau}.
    \ee
\end{proposition}

\proof Since
    \be
\gamma_n<\max\limits_{v\in\iminus(\VV_{n-1}),\;|v|<8\rho_{n-1}/\sigma_n}\left\{6,\frac{2\sigma_n+8\rho_{n-1}}{|v_\spa-2\rho_{n-1}|}\right\},
    \ee
writing  $\nu=P_{n-1}^{-1}v$, we have
    \be
\gamma_n<\max\limits_{v\in\iminus(\VV_{n-1}),\;
|v|<8\rho_{n-1}/\sigma_n}\left\{6,\frac{2\sigma_n+8\rho_{n-1}}{\lambda_{n-1}^{-1}|\nu_\spa-2\rho|}\right\}
<\max\limits_{v\in\iminus(\VV_{n-1}),\;|v|<8\rho_{n-1}/\sigma_n}\left\{6,\frac{2+8\rho}{|\nu_\spa-2\rho|}\right\}.
    \ee
If $u$ is Diophantine with respect to $\omega$ then there exists a
universal constant $\CC>0$ such that
    \be
\gamma_n<\max\limits_{v\in\iminus(\VV_{n-1}),\;|v|<8\rho_{n-1}/\sigma_n}\left\{6,\frac{2+8\rho}{\CC}|\nu|^\tau\right\}.
    \ee
Since
    \be
    |v|=|v_\spa|+|v_\spe|=\lambda_{n-1}^{-1}|\nu_\spa|+(\prod_{i=1}^{n-1}\beta_i)|\nu_\spe|\ge (\prod_{i=1}^{n-1}\beta_i)|\nu|,
    \ee
we have
    \be
\gamma_n<\max\limits_{v\in\iminus(\VV_{n-1}),|v|<8\rho_{n-1}/\sigma_n}\left\{6,\frac{2+8\rho}{\CC}(\prod_{i=1}^{n-1}\beta_i)^{-\tau}|v|^\tau\right\}.
    \ee
Thus,
    \be
\gamma_n<\xi(\lambda_{n-1}(\prod_{i=1}^{n-1}\beta_i)\sigma_n)^{-\tau},
    \ee
where $\xi$ is a constant that may depend only on $\rho$, $\CC$ and
$\tau$.\qed

For technical convenience we introduce the following quantities.

\begin{definition}
Let
    \bel{BmuDef} \mu_k=\exp\Bigl\{-\varrho
{1-\beta_k\over 2}L_{k-1}\Bigr\} =\exp\Bigl\{-{\varrho\over 2A_1}
2^{k+\kappa}a_k\Bigr\}\,,\qquad k\ge 1\,.
    \ee
\end{definition}

The following proposition provides important estimates for the
construction of the renormalization scheme.

\begin{proposition}\label{BmuBoundP} For all $k\ge 1$,
$\mu_{k+1}<\mu_k<\mu_{k+1}^{1/4}$. Furthermore, given $C,N>0$, if
$\kappa'$ and then $\kappa$ are chosen sufficiently large, then for
all $k\ge 1$,
    \bel{BmuBound}
\mu_k\le Ce^{-N2^{k+\kappa}a_k}\,,\qquad \mu_k\le C\eta_k^N\,,\qquad
\mu_k\le C(1-\beta_k)^N\,,\qquad\mu_k\le C\gamma_k^{-N}
    \ee
\end{proposition}

\proof The inequality $\mu_{k+1}<\mu_k<\mu_{k+1}^{1/4}$ follows from
the fact that $a_{k+1}/2<a_k<2a_{k+1}$. Let now $c=\varrho/2$. By
choosing $\kappa$ and $\kappa'$ sufficiently large, we have
$c/A_1\ge (2+3\tau)N$. Keeping $\kappa'$ fixed, and increasing
$\kappa$ further, if necessary, we obtain the first two bounds in
\eqref{BmuBound} by using that $2^{k+\kappa}a_k\ge
2^{k+\kappa}(k+\kappa')^{-1}\ge c'2^\kappa k$, for some constant
$c'>0$. The same inequality, together with
$1-\beta_k=a_k/A_k>(k+\kappa')^{-2}(2+3\tau)N/c$, implies the third
bound in \eqref{BmuBound}. These inequalities, together with
\clmp(Gamman), also imply the last bound. \qed

In particular, the first bound shows that the $\mu_k$ decrease at least
exponentially with $k$.

Note that in the definition of $\Omega_n'$~\eqref{BanDef}, we can replace $\min$ with a ``lower bound''. Our estimates are then uniform within classes of Brjuno vectors $\omega$ of unit length that admit the same sequence of lower bounds $n\mapsto\Omega_n'$ and matrices $u\in\sl(2,\Rr)$ Diophantine with respect to $\omega$ with the same $\CC$, $\rho$ and $\tau$.

\subsection{Stable manifold}\label{stablemanifold}

Let $K_0=K$ and let $K_n$ be such that $K_n(x,y)=(\omega,\lambda_n^{-1}uy)$, for $(x,y)\in D_\varrho\times\G$.
Having verified all of the assumptions made in the previous section,
we can now apply \clmt(BRRBoundsT) to the $n$-th step
renormalization transformation $\RR_n\,$, defined by the parameters
introduced above. Denote by $\LL_n$ the corresponding linear
operator $\LL$ from $\Ee\FF_\varrho(\gg,\VV_{n-1})$ to
$\Ee\FF_\varrho(\gg,\VV_n)$.

Define $\AA_{\varrho,k}=\AA_\varrho(\VV_k)$ and
$\AA_{\varrho,k}^0=\AA_\varrho^0(\VV_k)$, for all non-negative
integers $k$. To simplify notation, the norms in $\AA_{\varrho,k}$
and $\AA_{\varrho,k}^0$ will not be given indices. From
\clmt(BRRBoundsT) we immediately obtain the following theorem.


\begin{theorem}\label{BRRnBoundsT} There exist
universal constants $r,C>0$ such that the following holds, for every
positive integer $n$. Let $B_{n-1}$ be the open ball in
$\AA_{\varrho,n-1}(\sl(2,\Cc))$ of radius $r\sigma_n^2/\gamma_n^2$, centered at
$K_{n-1}$. Then
$\RR_n$ is a bounded analytic map from  $B_{n-1}$ to
$\AA_{\varrho,n}(\sl(2,\Cc))\,$, satisfying $\|\LL_n^{-1}\|\le 1$ and
    \beal{BRRnBounds}
\|(\Id-\mean)\RR_n(X)\|_\varrho &\le& C\gamma_n\sigma_n^{-2}\mu_n
\|(\Id-\mean)X\|_\varrho\,,\nonumber\\
\|\mean\RR_n(X)-\RR_n(\mean X)\|_\varrho &\le&
C\gamma_n^3\sigma_n^{-4} \|(\Id-\mean)X\|_\varrho^2\,.
    \eea
Moreover, if $X$ takes values in $\{\omega\}\times\SL(2,\Rr)$ when restricted to $\Rr^d\times\SL(2,\Rr)$, so does $\RR_n(X)$.
\end{theorem}


In what follows, a domain $\DD_{n-1}$ for $\RR_n$ is a subset of the
ball $B_{n-1}$ described in \clmt(BRRnBoundsT), that is open in
$\AA_{\varrho,n-1}$ and contains the vector field $K_{n-1}$. Given a
domain $\DD_{n-1}$ for each $\RR_n\,$, the domain $\wt\DD_n$ of the
combined renormalization operator
$\wt\RR_{n+1}=\RR_{n+1}\circ\RR_n\circ\ldots\circ\RR_1$ is defined
recursively as the set of all vector fields in the domain of
$\wt\RR_n$ that are mapped under $\wt\RR_n$ into the domain $\DD_n$
of $\RR_{n+1}\,$. By \clmt(BRRnBoundsT), these domains are open and
non-empty, and the transformations $\wt\RR_n$ are analytic.

