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\begin{document}
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\title[Linearizations of Complex Dynamical Systems]
{Limit at Resonances of Linearizations=20
\\ of Some Complex Analytic Dynamical Systems}

\author{Alberto Berretti}
\address{Alberto Berretti\\ Dipartimento di Matematica\\ II Universit\`{a} =
di
Roma (Tor Vergata)\\ Via della Ricerca Scientifica, 00133 Roma, Italy and I=
NFN,
Sez. Tor Vergata}
\email{{\tt berretti@roma2.infn.it}}

\author{Stefano Marmi}
\address{Stefano Marmi\\ Dipartimento di Matematica ``U. Dini''\\ Universit=
\`{a}
di Firenze\\ Viale Morgagni 57a, 50134 Firenze, Italy and INFN, Sez. Firenz=
e}
\email{{\tt marmi@fi.infn.it}}

\begin{abstract}

We consider the behaviour near resonances of linearizations of=20
germs of holomorphic diffeomorphisms of $(\fC,0)$ and of=20
the semi--standard map.=20

We prove that there exists suitable scalings under which the=20
linearizations converge uniformly to some analytic function as the=20
multiplier, or rotation number, tends non--tangentially to a=20
resonance.  This limit functions are computed analytically in the case=20
of germs and are related to the formal classifications of germs with a=20
parabolic fixed point.  In the semi--standard map case we give a=20
heuristic argument to compute the limit.

\end{abstract}

\maketitle

\section{Introduction}

Problems in which series with small denominators appear are quite=20
frequent in the theory of dynamical systems: they typically appear=20
when one tries to conjugate the dynamics of the system to the one of=20
the corresponding linearized problem.  As such, they appear in the=20
theory of Hamiltonian systems, in the theory of area--preserving twist=20
maps, in the theory of circle maps, in the costruction of normal forms=20
of differential equations in the neighborhood of a fixed point, in the=20
theory of iterated maps of $\fC^{n}$, and in many other, often=20
unexpected, cases.  We refer the reader to the relevant, huge=20
literature on the topic for comprehensive references and background.

Small denominators have the effect that the convergence of the series=20
which expresses the linearization is problematic, and the=20
linearization itself has a complicated analytic structure as a=20
function of the dynamical variables, of the perturbative parameter,=20
and of the characteristic frequencies of the motion we are trying to=20
conjugate to linear. In some specific cases, this analytic structure=20
has been the subject of numerical studies (\eg \cite{BM2} and=20
references quoted therein, \cite{LT}, \cite{BTT}).=20

In this article we prove some simple results for the linearization of=20
germs of holomorphic diffeomorphisms of $(\fC,0)$ (see \eg=20
\cite{Yoccoz}) and for the so called ``semi-standard map''=20
\cite{GreenePercival}, \cite{Percival}.

In particular, for each of the systems we take into account, we prove=20
that the linearizations have well--defined limits as the multiplier,=20
or rotation number, tends to a rational value $p/q$, \textit{provided=20
the dynamical variable is suitably rescaled}.  In the case of germs,=20
we actually compute this limit: this generalizes a result contained in=20
\cite{Yoccoz}, as this reference computes the limit only for the case=20
of quadratic polynomials.  In the case of the semi-standard map, we=20
provide a heuristic argument for the limit function, obtaining a=20
result which fully explains the observed numerical evidence.  We note=20
also that our results explain the numerical results of \cite{BTT}.

Of course, it would be quite interesting to generalize these results=20
to more interesting, and complicated, systems like the standard map:=20
\cite{BM2} contains some partial results in this direction.=20

\section{Germs of holomorphic diffeomorphisms of $(\fC,0)$}

Let $G$ denote the group of germs of holomorphic diffeomorphisms=20
of $(\fC,0)$ with a fixed point at $0$. An important problem=20
in complex dynamics is to describe the conjugacy classes of this group.=20
Since $0$ is a fixed point, one has some natural conjugacy invariants like=
=20
the multiplier $\lambda=3Df'(0)$ of the germ $f$ at $0$ and=20
the holomorphic index:=20
$$
i(f,0) =3D \frac{1}{2\pi i}\oint\frac{\der z}{z - f(z)},
$$
where we integrate on a small loop in the positive direction around=20
$0$. If $\lambda \neq 1$, $0$ is a simple fixed point and one clearly has:
$$
i(f,0) =3D \frac{1}{1-\lambda}.
$$
Let $G_{\lambda}$ denote the set of $f \in G$ such that=20
$f'(0)=3D\lambda$. $f$ is \emph{linearizable} if it belongs to the=20
same conjugacy class of the rotation $R_{\lambda}(z)=3D\lambda z$.=20
If  $|\lambda | \neq 1$, by the Poincar\'e--K\oe{}nigs linearization theore=
m=20
\cite{Koenigs}, \cite{Poincare},
one knows that there exists a unique germ $h_{\lambda,f} \in G_{1}$ such th=
at:
\begin{equation}
=09f \circ h_{\lambda,f}=3D h_{\lambda,f} \circ R_{\lambda}.
=09\label{eq:conjugacy}
\end{equation}

The function $h_{\lambda,f}$ is called the \emph{linearization} of=20
$f$: attracting ($|\lambda| < 1$) and repelling ($|\lambda| > 1$)=20
fixed points of holomorphic diffeomorphisms are linearizable.  If one=20
keeps $f-R_{\lambda}$ fixed as $\lambda$ varies, the linearization=20
$h_{\lambda ,f}$ depends holomorphically on the multiplier $\lambda$. =20
If $f_{\lambda}$ is an entire function and $|\lambda| > 1$ then=20
$h_{\lambda,f}$ is an entire function.

