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} 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%\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 $\|X-K\|_\varrho0$). 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 $a_{k+1}/20$. 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 $\|g\|_0=\|f\|_0\le r_n\|f\|_n'\delta>0$. 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