The following theorem, valid for the case of $\G=\SL(2,\Cc)$, is the corollary of the stable manifold theorem
of~\cite{KK1} (Section 6, Theorem 6.1), for sequences of maps
between Banach spaces.

\begin{theorem}\label{BRRnComposeT} If $\kappa'$ and
then $\kappa$ are chosen sufficiently large, then there exist a
sequence of domains $\DD_{n-1}$, for the renormalization
transformations $\RR_n$, $n\in\Nn$, such that the set
$\WW=\cap_{n=0}^\infty\wt\DD_n$ is the graph of an analytic function
$W:(\Id-\Ee)\DD_0\to\Ee\DD_0\,$, satisfying $W(0)=K$ and $DW(0)=0$.
For every $X\in\WW$ and every $n\in\Nn$, if $\chi_n=\prod_{i=1}^n\mu_i\,$,
then
    \beal{BRRnCompose}
 \bigl\|\wt\RR_n(X)-K_n\bigr\|_\varrho
&\le&\chi_n^{1/2}\|(\Id-\Ee)X\|_\varrho\,,\nonumber\\
\bigl\|\Ee[\wt\RR_n(X)-K_n]\bigr\|_\varrho
&\le&\chi_n\|(\Id-\Ee)X\|_\varrho^2\,,\\
\bigl\|(\Id-\mean)\wt\RR_n(X)\bigr\|_\varrho
&\le&\chi_n^{1/2}\|(\Id-\mean)X\|_\varrho\,.\nonumber
    \eea
\end{theorem}

\proof Let us first rescale our transformations $\RR_n\,$. Let
$r_n=r_{n-1}\sigma_{n+1}^2/\gamma_{n+1}^2$ for every positive
integer $n$, with $r_0>0$ smaller than half the constant $r$ from
\clmt(BRRnBoundsT).

Consider the transformations
$R_n:\AA_{\varrho,n-1}^0\to\AA_{\varrho,n}^0\,$, with $n\in\Nn,$
given by
    \bel{BRnDef}
R_n(Z)=r_n^{-1}\bigl[\RR_n(K_{n-1}+r_{n-1}Z)-K_n\bigr]\,.
    \ee

In order to show explicitly that the assumptions of the stable
manifold theorem of~\cite{KK1} are verified, we begin by introducing
the projection operator $\Pp=\Ee$. The restriction $R_n\proj$
defines a linear map from $\proj\AA_{\varrho,n-1}^0$ to
$\proj\AA_{\varrho,n}^0\,$, which will be denoted by $L_n\,$. By
\clmt(BRRnBoundsT), $R_n$ is analytic and bounded on the ball
$\|Z\|_\varrho<2$, and satisfies
    \beal{BRBounds}
\|(\Id-\mean)R_n(Z)\|_\varrho
&\le&\eps_n\|(\Id-\mean)Z\|_\varrho\,,\nonumber\\
\|(\Id-\proj)R_n(Z)\|_\varrho
&\le&\vartheta_n\|(\Id-\proj)Z\|_\varrho\,,\\ \|\proj
R_n(Z)-R_n(\proj Z)\|_\varrho
&\le&\varphi_n\|(\Id-\mean)Z\|_\varrho^2\,,\nonumber
    \eea
where
    \bel{BetpOne}
\eps_n=\vartheta_n=C\sigma_n^{-2}\gamma_n\sigma_{n+1}^{-2}\gamma_{n+1}^2\mu_n\,,\quad
\varphi_n=C\sigma_n^{-4}\gamma_n^3\sigma_{n+1}^{-2}\gamma_{n+1}^2\,.
    \ee
Here, $C\ge 1$ is a universal constant. In addition, we have
$\|L_n^{-1}\|<1/4$. We will restrict $R_n$ to the domain
$D_{n-1}\subset\AA_{\varrho,n-1}^0\,$, defined by
    \bel{BRnDomainDef}
\|\proj Z\|_\varrho<1\,,\qquad \|(\Id-\proj)Z\|_\varrho<1\,,\qquad
\|(\Id-\mean)Z\|_\varrho<\delta_{n-1}\,,
    \ee
where $\delta_{n-1}=(6\varphi_n)^{-1}$. By \clmp(BmuBoundP), if
$\kappa'$ and $\kappa$ are chosen sufficiently large (such that the
bounds of the same proposition are valid with $N$ large enough), then
$C\sigma_n^{-2}\gamma_n\sigma_{n+1}^{-2}\gamma_{n+1}^2\mu_n^{1/2}\le
1/6$, for all positive integers $n$. This inequality implies
    \bel{BetpTwo}
\eps_n=\vartheta_n\le\mu_n^{1/2}/6\,,\qquad
\eps_n\delta_{n-1}\le\delta_n\,,
    \ee
for all $n\ge 1$. The hypotheses of the stable manifold theorem (see
Theorem 6.1 of Section 6 in~\cite{KK1}) have now been verified, with
$\eps=1/6$ and $\vartheta=1/3$, and the conclusions of this theorem
imply the statements in \clmt(BRRnComposeT). \qed

\section{Reducibility of skew-product systems}\label{reducibility}

Let $K_0=K$, where $K(x,y)=(\omega,uy)$,  and let
$K_n=\RR_n(K_{n-1})$, i.e. $K_n(x,y)=(\omega,\lambda_n^{-1}uy)$, as before. Let $X_0=X\in\WW$, and let
$X_n=\RR_n(X_{n-1})$ be the renormalization orbit of $X$. For the case of $\G=\SL(2,\Cc)$, that we consider in this section, the stable manifold $\WW$ has been constructed in Section~\ref{stablemanifold}. We will
show that $X\in\WW$ is analytically reducible to $K$.

Recall that a vector field $X\in\AA_\varrho$ of the form~\eqref{skewproduct} is (analytically) reducible to $K$, if there is an (analytic) map $\psi:\Tt^d\times\G\to \Tt^d\times\G$, of the form $\psi(x,y)=(x,\Psi(x)^{-1}y)$, such that $\psi^*(X)=K$. Modulo smoothness assumptions, this is equivalent to
    \bel{conjugacyviaflows}
   \Psi(x+\omega t)e^{ut}\Psi(x)^{-1}=\Phi^t(x), \qquad  t\in\Rr\,,\quad x\in\Tt^d,
    \ee
where $\Phi^t(x)$ is the associated matrix solution of~\eqref{diffeq}, with $\Phi^0(x)$ the identity, as in~\eqref{cosflow}. $\Phi (x)$ satisfies the equation
    \bel{flowequation}
 \Phi^t(x)=\id +\int_0^t  f(x+\omega s) \Phi^s(x)\;ds.
    \ee
This equation can be used to construct $\Phi (x)$, using the contraction mapping principle.