When $|\lambda|=3D1$ the fixed point at the origin is \emph{indifferent}=20
and one must distinguish three different situations:
\begin{enumerate}
=09\item  \emph{parabolic} or \emph{resonant case}:=20
=09=09=09$\lambda=3D\Lambda=3D\exp\left(2\pi i \frac{p}{q}\right)$,
=09=09=09where $(p|q)$=3D1; \label{case:resonant}

=09\item  \emph{Brjuno's case}: $\lambda =3D \exp(2\pi i \om)$ and $\om$=20
=09=09=09is a \emph{Brjuno number} \cite{Brjuno}), \cite{MMY}: if=20
=09=09=09$\{p_{k}/q_{k}\}$ denotes the sequence of partial fractions of=20
=09=09=09the continued fraction expansion of $\om$, one requires the=20
=09=09=09convergence of the series $\sum_{k=3D0}^\infty \frac{\log=20
=09=09=09q_{k+1}}{q_{k}} < + \infty$; \label{case:brjuno}

=09\item  \emph{Cremer's case}: $\lambda =3D=09\exp(2\pi i=09\om)$
=09=09=09and=09$\om$ is \emph{not}=09a Brjuno number. \label{case:cremer}
\end{enumerate}
Actually, Cremer proved \cite{Cremer} that $G_\lambda$ is not a=20
conjugacy class if $\sup_{n}\frac{\log q_{n+1}}{q_{n}} =3D \infty$. =20
However we think that it is quite fair to give his name to the=20
complement of the Brjuno set.

In case (\ref{case:resonant}) Ecalle \cite{Ecalle} and Voronin=20
\cite{Voronin} have given a complete classification of the=20
conjugacy classes of $G$ contained in $G_{\Lambda}$; $R_{\Lambda}$=20
belongs to the class of $G_{\Lambda}$ with all the elements of order=20
$q$, but other classes are also possible.=20

In 1987 Yoccoz \cite{Yoccoz} proved that in case=20
(\ref{case:brjuno}) above $G_{\lambda}$ is a conjugacy class of $G$,=20
whereas in case (\ref{case:cremer}) it is not a conjugacy class and=20
there exists at least a non-linearizable germ $f\in G_\lambda$. A=20
remarkable example is given by the quadratic polynomial:=20
$$
P_{\lambda}(z) =3D e^{2\pi i\om} (z-z^{2})
$$=20
which is linearizable if and only if $\om$ is a Brjuno number.

An interesting property of the linearization $h_{\lambda,f}$ is that,=20
keeping $f-R_{\lambda}$ fixed as $\lambda$ varies in the unit disk=20
$\fD =3D \{ z \in \fC | |z| < 1 \}$, one can prove that it has=20
non-tangential limits at all Brjuno numbers. In what follows we shall=20
show the existence of suitable scalings under which $h_{\lambda,f}$=20
has non-tangential limits at resonances (\ie rational values of $\om$=20
or values equal to roots of unity for $\lambda$). We will also compute=20
these limits and show how they are reletad to the \emph{formal}=20
classification of conjugacy classes of $G_{\Lambda}$.=20

\subsection{The main theorem}

Let us thus consider the one-parameter family of germs $f_{\lambda}=20
\in G_{\lambda}$:
$$
f_{\lambda}(z) =3D \lambda z + \sum_{n=3D2}^{\infty}f_{n}z^{n},
$$
where we keep the coefficients $\{f_{n}\}_{n=3D2}^{\infty}$, \ie=20
$f_{\lambda} - R_{\lambda}$, fixed. We denote simply by $h_{\lambda}$=20
the corresponding linearization. We have the following theorem.

\begin{thm}
Let $V \subset \fD$ be a non-tangential sector with vertex=20
$\Lambda=3De^{2\pi i p/q}$; there exists a positive integer $k$ and a=20
complex number $A \in \fC^{*}$ such that, if we let:
\begin{equation}
=09\tilde{h}_{\lambda}(z) =3D \frac{1}{(\lambda-\Lambda)^{1/kq}}
=09=09=09=09=09=09 =09 h_{(\lambda-\Lambda)^{1/kq}\lambda}(z),
=09\label{eq:scallim}
\end{equation}
then, as $\lambda \rightarrow \Lambda$ in $V$, $\tilde{h}_{\lambda}$=20
tends uniformly on some small open disk around $0$ to the function:
\begin{equation}
=09\tilde{h}_{\Lambda}(z) =3D z \left(
=09=09=09=09=09=09=09=09=09 1 - \frac{A}{q\Lambda^{q-1}}z^{kq}
=09=09=09=09=09=09=09   \right)^{-1/kq}.
=09\label{eq:limitlin}
\end{equation}
\end{thm}