Now, observe that if $X_n$ is reducible to $K_n$, then $X_{n-1}$ is
reducible to $K_{n-1}$. Since
$X_n=\eta_n^{-1}\TT_n^*\UU_{n-1}^*(X_{n-1})$, where
$\UU_{n-1}=\UU_{\ssX_{n-1}}$, and $K_n=\eta_n^{-1}\TT_n^*(K_{n-1})$,
if there exists a map $\psi_n$ such that $\psi_n^*(X_n)=K_n$, then
we have $\psi_{n-1}^*(X_{n-1})=K_{n-1}$. The maps
$\psi_{n-1}$ and $\psi_n$ are related, by writing $\UU_{n-1}(x,y)=(x,U_{n-1}^{-1}(x)y)$, via
    \bel{BGammanSeq}
\Psi_{n-1}=U_{n-1}\Psi_n\circ T_n^{-1}\,, \qquad n\in\Nn.
    \ee
The proof of reducibility of the vector fields $X\in\WW$ that follows is based on this relationship.

Let $\FF_0(\VV_n)$ and $\FF_0'(\VV_n)$ be the spaces of continuous functions $f:\Tt^d\to
\Rr^{2\times 2}$, where $\Tt^d=\Rr^d/\ZZ_n$, lifted to $\Rr^d$, with frequencies in
$\VV_{n}$, equipped respectively with the norms
    \bel{norm0}
\|f\|_0=\sum_{v\in\VV_n}\|f_v\|\,,
    \ee
and
    \bel{normprime}
\|f\|'_n=r_n^{-1}\|f\|_0=r_n^{-1}\sum_{v\in\VV_n}\|f_v\|\,,
    \ee
where $r_n=\chi_n^{1/3}$, for $n\in\Nn$, and $r_0=1$.

%The subsets of maps in $\FF_0(\VV_n)$ and $\FF_0'(\VV_n)$ that take values in $\G$ will be denoted by
%$\FF_0(\G,\VV_n)$ and $\FF_0'(\G,\VV_n)$, respectively.

Let $\BB_0(\VV_n)$ be the set of maps $F:\Tt^d\to\G$, where $\G\subset\GL(2,\Rr)$, of the form $F=\id+f$ with $f\in\FF_0(\VV_n)$. Let $\BB_n$ be the set of maps $F:\Tt^d\to\G$ of the form $F=\id+f$ with $f\in\FF_0'(\VV_n)$. Let $B_n$ be the unit open ball in $\BB_n$, centered at the identity $\id$.

Define (formally) the map $M_n$ from $B_n$ into $\BB_{n-1}$, as
    \be
M_n(F)=U_{n-1}F\circ T_n^{-1}.
    \ee

\begin{proposition}\label{BMMnContracts} If $\kappa'$ and
$\kappa$ are sufficiently large, then there exists an
open neighborhood $B$ of $K$ in $\AA_\varrho\,$, such that for every
$X\in\WW\cap B$, and for every $n\ge 1$, the map $M_n$ is well
defined and analytic, as a function from $B_n$ to $\BB_{n-1}\,$.
Furthermore, $M_n$ takes values in the ball of radius $1/2$ centered at the identity in $B_{n-1}$,  and
$\|DM_n(F)\|\le\mu_n^{1/4}\,$, for all $F\in B_n\,$.
\end{proposition}

\proof $M_n$ is well-defined in some open neighborhood of $\id$ in
$\BB_n\,$, and
    \bel{BMMnExpr}
M_n(F)=\id+g+(U_{n-1}-\id)(\id+g)\,,\qquad g=f\circ T_{n}^{-1}\,,
    \ee
where $F=\id+f$. In order to
estimate $U_{n-1}-\id$, we can apply \clml(BElim). We will use
\clmp(BmuBoundP) and assume that $\kappa'$ and then $\kappa$ have
been chosen sufficiently large, without always mentioning it. By
\clml(BElim) and \clmt(BRRnComposeT), there exist a constant $C>0$,
such that
    \beal{BMMnCOne}
     \|U_{n-1}-\id\|_{\varrho} &\le& C\gamma_n\sigma_n^{-1}\|\Iminus
X_{n-1}\|_\varrho \le C\gamma_n\sigma_n^{-1}\chi_{n-1}^{1/2}
\|(\Id-\mean)X\|_\varrho\nonumber\\
&\le&\chi_{n-1}^{1/3}\|(\Id-\mean)X\|_\varrho\le \chi_{n}^{1/12} \,,
    \eea
for all $n>1$, and for all $X\in\WW\cap B$, provided that the
neighborhood $B$ of $K$ has been chosen sufficiently small
(depending on $\kappa'$ and $\kappa$). The first inequality in
\eqref{BMMnCOne} and the final bound also hold for $n=1$.

We further have,
    \bel{productbound}
\|(U_{n-1}-\id)(\id+g)\|_0\le\|U_{n-1}-\id\|_\varrho(1+\|g\|_{0}),
    \ee
and $\|g\|_0=\|f\|_0\le r_n\|f\|_n'<r_n$. Here, and in what
follows, we assume that $F\in B_n\,$. By using that
$r_n/r_{n-1}=\mu_n^{1/3}\,$, we obtain
$\|g\|'_{n-1}\le\mu_n^{1/3}\,$. When combined with \eqref{BMMnCOne}
and \eqref{productbound}, this shows that $M_{n-1}$ maps $B_n$
into $B_{n-1}/2$. Using the fact that the inclusion map from $B_n$ into
$B_{n-1}$ is bounded in norm by $\mu_n^{1/3}$, together with the property $\|f_1f_2\|_{n-1}'\le r_{n-1}\|f_1\|_{n-1}'\|f_2\|_{n-1}'$, we obtain that
$\|D M_n(F)\|\le\mu_n^{1/4}\,$, for all $n\ge 1$, and for all $F\in
B_n\,$. \qed

\begin{proposition}\label{BGammaLimits} Under the same assumptions as in
\clmp(BMMnContracts), there exists an open neighborhood $B$ of $K$
in $\AA_\varrho\,$, such that the following holds. Given any
$X\in\WW\cap B$, and any sequence of functions $F_k\in B_k\,$, $k\in\Nn$,
define
    \bel{BGammanmDef}
\Psi_{n,k}=M_{n+1}\circ\ldots \circ M_k (F_k)\,,\qquad
0\le n<k\,.
    \ee
Then the limits $\Psi_n=\lim_{k\to\infty}\Psi_{n,k}$ exist in
$\BB_n\,$, are independent of the choice of the maps
$F_0,F_1,\ldots$, and satisfy the identities \eqref{BGammanSeq}. The
map $X\mapsto\Psi_0$ is analytic and bounded on $\WW\cap B$.
\end{proposition}

\proof By \clmp(BMMnContracts), the maps $M_n:
B_n\to B_{n-1}/2$ contract distances by a factor of at least $1/2$.
Thus, if $1\le n<k<k'$, then the difference $\Psi_{n,k'}-\Psi_{n,k}$
is bounded in norm by $2^{n-k+1}$. This shows that the sequence
$k\mapsto\Psi_{n,k}$ converges uniformly in $\BB_n$ to a limit $\Psi_n\,$,
which is independent of the choice of the functions $F_k\,$. By
choosing $F_k=\Psi_k$ for all $k$, we obtain the identities
\eqref{BGammanSeq}. The analyticity of $X\mapsto\Psi_0$ follows via
the chain rule from the analyticity of the maps used in our
construction, and from uniform convergence.
 \qed