This theorem generalizes to germs a result of Yoccoz \cite{Yoccoz} for the=
=20
linearization of the quadratic polynomial, obtained by different=20
techniques.=20

\begin{rem}
Note that one may use any scaling $z \rightarrow=20
s((\lambda-\Lambda)^{\frac{1}{kq}}) z$ instead of $z  \rightarrow=20
(\lambda-\Lambda)^{\frac{1}{kq}} z$, provided $s$ is analytic,=20
$s(0)=3D0$ and $s'(0) \neq 0$. In this case one finds:
$$
\tilde{h}_{\Lambda} =3D z \left(
=09=09=09=09=09=09=091 - \frac{A s'(0)}{q \Lambda^{q-1}} z^{kq}
=09=09=09=09=09=09\right)^{-1/kq}.
$$
\end{rem}

Before proving this theorem we need to recall some properties of=20
parabolic fixed points.=20

\subsection{Parabolic Fixed Points}

It is wel known \cite{Beardon} that if $f_{\Lambda}$ is a rational=20
function of degree $d \geq 2$ then $0$ belongs to the Julia set of=20
$f_{\Lambda}$. In a neighborhood of $0$ the dynamics is described in=20
terms of attracting and repelling petals.=20

We actually need a much weaker result, which holds for all germs=20
$f_{\Lambda} \in G_{\Lambda}$.  In the following, we denote with=20
$f^{\circ q}$ the composition of the function $f$ with itself $q$=20
times.

\begin{lem}\label{lem:mainlemma}
There exists a positive integer $k \geq 1$ and a nonzero complex=20
number $A \in \fC^{*}$ such that:
\begin{equation}
=09f^{\circ q}_{\Lambda}(z) - z + A z ^{kq+1} + O(z^{kq+2}).
=09\label{eq:mainlemma}
\end{equation}
If $f_{k}=3D0$, $k=3D2,\ldots,n$, then $kq+1 \geq n$.=20
\end{lem}
\begin{proof}
This is quite easily proved by comparing the series developments of=20
both sides in the equation $f_{\Lambda} \circ f^{\circ q}_{\Lambda} =3D=20
f^{\circ q}_{\Lambda}\circ f_{\Lambda}$.=20
\end{proof}

\begin{rem}
If $f_{\lambda}$ is a polynomial of degree $d$, then, by a result of=20
Douady and Hubbard (see=20
(\cite{DH}), sect. XX, prop. XX), $1 \leq k \leq d-1$.=20
\end{rem}

\begin{rem}
There exists a unique $B \in \fC^{*}$ such that $f^{\circ q}_{\Lambda}$=20
belongs to the same analytic conjugacy class of a germ $g_{B}$ of the=20
form:
$$
g_{B}(z) =3D z + z^{kq+1} + B z^{2kq+1} + O(z^{2kq+2}).
$$
Moreover $f^{\circ q}_{\Lambda}$ is \emph{formally} and=20
\emph{topologically} conjugated to the polynomial $z+z^{kq+1}+Bz^{2kq+1}$.=
=20

Indeed, the existence of an analytic conjugation of $f^{\circ q}_{\Lambda}$=
=20
to $g_{B}$ is proved by induction over a finite number of steps,
starting with $f_{\lambda}$ and iterating conjugations by=20
polynomials $\phi_{j}(z)=3Dz+\beta_{j}z^{kq+j+1}$, with=20
$j=3D1,\ldots,kq-1$ and suitably chosen $\beta_{j}$. The formal=20
conjugation is proved by iterating conjugations by polynomials=20
$\phi_{l}(z)=3Dz+\beta_{l}z^{l}$, with $l \geq 2k+2$ and suitably=20
chosen $\beta_{l}$.=20
\end{rem}

\begin{rem}
Let:
$$
f^{\circ q}_{\Lambda} =3D z + A z^{kq+1} + \sum_{j=3Dkq+2}^{\infty}A_{j}z^{=
j}.
$$
Then, since $i(f^{\circ q}_{\Lambda},0) =3D i(g_{B},0) =3D B$, we have:
$$
B =3D -\frac{1}{A}\sum_{j=3D1}^{kq}\frac{(-1)^{j}}{A^{j}}
=09\sum_{n_{1}+\ldots+n_{j}=3Dkq}A_{n_{1}+kq+1} \cdots A_{n_{j}+kq+1}.
$$
\end{rem}

\subsection{Germs with Almost Resonant Linear Part}

We now consider the one-parameter family of germs $f_{\lambda} \in=20
G_{\lambda}$, with $\lambda$ close to $\Lambda$, and their iterates=20
$f^{\circ q}_{\lambda}$. Without any loss of generality, we can assume the=
=20
following normalization condition: there exists $r > 0$ and $\rho > 0$=20
such that for all $\lambda \in \fC$, $|\lambda - \Lambda| < \rho$:

\begin{enumerate}
=09\item  $f_{\lambda}, \; f^{\circ q}_{\lambda}: \fD_{r} \mapsto \fC$=20
=09injectively, where $\fD_{r} =3D \{ z \in \fC | |z| < r \}$;

=09\item  $f^{\circ q}_{\lambda} =3D \lambda^{q}z +=20
=09\sum_{j=3D2}^{\infty}f^{(q)}_{j}(\lambda)z^{j}$ and=20
=09$|f^{(q)}_{j}(\lambda)| \leq r^{1-j}$.
\end{enumerate}

\begin{rem}\label{rem:pollambda}
$f^{(q)}_{j}(\lambda)$ is a polynomial in $\lambda$, with a zero at=20
$\Lambda$ if $2 \leq j \leq kq$. Thus we can assume that there exists=20
$c_{1}>0$ such that for all $|\lambda - \Lambda|<\rho$:
$$
|f^{(q)}_{j}(\lambda)| \leq c_{1}|\lambda - \Lambda|r^{1-j}
$$
for all $j=3D2,\ldots,kq$.
\end{rem}