In what follows, the maps $M_1$ and $\RR_1$ associated to a vector field $X\in\WW$ or the form $X(x,y)=(\omega,f(x)y)$
will be denoted by $M_\ssX$ and $\RR_\ssX$, respectively, in order
to show explicitly the dependence on the vector field $X$. The map
$\Psi_0$ associated with the vector field $X\in\WW$ will be denoted by
$\Psi_\ssX\,$.

\begin{proposition}\label{BAnalyticTori} Let $\varrho>\delta>0$.
Under the same assumptions as in \clmp(BMMnContracts), there exists
an open neighborhood $B$ of $K$ in $\AA_{\varrho}(\VV_0)$, such
that $\Psi_\ssX$ has an analytic continuation to $\|\im x\|<\delta$,
for each $X\in B\cap\WW$. With this continuation, $X\mapsto\Psi_\ssX$
defines a bounded analytic map from $B\cap\WW$ to
$\FF_\delta(\SL(2,\Cc),\VV_0)$.
\end{proposition}

\proof Consider the translations $\JJ_q(x,y)=(J_q(x),y)$, with $J_q(x)=x+q$ and $q\in\Rr^d$, defined on
$\Cc^d\times \SL(2,\Cc)$. For vector fields
$X\in\AA_\varrho(\VV_0)$, with $X(x,y)=(\omega,f(x)y)$, we denote by $\JJ_q^\ast X$  the pullback under $\JJ_q$,
as we did earlier. For maps $F\in\FF_\varrho(\VV_0)$, define
$J_q^\star$ by $J_q^\star(F)=F\circ J_q$. By examining the
construction of the map $\UU_\ssX$ and the scaling map $\TT$, one
can verify that
    \be
\JJ_q\circ\UU_{\JJ_q^\ast X}=\UU_\ssX\circ \JJ_q\,,\qquad \JJ_{Tq}\circ\TT=\TT\circ\JJ_q \,.
    \ee
Thus, one further has
    \be
\RR_{\JJ_q^\ast X}\circ\JJ_q^\ast=\JJ_{T^{-1}q}^\ast\circ\RR_X\,,\qquad
M_{\JJ_q^\ast X}=J_q^\star\circ M_X\circ (J_{T^{-1}q}^\star)^{-1}.
    \ee
We have also used that the translations $\JJ_q^\ast$ are isometries on
the spaces $\AA_\varrho(\VV_0)$, and that the domain of $\RR$ and the
manifold $\WW$ are invariant under these translations.

For convenience, we extend the map $X\mapsto\Psi_\ssX$ to an open
neighborhood of $K$, by setting $\Psi_\ssX=\Psi_{\ssX'}\,$, where
$X'=(\Id+W)(X-\Ee X)$. With this extension, the map
$X\mapsto\Psi_\ssX$ is analytic and bounded on a sufficiently small
open ball $B\subset\AA_\varrho(\VV_0)$.

By examining the construction of $\Psi_\ssX$ and using the
invariance property of $\WW$ under $\JJ_q^\ast$, we can verify that, for any
$q\in\real^d$, we have $\Psi_{\JJ^\ast_q X}=J_q^\star\Psi_\ssX$, and therefore
    \bel{BGammaXu}
\Psi_\ssX(q)=\Psi_{\JJ_q^\ast X}(0)\,.
    \ee
The idea now is to use the analyticity of map $X\mapsto\Psi_\ssX\,$,
to extend the right hand side of equation \eqref{BGammaXu} to the
complex domain $\|\im q\|<\delta$. Let $0<\varrho'<\varrho-\delta$. Let $B'$ be an open ball around $K$
in $\AA_{\varrho'}(\VV_0)$ such that the function $X\mapsto \Psi_{\ssX}$ is well-defined for every $X\in B'$. Now, let $B$ be an open ball in $\AA_{\varrho}(\VV_0)$ such that
$\JJ_q^\ast B\subset B'$, for all $q\in\Cc^d$
of norm smaller than or equal to $\varrho'-\varrho$. Regarded as a
function of $(X,q)$, the right hand side of \eqref{BGammaXu}, is
analytic and bounded on the product of $B$ with the strip $\|\im
q\|<\varrho-\varrho'$. This show that  $\Psi_\ssX\,$ belongs to
$\FF_\delta(\VV_0)$ for all $X\in B$. \qed

\begin{theorem}\label{AnalyticConjugacy}
If $\kappa'$ and $\kappa$ are chosen sufficiently large then
there exists an open neighborhood $B$ of $K$ in
$\AA_\varrho(\VV_0)$ and for every $X\in\WW\cap B$, of the form $X(x,y)=(\omega,f(x)y)$,
there exists an analytic map $\psi_\ssX:D_\delta\times\SL(2,\Cc)\to D_\delta\times\SL(2,\Cc)$ of the form $\psi_\ssX(x,y)=(x,\Psi_\ssX(x)^{-1}y)$ with $ \Psi_\ssX\in
\FF_{\delta}'(\SL(2,\Cc),\VV_0)$, and $0<\delta<\varrho$, such that $\psi_\ssX^\ast X=K$. If $f$ is real-matrix valued when restricted to $\Rr^d$, so is $\Psi_\ssX$.
\end{theorem}

\proof  Let $\G=\SL(2,\Cc)$. By \clmp(BGammaLimits) and \clmp(BAnalyticTori), there
exists an open neighborhood $B$ of $K$ in $\AA_{\varrho}(\VV_0)$
and, for every $X\in\WW\cap B$, the map $\Psi_\ssX\in
\FF_\delta(\G,\VV_0)$, defined in \clmp(BGammaLimits), such that
the map $X\mapsto\Psi_\ssX$ is analytic from $B\cap\WW$ to
$\FF_\delta(\G,\VV_0)$. Since for every $x\in D_\delta$, $\Psi_\ssX (x)\in\G$ is invertible,
the map $\psi_\ssX:D_\delta\times\G\to D_\delta\times\G$ is well defined by $\psi_\ssX (x,y)=(x,\Psi_\ssX (x)^{-1}y)$, and also invertible.
Notice also that the conclusions of \clmp(BMMnContracts) and \clmp(BGammaLimits) are valid if in the definitions of the norms~\eqref{norm0} and \eqref{normprime}, $\|f_v\|$ is replaced by $(1+|\omega\cdot v|)\|f_v\|$, since $\|U_{n-1}-\id\|_{\varrho}$ can be replaced by  $\|U_{n-1}-\id\|_{\varrho}'$ in \eqref{BMMnCOne}. Therefore, $\Psi_\ssX\in\FF_\delta'(\G,\VV_0)$.