Let $\alpha \in (0,\pi/2)$ and let $C(\Lambda,\alpha)$ denote the=20
complex cone with vertex at $\Lambda$ defined by:
\begin{equation}
=09C(\Lambda,\alpha) =3D \{\lambda | \arg\Lambda+\pi-\alpha <
=09=09=09=09=09=09\arg(\lambda - \Lambda) <
=09=09=09=09=09=09\arg\lambda+\pi+\alpha.
=09\label{eq:sector}
\end{equation}
Let $C(\Lambda,\alpha,\rho) =3D C(\Lambda,\alpha) \cap=20
\{|\lambda-\Lambda|<\rho\}$, where $\rho>0$. We then have the=20
following.

\begin{lem}\label{lem:estimlambda}
For all $\alpha \in (0,\pi/2)$ and for all sufficiently small $\rho>0$=20
there exists $c_{2}>c_{1}>0$ such that, if $\lambda \in=20
C(\Lambda,\alpha,\rho)$, then:
\begin{equation}
=09|\lambda^{jq}-\lambda^{q}| \geq c_{2}|\lambda-\Lambda|
=09\label{eq:estimlambda}
\end{equation}
for all $j \geq 2$.
\end{lem}
\begin{proof}
If $j=3D2$ one has:
$$
\frac{|\lambda^{2q}-\lambda^{q}|}{|\lambda-\Lambda|} =3D
|\lambda^{q}|\frac{|\lambda^{q}-1|}{|\lambda-\Lambda|} \geq
c_{3}(1-\rho)^{q}|q\Lambda^{q-1}| \geq c_{2};
$$
if $j>2$ instead:
$$
\frac{|\lambda^{jq}-\lambda^{q}|}{|\lambda-\Lambda|} =3D
|\lambda^{q}|\frac{|\lambda^{q}-1|}{|\lambda-\Lambda|}
|1+\lambda^{q}+\ldots+\lambda^{(j-2)q}| \geq
c_{4}(1-\rho)^{q}|q\Lambda^{q-1}|(j-2) \geq c_{2},
$$
where $c_{3}$ and $c_{4}$ are some positive constants.
\end{proof}
Note that one needs the assumption that $\lambda \rightarrow \Lambda$=20
non-tangentially in order to control the sum=20
$|1+\lambda^{q}+\ldots+\lambda^{(j-2)q}|$ which is clearly not=20
bounded below if $|\lambda|=3D1$.=20

The power series expansion:
\begin{equation}
=09h_{\lambda}(z) =3D z + \sum_{j=3D2}^{\infty}h_{j}(\lambda)z^{j}
=09\label{eq:siegelseries}
\end{equation}
for the linearization $h_{\lambda}$ of $f_{\lambda}$ can be=20
recursively determined by means of (\ref{eq:conjugacy}). However one=20
also has:
\begin{equation}
=09f^{\circ q}_{\lambda} \circ h_{\lambda} =3D h_{\lambda} \circ f^{\circ q=
}_{\lambda},
=09\label{eq:conjugacyqtimes}
\end{equation}
thus one gets:
\begin{equation}
=09h_{j}(\lambda) =3D \frac{1}{\lambda^{jq}-\lambda^{q}}
=09=09\sum_{i=3D2}^{j} f^{\circ q}_{j}=20
=09=09\sum_{j_{1}+\ldots+j_{i}=3Dj}=20
=09=09h_{j_{1}}(\lambda) \cdots h_{j_{i}}(\lambda),
=09\label{eq:siegelrecurqtimes}
\end{equation}
for all $j \geq 2$.

Now let:
\begin{eqnarray*}
=09\sigma_{1} & =3D & 1 \\
=09\sigma_{j} & =3D & \sum_{i=3D2}^{j}\sum_{j_{1}+\ldots+j_{i}=3Dj}
=09\sigma_{j_{1}}\cdots\sigma_{j_{i}} \quad \text{for all } j \geq 2.
\end{eqnarray*}

\begin{lem}
There exists a positive constant $c_{5}$ such that:
\begin{equation}
=09\sigma_{j} \leq c_{5} (3-2\sqrt{2})^{1-j},
=09\label{eq:lemmasigma}
\end{equation}
for all $j \geq 1$.
\end{lem}

\begin{proof}
The generating function=20
$\sigma(z)=3D\sum_{i=3D1}^{\infty}\sigma_{i}z^{i}$ satisfies the=20
functional equation:
$$
\sigma(z)=3Dz+\frac{\sigma(z)^{2}}{1-\sigma(z)},
$$
so that:
$$
\sigma(z)=3D\frac{1+z-\sqrt{1-6z+z^{2}}}{4}
$$
is analytic in the disk $|z|<3-2\sqrt{2}$ and bounded and continuous=20
on its closure; (\ref{eq:lemmasigma}) then follows by Cauchy's=20
estimate.=20
\end{proof}

\begin{lem}\label{lem:lemmah}
Let $\rho>0$ be sufficiently small. For all $\lambda \in=20
C(\Lambda,\alpha,\rho)$ and for all $j \geq 1$ one has:
\begin{equation}
=09|h_{j}(\lambda)| \leq \sigma_{j}r^{j-1}\left(
=09=09=09=09=09=09\frac{1}{c_{2}|\lambda-\Lambda|}
=09=09\right)^{\lfloor\frac{j-1}{kq}\rfloor},
=09\label{eq:lemmah}
\end{equation}
where $\lfloor x \rfloor$ denotes the integer part of $x$.
\end{lem}

\begin{proof}
We proceed by induction: (\ref{eq:lemmah}) is clearly true if $j=3D1$.=20
Assume it holds for all $j=3D1,\ldots,n-1$. If $n \leq kq$ then from=20
remark \ref{rem:pollambda} and lemma \ref{lem:estimlambda} it follows=20
that:
\begin{multline*}
|h_{n}(\lambda)| \leq \frac{1}{c_{2}|\lambda-\Lambda|}
=09=09=09\sum_{i=3D2}^{n}c_{1}|\lambda-\Lambda|r^{i-1} \\
=09\sum_{n_{1}+\ldots+n_{i}=3Dn}\sigma_{n_{1}}\cdots\sigma_{n_{i}}
=09=09=09r^{i-(n_{1}+\ldots+n_{i}} \leq \sigma_{n}r^{1-n}.
\end{multline*}
If $n>kq$ then we split the previous sum into two parts:=20
$2 \leq i \leq kq$ and $i > kq$. For the first part we remark that:
$$
\lfloor\frac{n_{1}-1}{kq}\rfloor + \ldots +
\lfloor\frac{n_{i}-1}{kq}\rfloor \leq
\lfloor\frac{n-i}{kq}\rfloor \leq
\lfloor\frac{n-1}{kq}\rfloor.
$$
The induction assumption applied to the second part gives:
$$
=09\frac{1}{c_{2}|\lambda-\Lambda|}\sum_{i=3Dkq+1}^{n}r^{1-i}
=09=09\sum_{n_{1}+\ldots+n_{i}=3Dn}\sigma_{n_{1}}\cdots\sigma_{n_{i}}
=09=09r^{i-(n_{1}+\ldots+n_{i})}
=09=09\left(
=09=09=09\frac{1}{c_{2}|\lambda-\Lambda|}
=09=09\right)^{\lfloor\frac{n-i}{kq}\rfloor}.
$$
Since $i>kq+1$ then=20
$\lfloor\frac{n-i}{kq}\rfloor\leq\lfloor\frac{n-1}{kq}\rfloor-1$, and=20
the proof is complete.
\end{proof}

As lemma \ref{lem:lemmah} clearly shows, as $\lambda \rightarrow=20
\Lambda$ non tangentially, the radius of convergence of $h_{\lambda}$=20
tends to $0$ as $|\lambda-\Lambda|^{1/kq}$. The dynamical reason for=20
this fact is given by the following simple lemma.

\begin{lem}
Suppose that $f_{2} \neq 0$. Then there exists $\eps_{1}>0$ and=20
$\rho_{1}>0$ such that $f^{\circ q}_{\lambda}$ has exactly $kq$ fixed point=
s=20
in $\fD^{*}_{\eps_{1}}$ for all $\lambda\in\fC$ such that=20
$0<|\lambda-\Lambda|<\rho_{1}$. These $kq$ fixed points form $k$=20
cycles of period $q$ for $f_{\lambda}$.=20
\end{lem}

\begin{proof}
Note that $0$ is a simple fixed point for $f^{i}_{\lambda}$,=20
$i=3D1,\ldots,q-1$, whereas it is a fixed point of multiplicity $kq+1$=20
for $f^{\circ q}_{\lambda}$. The zeros of an analytic function are=20
isolated, and $f_{\lambda}$ depends analytically on $\lambda$, so=20
there exist $\eps_{1}>0$ and $\rho_{1}>0$ such that=20
$|f^{i}_{\lambda}(z)-z| \geq c_{5} > 0$ for all $z$ such that=20
$|z|=3D\eps_{1}$ or $|z|=3D2\eps_{1}$ and for all $\lambda$ such that=20
$\lambda-\Lambda|<\rho_{1}$. By the argument principle the number=20
$N(f^{i}_{\lambda},\delta)$ of fixed points of $f^{i}_{\lambda}$=20
contained in the disk $|z|<\delta$ is given by:
$$
N(f^{i}_{\lambda},\delta) =3D \frac{1}{2\pi i}
=09\oint_{|z|=3D\delta}\der z=20
=09\frac{(f^{i}_{\lambda})'(z)-1}{f^{i}_{\lambda}(z)-z}.
$$
$N(f^{i}_{\lambda},\eps_{1})$ and $N(f^{i}_{\lambda},2\eps_{1})$ are=20
continuous functions of $\lambda$ with integer values, so they are=20
necessarily constant for $|\lambda-\Lambda|<\rho_{1}$ and $1 \leq i=20
\leq q$. Thus $N(f^{\circ q}_{\lambda},\eps_{1}) =3D=20
N(f^{\circ q}_{\lambda},2\eps_{1}) =3D N(f^{\circ q}_{\Lambda},2\eps_{1}) =
=3D=20
N(f^{\circ q}_{\Lambda},\eps_{1}) =3D kq+1$. Therefore $f^{\circ q}_{\lambd=
a}$ has=20
exactly $kq$ fixed points in $0<|z|<\eps_{1}$ and no other fixed=20
point in the annulus $\eps_{1}<|z|<2\eps_{1}$. On the other hand one=20
can choose $\eps_{1}$ and $\rho_{1}$ such that $|f_{\lambda}(z)| \leq=20
2\eps_{1}$ if $|z|<\eps_{1}$ and $|\lambda-\Lambda|<\rho_{1}$, thus=20
the $kq$ fixed points must form $k$ cycles of period $q$ for=20
$f_{\lambda}$.=20
\end{proof}