It remains to show that $\psi_\ssX^*(X)=K$, or equivalently~\eqref{conjugacyviaflows}.
The maps $\Phi^t_\ssX$ and $\Phi^t_{\RR(X)}$, associated to vector fields $X$ and $\RR(X)$, respectively, are related via
    \be
\Phi^t_{\RR(X)}=[U(\cdot +\eta^{-1}\omega t)^{-1}\Phi^{\eta^{-1}t}_{\ssX}U]\circ T.
    \ee
Using this identity we can obtain the following relation, for all integer $m\ge 0$,
    \be
(\Phi_m^t)^{-1}\Psi_{m}(\cdot +\omega t)e^{tu}=U_{m}[(\Phi_{m+1}^{\eta_{m+1} t})^{-1}\Psi_{m+1}(\cdot+\eta_{m+1}\omega t)e^{tu}]\circ T_{m+1}^{-1}.
    \ee
Iterating this identity we find
    \beal{iteratedidentity}
(\Phi_0^t)^{-1}\Psi_{0}(\cdot +\omega t)e^{-tu}=[U_{0}U_1\circ T_1^{-1}\cdots U_m\circ T_m^{-1} \circ\cdots \circ T_{1}^{-1}] \nonumber \\
\cdot[(\Phi_{m+1}^{\lambda_{m+1} t})^{-1}\Psi_{m+1}(\cdot +\lambda_{m+1}\omega t)e^{tu}]\circ T_{m+1}^{-1}\circ \cdots \circ T_1^{-1}.
    \eea
We know that the expression in the first square bracket converges to $\Psi_0$ in $\FF_0(\G,\VV_0)$, as $m\to\infty$, and that
$\|U_{0}U_1\cdots U_m-\id\|_0<1$. Here, we have used that $\|F\circ T\|_0=\|F\|_0$.
We have previously obtained that $\Psi_m\in B_m$, and thus
    \be
    \|\Psi_m-\id\|_0<\chi_m^{1/3}.
    \ee
Moreover, by writing the integral equation that $\varphi^t(x)=e^{-ut}\Phi^t(x)-\id$ satisfies,
    \be
 \varphi^t(x)=\int_0^t e^{-us}[f(x+\omega s)-u]e^{us} [\varphi^s(x)+\id]\;ds \,,
    \ee
and applying the contraction mapping principle, we can obtain the following estimate
    \be
    \|e^{-ut}\Phi^t(x)-\id\|_0\le 2e^{2\|u\||t|}\|f-u\|_0,
    \ee
for $t$ in a small open interval of $\Rr$ containing zero.

Similarly, starting from the equation~\eqref{flowequation} one can obtain, for a sufficiently small open interval of values $t$ around zero, an estimate $\|\Phi^t(x)-\id\|_0\le 2|t| \|f\|_0.$ Finally, using $u_{m+1}=\lambda_{m+1}^{-1}u$, we obtain the following estimate
    \bel{PhiPsiE}
\|(\Phi_{m+1}^{\lambda_{m+1} t})^{-1}\Psi_{m+1}(\cdot +\lambda_{m+1}\omega t)e^{tu}-\id\|_0\le c_1\|f_{m+1}-u_{m+1}\|_0+c_2\|\Psi_{m+1}-\id\|_0< c\chi_{m+1}^{1/3}.
    \ee
Here $c,c_1,c_2>0$ are universal constants that depend only on $\|u\|$, an upper bound on $|t|$ and an upper bound on $\|f\|_0$.

Using~\eqref{PhiPsiE}, we find that the right hand side of~\eqref{iteratedidentity}, and thus the left one, converges to $\Psi_0(x)$ in $\FF_0(\G,\VV_0)$. Since the convergence in $\FF_0(G,\VV_0)$ also applies pointwise convergence (see~\clmp(facts1)), we have
    \be
    \Phi_0^t(x)^{-1}\Psi_{0}(x+\omega t)e^{-tu}=\Psi_0(x),
    \ee
for $t$ in a small open interval containing zero, and all $x\in\Rr^d$. The identity can now be extended for all $t\in\Rr$, using the cocycle property~\eqref{cocycleidentity}.
 \qed

\proofoftheorem This theorem follows from \clmt(main1). To
see this, consider a family of vector fields $X_v$ with $X_v(x,y)=X(x,y)+(0,vy)$, with
$v\in \sl(2,\Rr)$. $X_v$ belongs to $\WW$ if and only if
    \be
\Ee(X_v)-W((\Ii-\Ee)X_v)=0\,,
    \ee
or equivalently, for every $(x,y)\in\Tt^d\times\SL(2,\Rr)$,
    \be
    \Ee(X)(x,y)+(0,vy)-W((\Ii-\Ee)X)(x,y)=0\,.
    \ee
Clearly, this equation has a unique solution $v$, for $X$ close to $K$ (such that $(\Ii-\Ee)X$ is in the
domain of $W$). The first assertion now follows from the fact that the
vector fields on $\WW$ are analytically reducible to $K$. The analyticity of the map $X\mapsto v$ follows from the analyticity of $W$ (\clmt(main1)). \qed

\section{A normal form theorem for skew-product flows}\label{NF}

Let $\G$ be an arbitrary Lie subgroup of $\GL(n,\Cc)$.
Let $\Iminus$ be a fixed projection operator on $\FF_\varrho$, and
let $K(x,y)=(\omega,{u}(x)y)$ be a fixed vector field in $\AA_\varrho$, with ${u}\in\FF_\varrho(\gg)$, that satisfies $\Iminus u=0$.
Recall the definition of the operator
    \be
\hat f g(x)=D_\omega g(x)-\ad_{f(x)}g(x),
    \ee
where $\ad_{f(x)}g(x)=f(x)g(x)-g(x)f(x)$, for an arbitrary $g\in\FF_\varrho$.

We will assume that the operator $\hat{u}$ maps $\Iminus{\FF'_\varrho}(\gg)$ into $\Iminus\FF_\varrho(\gg)$, and has an inverse which is bounded in norm by a positive
constant $\zeta^{-1}>1$, i.e. for every
$g\in\Iminus\FF_\varrho(\gg)$,
    \bel{Nonres}
\bigl\|\hat {u}^{-1}g\bigr\|'_\varrho\le
\zeta^{-1}\|g\|_\varrho\,.
    \ee
In the case when $\G=\SL(2,\Cc)$, $K(x,y)=(\omega,uy)$, with $u\in\gg$, and $\Iminus$ is the projection operator onto the non-resonant subspace of $\FF_\varrho(\gg)$, this assumption is verified by~\clmp(BKXBound).

Given a vector field $X:D_\varrho\times\G\to D_\varrho\times\G$, of the form $X(x,y)=(\omega,f(x)y)$, near $K$, with  $f\in\FF_\varrho(\gg)\,$, the goal of
this section is to construct a change of variables $\UU_\ssX (x,y)=(x,U_f(x)^{-1}y)$, with
$U_f:D_{\varrho}\mapsto\G\,$, such that the pullback
$\UU_\ssX^\ast X(x,y)=(\omega, U_f^\star f(x)y)$ of $X$ under $\UU_\ssX$, belongs to $\AA_{\varrho}$
and satisfies
    \bel{ElimCondP}
\Iminus U_f^\star f=0\,.
    \ee

We will construct a solution to \eqref{ElimCondP} in the following
way. First, we will construct an approximation to $\UU_\ssX$ given
by the time-one map $\phi_Z^1$ of the flow of a vector field
$Z$ of the form $Z(x,y)=(0,g(x)y)$ with $g\in\Iminus\FF_\varrho(\gg)$ that satisfies the equation
    \bel{ElimAppEqu}
\Iminus (f-\hat f g)=0\,.
    \ee
Here and in the following the flow of a vector field $Z$ is denoted by $\phi_Z$ and satisfies the
equation $\frac{d}{dt}\phi_Z^t=Z\circ\phi_Z^t,$ with the initial
condition $\phi_Z^0=\id$. Then, we will find the pullback of $X$
under $\phi_Z^1$, and iterate this procedure. Notice that $\phi_Z^t(x,y)=(x,\Phi_g^{-t}(x)y)$, where $\Phi_g^t(x)=e^{g(x)t}$ is a matrix exponential, defined for an arbitrary $A\in\Cc^{n\times n}$ by
    \be
e^{A}=\sum_{n=0}^\infty \frac{A^n}{n!}.
    \ee

We begin by proving some estimates for pullbacks under a
time $t$ map of the flow of a vector field $Z(x,y)=(0,g(x)y)$, with $g\in\FF_\varrho(\gg)$.

\begin{proposition}\label{FieldFlowBound} Let
$t\in\real$ be fixed. Let $g\in\FF_\varrho(\gg)$ and $f\in\FF_\varrho(\gg)$ satisfy $\|tg\|_\varrho\le \eps$ and
$\|tD_\omega g\|_\varrho\le s\eps$, with $\eps\le 1/6$, and $s>0$. Then
$(\Phi_g^t)^\star f$ belongs to $\FF_{\varrho}(\gg)$, and satisfies
    \beal{BasicFlowBound}
 \bigl\|(\Phi_g^t)^\star f-f\bigr\|_{\varrho} &\le&
2(2\|f\|_\varrho+s)\,\eps\,,\nonumber\\ \bigl\|(\Phi_g^t)^\star
f-f+t\hat f g\bigr\|_{\varrho} &\le& (2\|f\|_\varrho+s)\,\eps^2\,.
    \eea
\end{proposition}

\proof  Writing $\Phi_g^{t}$ explicitly and evaluating
    \be
    {(\Phi_g^{t})}^\star f=\Phi_g^{-t} (f-D_\omega )\Phi_g^{t}\,,
    \ee
at point $x\in D_\varrho$, we find
    \be
    {(\Phi_g^t)}^\star f(x)=e^{-g(x)t} (f(x)-D_\omega )e^{g(x)t}.
    \ee
Differentiating with respect to $t$, one can verify the identities
    \be
    e^{-g(x)t} f(x)e^{g(x)t}=e^{\ad_{-g(x)t}}f(x)=\sum_{n=0}^\infty \frac{1}{n!}(\ad_{-g(x)t})^n f(x),
    \ee
and
    \be
    e^{-g(x)t} D_\omega e^{g(x)t}=\sum_{n=1}^\infty \frac{1}{n!}(\ad_{-g(x)t})^{n-1} D_\omega g(x)t.
    \ee
These two equalities guarantee that $(\Phi_g^t)^\star f\in\FF_\varrho(\gg)$, since $D_\omega g\in\FF_\varrho(\gg)$. Moreover,
    \be
    [{(\Phi_g^t)}^\star f -f](x)=\sum_{n=1}^\infty \frac{1}{n!}(\ad_{-g(x)t})^{n-1} (\ad_{-g(x)t}f(x)-D_\omega g(x)t).
    \ee

Using~\clmp(facts1), the last identity implies
    \be
    \|{(\Phi_g^t)}^\star f -f\|_\varrho\le\frac{e^{2\|tg\|_\varrho}-1}{2\|g\|_\varrho}(2\|f\|_\varrho\|g\|_\varrho+\|D_\omega g\|_\varrho).
    \ee
Similarly, from the same identity, we find that
    \be
    \|{(\Phi_g^t)}^\star f -f+t\hat fg\|_\varrho\le\frac{e^{2\|tg\|_\varrho}-2\|tg\|_\varrho-1}{2\|g\|_\varrho}(2\|f\|_\varrho\|g\|_\varrho+\|D_\omega g\|_\varrho).
    \ee

The first and second bound in~\eqref{BasicFlowBound} follow from the last two inequalities.  \qed

We start with our first step which is to solve the
equation~\eqref{ElimAppEqu}.

\begin{proposition}\label{ElimFirst}
If $f\in\FF_\varrho(\gg)\,$, where $\varrho>0$,
satisfies
    \bel{ElimFirstCond}
\|f-{u}\|_\varrho\le{\textstyle{1\over 4}}\zeta \,,\qquad \|{\Iminus}
f\|_\varrho\le{\textstyle{1\over 12}}\zeta \,,
    \ee
then the equation~\eqref{ElimAppEqu} has a unique solution
$g\in{\Iminus}\FF'_\varrho(\gg)\,$, that satisfies
$\|g\|'_\varrho\le{2\over \zeta}\|{\Iminus} f\|_\varrho\,$.
Moreover, $(\Phi_g^1)^\star f$ belongs to $\FF_{\varrho}(\gg)$ and
satisfies
    \beal{ElimFirstBound}
\bigl\|(\Phi_g^1)^\star f-f\bigr\|_{\varrho} &\le&{4\over
\zeta }(2\|f\|_\varrho+1)\|{\Iminus} f\|_\varrho \,,\nonumber\\
 \bigl\|(\Phi_g^1)^\star
f-f+\hat fg\bigr\|_{\varrho} &\le&{4\over\zeta^2}(2\|f\|_\varrho+1)\|{\Iminus}
f\|_\varrho^2\,.
    \eea
\end{proposition}

\proof Let $\tilde f=f-{u}$. We can rewrite the equation~\eqref{ElimAppEqu} in the form
    \bel{ElimAppEqu2}
    \Iminus(\Ii-\ad_{\tilde f}\hat {u}^{-1})\hat u g=\Iminus f.
    \ee

For every $g\in\FF_\varrho(\gg)\,$, using the first condition in~\eqref{ElimFirstCond}, we
have that
    \bel{XminusKhat}
\|\ad_{\tilde f}g\|_\varrho \le{2}\|g\|_\varrho\|\tilde f\|_\varrho
\le{\zeta\over 2}\|g\|_\varrho\,.
    \ee
Also, for every $g\in\Iminus\FF_\varrho(\gg)$,
    \bel{}
\|\Iminus\ad_{\tilde f}\hat {u}^{-1}g\|_\varrho\le \frac{\zeta}{2}
\|\hat {u}^{-1} g\|_\varrho\le\frac{1}{2}\|g\|_\varrho.
    \ee
Thus, the operator $\Iminus\ad_{\tilde f}\hat {u}^{-1}$ is
bounded on $\Iminus\FF_\varrho(\gg)$, with the operator norm smaller than or
equal to $1/2$. The equation~\eqref{ElimAppEqu2} can then be solved by inverting the operator $(\Ii-\ad_{\tilde f}\hat {u}^{-1})\Iminus$ by means of Neumann series. The  solution  $g\in\Iminus\FF'_\varrho(\gg)$ is
unique and satisfies the desired bound. The second bound in~\eqref{ElimFirstCond}, now allows us to apply
\clmp(FieldFlowBound) and the bounds~\eqref{ElimFirstBound} follow directly from it, by setting $\eps=(2/\zeta)\|{\Iminus}
f\|_\varrho$, $s=1$ and $t=1$. \qed

Our next step is to iterate the map $f\mapsto(\Phi_g^1)^\star f$
described in Proposition~\ref{ElimFirst}. We start with  $f=f_0$ and set
    \bel{Xnplusone}
f_{n+1}=(\Phi_{g_n}^1)^\star f_n\,,\qquad
{\Iminus}(f_n-\hat f_n g_n)=0\,,
    \ee
for all non-negative integer $n$. The expectation is that the maps $U_n$ defined by
    \bel{UnDef}
U_n=\Phi_{g_0}^1\Phi_{g_1}^1\ldots\Phi_{g_{n-1}}^1,
    \ee
for $n\in\Nn$ and $U_0=\id$, converge to a solution $U_f$ of the equation~\eqref{ElimCondP},
as $n$ tends to infinity. This leads to the main result of this
section. To prove it, the following estimate will be needed.