\subsection{Proof of the Main Theorem}

\begin{proof}
Since $\lambda \rightarrow \Lambda$ non--tangentially, we can assume=20
that $\lambda \in C(\Lambda,\alpha,\rho)$ and $\rho>0$, sufficiently=20
small.=20

If $\tilde{h}_{j}(\lambda)$ denotes the $j$-th coefficient of the=20
power series expansion of $\tilde{h}_{\lambda}$, then:
$$
|\tilde{h}_{J}(\lambda)| \leq \sigma_{j}r^{1-j}
=09(\lambda-\Lambda)^{\frac{j-1}{kq}}\left(
=09=09\frac{1}{c_{2}|\lambda-\Lambda|}
=09\right)^{\lfloor \frac{j-1}{kq}\rfloor},
$$
and by (\ref{eq:siegelseries}) we have the uniform convergence of=20
$\tilde{h}_{\lambda}$ on a disk around $0$ of radius at least=20
$(3-2\sqrt{2})rc_{2}^{1/kq}$.=20

To compute the limit function $\tilde{h}_{\Lambda}$, note that:
$$
\tilde{f}^{\circ q}_{\lambda} =3D=20
\lambda^{q}z + \sum_{j=3D2}^{kq}
f^{\circ q}_{j}
(\lambda)(\lambda-\Lambda)^{\frac{j-1}{kq}}z^{j} +
f^{\circ q}_{kq+1}
(\lambda)(\lambda-\Lambda)+O((\lambda-\Lambda)^{1+1/kq}),
$$
and by lemma \ref{lem:mainlemma} and remark \ref{rem:pollambda}
we have that $f^{\circ q}_{j}(\lambda) \rightarrow 0$ for=20
$j=3D2,\ldots,kq$ and $f^{\circ q}_{kq+1}(\lambda) \rightarrow A \neq=20
0$. Since:
$$
\tilde{f}^{\circ q}_{\lambda} \circ \tilde{h}_{\lambda} =3D
\tilde{h}_{\lambda} \circ \tilde{f}^{\circ q}_{\lambda},
$$
we easily find that $\tilde{h}_{\Lambda}$ is the solution of the=20
ordinary differential equation:
$$
\tilde{h}_{\Lambda}(z) =3D z \tilde{h}'_{\Lambda}(z) -=20
\frac{A}{q\Lambda^{q-1}}(\tilde{h}_{\Lambda}(z))^{kq+1},
$$
with initial condition $\tilde{h}_{\Lambda}(0)=3D0$, whose solution is=20
just (\ref{eq:limitlin}).
\end{proof}

\section{The Semi--Standard Map}

Now we generalize the results on the scaling at resonances of the=20
linearizations for holomorphic germs of $(\fC,0)$ to the one-parameter=20
family of biholomorphic symplectic diffeomorphisms=20
$(x',y')=3DF_{\eps}(x,y)$ of $\fC/2\pi\fZ\times\fC$=20
\cite{GreenePercival},\cite{Percival} given by:
\begin{equation}
=09\begin{cases}x' & =3D x+y+\eps e^{ix},\\
=09=09=09=09 y' & =3D y + \eps e^{ix}.
=09\end{cases}
=09\label{eq:ssm}
\end{equation}
Clearly $\der x' \wedge \der y' =3D \der x \wedge \der y$.=20

We look for an analytic embedding $H_{\lambda}$ of a one-dimensional=20
complex pointed disk $\fD^{*}_{r} =3D \{z \in \fC | 0<|z|<r\}$ on which=20
the dynamics is given by multiplication by $\lambda$: $z \mapsto=20
R_{\lambda}(z) =3D \lambda z$; then:
\begin{equation}
=09F_{\eps} \circ H_{\lambda} =3D H_{\lambda} \circ R_{\lambda}.
=09\label{eq:ssmconj1}
\end{equation}

Let:
\begin{eqnarray}
=09z & =3D & -i\eps e^{i\th},\notag\\
=09x & =3D & \th - i\phi_{\lambda}(z),
=09\label{eq:complexssm}  \\
=09y & =3D & -i (\phi_{\lambda}(z)-\phi_{\lambda}(\lambda^{-1}z))=20
=09-i\log\lambda. \notag
\end{eqnarray}
Then:
$$
H_{\lambda} =3D \left(
=09=09=09=09=09-i\left(
=09=09=09=09=09=09=09\log\frac{iz}{\eps}+\phi_{\lambda}(z)
=09=09=09=09=09  \right),
=09=09=09=09=09-i(\phi_{\lambda}(z)-\phi_{\lambda}(\lambda^{-1}z))
=09=09=09=09=09=09-i\log\lambda
=09=09=09   \right)
$$
and our conjugacy equation becomes:
\begin{equation}
=09(D^{2}_{\lambda}\phi)(z) =3D \phi_{\lambda}(\lambda z) -=20
=09=09=09=09=09=09=09   2\phi_{\lambda}(z) +
=09=09=09=09=09=09=09   \phi_{\lambda}(\lambda^{-1}z) =3D
=09=09=09=09=09=09=09   -z e^{\phi_{\lambda}(z)}.
=09\label{eq:ssmconj2}
\end{equation}