\begin{proposition}\label{YtBound} If $\varrho,t>0$ and $g\in\FF_\varrho'$ is given, then $\Phi_g^t-\id\in\FF_\varrho'$ and
    \bel{YBound}
\|\Phi_g^t-\id\|_{\varrho}' \le e^{\|tg\|_\varrho'}-1\,.
    \ee
\end{proposition}

\proof The claim follows directly from $\Phi_{g}^t(x)=e^{g(x)t}$, using the Banach algebra property of $\FF_\varrho'$. \qed

Let $\varrho>0$, and let us choose
$R\ge\|{u}\|_\varrho+\zeta/4+1/2$ and $\eps> 0$, subject to the constraints
    \bel{EpsCond}
\eps\le 2^{-4}\zeta \,,\qquad \eps\le 2^{-6}\zeta^2R^{-1}\,.
    \ee

In the following, we state and prove a normal form theorem that is
the main result of this section.

\begin{theorem}\label{ElimFull} {\rm\bf }
If $f\in\FF_\rho(\gg)$ such that
    \bel{ElimFullCond}
\|f-u\|_\varrho\le 2^{-3}\zeta \,,\qquad \|\Iminus
f\|_\varrho\le\eps\,,
    \ee
with $\eps>0$ satisfying conditions \equ(EpsCond), then there exists
an analytic change of coordinates $U_f: D_{\varrho}\to\G\,$, such that $U_f^\star f$ belongs to
$\FF_{\varrho}(\gg)$ and satisfies equation \equ(ElimCondP). The map
$f\mapsto U_f$ is continuous in the region defined by
inequalities \equ(ElimFullCond), analytic in the interior of this
region, and satisfies the bounds
    \beal{ElimFullBound}
 \|U_f-\id\|_{\varrho}' &\le&{3\over\zeta}\|\Iminus f\|_\varrho\,,\nonumber\\
\|U_f^\star f-f\|_{\varrho} &\le&
\frac{64R}{7\zeta}\|\Iminus f\|_\varrho\,,\\
\bigl\|U_f^\star f-f+\hat f g\bigr\|_{\varrho}
&\le&\Bigl(\frac{64R}{7\zeta}+1\Bigr){8R\over\zeta^2}
\|\Iminus f\|_\varrho^2\,.\nonumber
    \eea
Here, $g\in \Iminus\FF'_\varrho(\gg)$ is defined by \equ(ElimAppEqu) and
satisfies the bound $\|g\|'_\varrho\le{2\over\zeta}\|\Iminus
f\|_\varrho\,$. If $f$ is real matrix valued when restricted to $\Rr^d$, so is $U_f$.
\end{theorem}

\proof Our first goal is to prove that \eqref{Xnplusone} defines a
sequence of maps $f_m\in\FF_{\varrho}\,$, satisfying
    \bel{EFindhyp}
\|f_m-f_{m-1}\|_{\varrho}\le2^{-m-3}\zeta \,,\qquad \|{\Iminus}
f_m\|_{\varrho}\le 8^{-m}\eps\,.
    \ee
We proceed by induction. If we define $f_{-1}=u$ and $f_0=f$, then these bounds hold for
$m=0$ by~\eqref{ElimFullCond}. Assume now that~\eqref{EFindhyp}
holds for all integer $m$ with $0\le m\le n$. Then, by summing up the
bounds on $f_m-f_{m-1}$ for $0\le m\le n$, we obtain the first of the
inequalities
    \bel{ElimCondn}
\|f_n-{u}\|_{\varrho}\le{1\over 4}\zeta \,,\qquad \|{\Iminus}
f_n\|_{\varrho}\le 4^{-n-2}\zeta \,.
    \ee
The second inequality follows from~\eqref{EFindhyp}, by substituting
the first bound in~\eqref{EpsCond} on $\eps$. Thus, \clmp(ElimFirst)
guarantees a unique solution to~\eqref{Xnplusone}, and yields the
bounds
    \bel{EFindnext}
\|f_{n+1}\!-\!f_n\|_{\varrho} \le {1\over
\zeta}8^{-n+1}R\eps\,,\qquad \|{\Iminus} f_{n+1}\|_{\varrho} \le
{1\over\zeta^2}8^{-2n+1}R\eps^2\,.
    \ee
Here, we have also used that $2\|f_n\|_{\varrho}+1\le 2R$, which follows
from the first inequality in~\eqref{ElimCondn}. By using the second
condition in~\eqref{EpsCond}, we now obtain~\eqref{EFindhyp} for
$m=n+1$ from the bounds~\eqref{EFindnext}.

Next, consider the functions $\varphi_j=\Phi_{g_j}^1-\id$. By
\clmp(YtBound) and \clmp(ElimFirst),
    \bel{psikboundetc}
\|\varphi_j\|_{\varrho}'\le e^{\|g_j\|_{\varrho}'}-1 \le e^{{2\over \zeta
}\|{\Iminus} f_j\|_{\varrho}}-1\,.
    \ee
Using \eqref{EFindhyp} we have the bounds
    \bel{myflowbounds}
\|\varphi_j\|_{\varrho}'\le e^{{2\eps\over \zeta }8^{-j}}-1\,.
    \ee
Define the maps
$U_{m,n}=\Phi_{g_m}^1\Phi_{g_{m+1}}^1\ldots\Phi_{g_{n-1}}^1$
in $\FF_{\varrho}'(\G)$. Here, and in what follows, it is assumed
that $0\le m<n$. Setting $U_{j,j}=\id$, and using the bounds
\eqref{myflowbounds} we obtain
    \bel{UnkDiffBound}
\|U_n-U_m\|_{\varrho}' =\left\|\sum_{j=m}^{n-1}\varphi_j
U_{j+1,n}\right\|_{\varrho}' \le e^{{16\eps\over 7\zeta}8^{-m}}-1\,.
    \ee
This shows that the sequence $n\mapsto U_n$ converges uniformly in
$\FF_{\varrho}'(\G)$ to a limit $U_f\in \FF_{\varrho}'(\G)$.
The analyticity of the maps $U_n$ and the uniform convergence of the
sequence $n\mapsto U_n$ implies the analyticity of the map
$f\mapsto U_f$. Clearly, $f_n$ converge to $U_f^\star (f)$ in
$\FF_\varrho(\gg)$. The first inequality in~\eqref{ElimFullBound} follows
from the estimate \eqref{UnkDiffBound}, if we
set $m=0$ and $\eps=\|{\Iminus} f\|_\varrho\,$, and use the first bound in~\eqref{EpsCond}. The second inequality
in~\eqref{ElimFullBound} is now obtained by summation over $n$ of the first bound
in~\eqref{EFindnext}.

Since $U_f^\star f=U_{f_1}^\star f_1$ with $f_1=(\Phi_g^1)^\star
f$, we have
    \bel{EFBTwo}
U_f^\star f-f+\hat f g =\bigl(U_{f_1}^\star f_1-f_1\bigr)
+\bigl((\Phi_g^1)^\star f-f+\hat f g\bigr)\,.
    \ee
The norm of the term in the first parentheses of~\eqref{EFBTwo} can
be estimated in the same way as the term $\|U_f^\star
f-f\|_{\varrho}$, yielding the bound
    \bel{EFBThree}
\bigl\|U_{f_1}^\star f_1-f_1\bigr\|_{\varrho} \le
\frac{64R}{7\zeta}\|\Iminus f_1\|_\varrho\,.
    \ee
Since $\|{\Iminus} f_1\|_{\varrho}\le\|(\Phi_g^1)^\star
f-f+\hat f g\|_{\varrho}\,$, the third bound in~\eqref{ElimFullBound}
now follows from the second inequality in~\eqref{ElimFirstBound}.
 \qed

\section*{Acknowledgements}
I would like to thank Professor Hans Koch and Professor Wellington
de Melo for discussions. I am also grateful to the staff of IMPA,
where this work has been done, for their hospitality. This work has
been financially supported by CNPq.
\begin{thebibliography}{99}


\bibitem{Amor} S. H. Amor, {\it Sur la densit\' e d'\' etat de l'operateur de Schr\" odinger quasi-p\' eriodique unidimensionnel},
C.R. Acad. Sci. Paris, Ser. I {\bf 343}  (2006), 423--426.