If $|\lambda| \neq 1$ then (\ref{eq:ssmconj2}) has a unique analytic=20
solution such that $\phi_{\lambda}(0)=3D0$. When $|\lambda| =3D 1$ there=20
are the same three possibilities we discussed for germs. In the=20
\emph{resonant case} $F_{\eps}$ is in general not even formally=20
linearizable. Our conjugacy equation (\ref{eq:ssmconj2}) has an=20
analytic solution \emph{if and only iff} $\lambda =3D e^{2\pi i\om}$ and=20
$\om$ is a \emph{Brjuno number} \cite{Marmi}, \cite{Davie}.=20

\begin{rem}
Let=09$H(x,y)=09=3D (e^{ix},y)$. Then=09$H$=09conjugates $F_{\eps}$ to the
biholomorphic diffeomorphism $G_{\eps}$=09of $\fC^{2}$ defined by:
$$G_{\eps}(z_{1},z_{2})=09=3D (z_{1}e^{i(z_{2}+\eps=09z_{1})},
z_{2}+\eps z_{1}).
$$
The=09points $(0,z_{2})$ are fixed by=09$G_{\eps}$ and the image
$H_{\lambda}(\fD^{*}_{r})$ is associated \cite{Herman} with=09the=09center
manifold passing through the point $(0,-i\log\lambda)$ and tangent at
this point to $\{z_{2}=3D-i\log\lambda\}$.
\end{rem}

As in the case of germs, $\phi_{\lambda}$ depends holomorphically on=20
$\lambda \in \fD$ and it has non-tangential limits at all Brjuno=20
numbers. In what follows we will prove the existence of suitable=20
scalings under which $\phi_{\lambda}$ has non-tangential limits at all=20
resonances. We will compute exactly the limit when $\lambda=20
\rightarrow 1$, and we will give a heuristic argument to calculate=20
the limit at all other roots of unity.=20

\subsection{Existence of Non--Tangential Limits}

We now prove the following theorem.

\begin{thm}\label{thm:ssm}
Let $V$ be a non-tangential sector with vertex $\Lambda$ and let:
\begin{equation}
\tilde{\phi}_{\lambda}(z) =3D \phi_{\lambda}((\lambda-\Lambda)^{2/q}z).
\label{eq:phitilde}
\end{equation}
Then:
$$
\tilde{H}_{\lambda}(z) =3D \left(-i\left(
=09\log\frac{iz}{\eps}+\tilde{\phi}_{\lambda}(z)\right),
=09-i(\tilde{\phi}_{\lambda}(z)-\tilde{\phi}_{\lambda}(\lambda^{-1}z))
=09-i\log\lambda\right)
$$
conjugates $F_{(\lambda-\Lambda)^{2/q}\eps}$ to the rotation=20
$R_{\lambda}$. As $\lambda\rightarrow\Lambda$ in $V$,=20
$\tilde{\phi}_{\lambda}$ converges uniformly on some small open disk=20
around the $0$ to an analytic function $\tilde{\phi}_{\Lambda}$.=20
\end{thm}

\begin{proof}
The first statement is obvious.

Expanding $\phi_{\lambda}$ into powers of $z$:
$$
\phi_{\lambda}(z) =3D \sum_{j=3D1}^{\infty}\phi_{j}(\lambda)z^{j}
$$
and substituting into (\ref{eq:ssmconj2}) one finds:
\begin{align*}
=09\phi_{1}(\lambda) & =3D \frac{1}{D_{1}(\lambda)}  \\
\intertext{and, for all $j \geq 2$:}
=09\phi_{j}(\lambda) & =3D \frac{1}{D_{j}(\lambda)}\sum_{k=3D1}^{j}
=09=09\sum_{j_{1}+\ldots+j_{k}=3Dj-1}
=09=09\phi_{j_{1}}(\lambda)\cdots\phi_{j_{k}}(\lambda),
\end{align*}
where $D_{j}(\lambda) =3D -(\lambda^{j/2}-\lambda^{-j/2})^{2}$.

Since $\lambda\rightarrow\Lambda$ in $V$ there exists a positive=20
constant $c_{6}$ such that:
$$
|D_{j}(\lambda)| \geq=20
\begin{cases}
=09c_{6} & \text{if $j \neq 0 \pmod q$,}\\
=09c_{6}|\lambda-\Lambda|^{2} & \text{if $j =3D 0 \pmod q$.}
\end{cases}
$$
It is now easy to prove by induction that:
$$
|\tilde{\phi}_{j}(\lambda)| =3D=20
|\phi_{j}(\lambda)|\,|\lambda-\Lambda|^{2j/q} \leq
c_{7}\sigma_{j}c_{6}^{-2\lfloor\frac{j}{q}\rfloor}
=09|\lambda-\Lambda|^{2\frac{j}{q}-2\lfloor\frac{j}{q}\rfloor}
$$
for some constant $c_{7}>0$, and from this estimate the theorem=20
clearly follows.
\end{proof}

\begin{rem}
Note that $\tilde{\phi}_{\lambda}$ is actually a function of $z^{q}$.=20
\end{rem}

\begin{rem}
If $\lambda \rightarrow 1$ non-tangetially, one can easily compute=20
the limit $\tilde{\phi}_{1}$. Note that $\tilde{\phi}_{\lambda}$ is=20
the solution of:
$$
D^{2}_{\lambda}\tilde{\phi}_{\lambda}(z) =3D=20
=09-(\lambda - 1)^{2}z e^{\tilde{\phi}_{\lambda}(z)}.
$$