\bibitem{AK} A. Avila and R. Krikorian, {\it Reducibility or nonuniform hyperbolity of quasiperiodic Schr\" odinger cocycles},
 Ann. Math. {\bf 164} 4 (2006), 911--940.

%\bibitem{AJ1} A. Avila and S. Jitomirskaya, {\it The Ten Martini Problem},
% Ann. Math., to appear.

\bibitem{AJ2} A. Avila and S. Jitomirskaya, {\it Almost localization and almost reducibility},
Journal of the European Math. Soc., to appear (2009).

\bibitem{Bogoljubov} N. N.~Bogoljubov, Ju. A. Mitropolitskii and A. M. Samoilenko, {\it Methods of Accelarated Convergence in Nonlinear Mechanics}, Springer Verlag (1976).

\bibitem{Bourgain} J. Bourgain, {\it On the spectrum of lattice Schr\" odinger operators with deterministic potential},
Dedicated to the memory of Tom Wolff, J. Anal. Math. {\bf 88} (2002), 221--254.

\bibitem{Brjuno1} A. D.~Brjuno, {\it Analytic form of
differential equations I}, Trudy Moskov. Mat. Obshch. {\bf 25}
(1971), 119-262; Trans. Moscow Math. Soc. {\bf 25} (1973),
131--288.

\bibitem{Brjuno2} A. D.~Brjuno, {\it Analytic form of differential
equations II},  Trudy Moskov. Mat. Obshch. {\bf 26} (1972),
199-239; Trans. Moscow Math. Soc. {\bf 26} (1974), 199--239.

\bibitem{DS} E. I. Dinaburg and Ja. G. Sinai, {\it The one-dimensional Schr\" odinger equation with quasiperiodic potential},
 Funkcional. Anal. i Prilo\v zen. {\bf 9} 4 (1975), 8--21.

\bibitem{Eliasson} L. H. Eliasson, {\it Floquet Solutions for the 1-dimensional Quasi-Periodic Schr\" odinger equation},
Comm. Math. Phys. {\bf 146} (1992), 447--482.

\bibitem{Eliasson2} L. H. Eliasson, {\it Linear quasi-periodic systems - reducibility and almost reducibility},
XIVth International Congress on Mathematical Physics (2006), 195--205.

\bibitem{Gallavotti} G. Gallavotti, {\it Twistless KAM tori}, Comm. Math. Phys. {\bf 164} 1
(1994), 145--156.

\bibitem{GG} G. Gallavotti and G. Gentile, {\it Degenerate elliptic resonances}, Comm. Math. Phys. {\bf 257}
2 (2005), 319--362.

\bibitem{Gentile} G. Gentile, {\it Resummation of Perturbation Series and Reducibility for Bryuno
Skew-Product Flows}, J. Stat. Phys. {\bf 125} 2 (2006), 317--357.

\bibitem{HP} E. Hille and R. S. Phillips, {\it Functional analysis and semi-groups},
AMS Colloquium Publications {\bf 31} (1974).

\bibitem{HouYou} X. Hou and J. You, {\it The rigidity of reducibility of cocycles on $\SO (N,\Rr)$},
Nonlinearity {\bf 21} (2008), 2317--2330.

\bibitem{Katok} S. B. Katok, {\it Linear Extensions of Dynamical Systems and the Reducibility Problem}, Matematicheskie Zametki {\bf 8}
4 (1970), 451--462.

\bibitem{KLM} K. Khanin, J. Lopes Dias, J. Marklof, {\it Multidimensional continued fractions, dynamic renormalization and KAM theory}, Commun. Math. Phys. {\bf 270} (2007), 197--231.

\bibitem{KK1} H. Koch and S. Koci\' c, {\it Renormalization of vector fields and Diophantine invariant tori}, {
Ergod. Theor. Dynam. Sys.} {\bf 28} 5 (2008), 1559--1585.

\bibitem{KK2} H. Koch and S. Koci\' c, {\it A renormalization group aproach to quasiperiodic motion with Brjuno frequencies}, preprint: mp\_arc 06-269
(2006), Ergod. Theor. Dynam. Sys. to appear.

\bibitem{KLD} H.~Koch and J.~Lopes Dias, {\it Renormalization of
Diophantine skew flows, with applications to the reducibility
problem}, Discrete Cont. Dyn. Sys. {\bf 21} (2008), 477--500.

\bibitem{Kocic} S. Koci\' c, {\it Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori},  Nonlinearity {\bf 18} (2005), 1--32.

\bibitem{Krikorian1} R. Krikorian, {\it R\' educibilit\' e presque partout des flots fibr\' es quasi-p\' eriodiques \`a valeurs dans des groupes compacts},
Ann. Sci. de l'\'E.N.S. $4^e$ s\' erie {\bf 32} 2 (1999), 187--240.

\bibitem{Krikorian2} R. Krikorian, {\it R\' educibilit\' e des syst\` emes produits-crois\' es \`a valeurs dans des groupes compacts},
Ast\' erisque {\bf 259} (1999), 1--216.

\bibitem{Krikorian3} R. Krikorian, {\it $C^0$-densit\' e globale des syst\` emes produits-crois\' es sur le cercle r\' eductibles},
Ergod. Theor. Dyn. Sys. {\bf 19} (1999), 61--100.

\bibitem{Krikorian4} R. Krikorian, {\it Global density of reducible quasi-periodic cocycles on $\Tt^1\times\SU(2)$},
Ann. of Math. {\bf 154} 2 (2001), 269--326.

\bibitem{Lagarias} J. C. Lagarias, {\it Geodesic multidimensional continued fractions}, Proc. London Math. Soc., {\bf 3} 69 (1994), 464--488.

\bibitem{LopesDias06} J. Lopes Dias, {\it A normal form theorem
for Brjuno skew-systems through renormalization}, J. Differential
Equations {\bf 230} (2006), 1--23.

\bibitem{LopesDias08} J. Lopes Dias, {\it Local conjugacy classes for analytic torus flows}, J. Differential
Equations {\bf 245} (2008), 468--489.

\bibitem{Moser} J. Moser, {\it Convergent Series Expansions for Quasi-Periodic Motions}, Mathematische Annalen {\bf 169} (1967), 136--176.

\bibitem{PS} J. Puig and C. Sim\' o, {\it Analytic families of reducible linear quasi-periodic differential equations},
Ergod. Th. \and Dynam. Sys. {\bf 26} (2006), 481--524.

\bibitem{Rychlik} M. Rychlik, {\it Renormalization of cocycles and linear ODE with almost-periodic coefficients}, Invent. Math. {\bf 110} (1992), 173--206.

\end{thebibliography}
\end{document}
---------------0908071014531--