Since:
\begin{eqnarray*}
=09\tilde{\phi}_{\lambda}(\lambda z) & =3D &
=09=09\tilde{\phi}_{\lambda}(z) +=20
=09=09(\lambda-1) z \tilde{\phi}'_{\lambda}(z) +
=09=09\frac{1}{2}(\lambda-1)^{2}z^{2}\tilde{\phi}''_{\lambda}(z) +
=09=09\ldots,  \\
=09\tilde{\phi}_{\lambda}(\lambda^{-1} z) & =3D &=20
=09=09\tilde{\phi}_{\lambda}(z) +
=09=09\frac{1-\lambda}{\lambda} z \tilde{\phi}'_{\lambda}(z) +
=09=09\frac{1}{2}\frac{(1-\lambda)^{2}}{\lambda^{2}}
=09=09=09z^{2}\tilde{\phi}''_{\lambda}(z) +
=09=09\ldots,
\end{eqnarray*}
taking the limit $\lambda \rightarrow 1$ one finds:
\begin{equation}
=09z\tilde{\phi}'_{1}(z)+z^{2}\tilde{\phi}''_{1}(z) =3D=20
=09=09-ze^{\tilde{\phi}_{1}(z)},
=09\label{eq:eqforphi1}
\end{equation}
with the initial condition $\tilde{\phi}_{1}(0)=3D0$. Thus:
\begin{equation}
=09\tilde{\phi}_{1}(z) =3D -2\log\left(1+\frac{z}{2}\right).
=09\label{eq:solphi1}
\end{equation}
\end{rem}

\begin{rem}
Replacing $\eps e^{ix}$ in (\ref{eq:ssm}) with an entire function $f$=20
of $\eps e^{ix}$ such that $f(0)=3D0$ the conjugacy equation=20
(\ref{eq:ssmconj2}) becomes:
$$
(D^{2}_{\lambda}\phi)(z) =3D f(-ze^{\phi_{\lambda}(z)})
$$
and theorem \ref{thm:ssm} can be immediately proved also in this=20
(slightly) more general case.=20
\end{rem}

\subsection{Calculation of the Limits at Resonances}

We would like to compute exactly $\tilde{\phi}_{\Lambda}$ for all=20
$\Lambda =3D \exp\left(2\pi i \frac{p}{q}\right)$ with $(p,q)=3D1$. As a=20
preliminary step toward this result, we give an \emph{heuristic}=20
argument, reminiscent of the renormalization group method, which=20
provides just this. While all the combinatorics is handled rigorously,=20
we can't estimate the error terms that we neglect systematically at=20
each step of the ``renormalization''. We believe, though, that our=20
result are qualitatively accurate and quite likely exact, and that=20
the method could be turned into a tool for the rigorous analysis of=20
such maps.=20

The basic idea behind the procedure is similar to the one previously=20
used for germs, \ie that the behaviour of the semistandard map around=20
a resonance $p/q$ iterated $q$ times is ``roughly the same'' as the one=20
of the semistandard map around the ``main resonance'' $0/1$. The same=20
idea was used in \cite{BM2} to handle the resonance $1/2$ for the=20
standard map.=20

We need first the following combinatorial lemma.

\begin{lem}\label{lem:arith}
Consider the following recursively defined sequences:
\begin{align}
a^{(0)}_{k} & =3D \mu \delta_{k,0}, \quad \text{for } \mu \in \fC,\notag\\
b^{(r+1)}_{k} & =3D \Lambda^{k}a^{(r)}_{k},\\
a^{(r)}_{k} & =3D b^{(r)}_{k+1} - 2b^{(r)}_{k} + b^{(r)}_{k-1}.\notag
\end{align}
Then:
\begin{enumerate}
=09\item  $a^{(r)}_{k} =3D 0$ if $|k|>r$; \label{case:a}

=09\item  $\sum_{k\in\fZ}a^{(r)}_{k} =3D 0$; \label{case:b}

=09\item  $\sum_{k\in\fZ}ka^{(r)}_{k} =3D 0$; \label{case:c}

=09\item  $a^{(q)}_{q} =3D a^{(q)}_{-q}=3D\mu$, $a^{(q)}_{0} =3D -2\mu$, an=
d in all=20
=09for all other $k$ $a^{(q)}_{k} =3D 0$. \label{case:d}
\end{enumerate}
\end{lem}

\begin{proof}
(\ref{case:a}), (\ref{case:b}) and (\ref{case:c}) are most easily=20
proved by induction. In order to prove (\ref{case:d}) we use the=20
generating function:
$$
f^{(r)}(z) =3D \sum_{k\in\fZ}a^{(r)}_{k}z^{k}.
$$
$f^{(r)}$ is a rational function because of (\ref{case:a}). We also=20
have:
$$
a^{(r)}_{k} =3D \Lambda^{k}\left(
=09=09\Lambda a^{(r-1)}_{k+1}-2a^{(r-1)}_{k}+\Lambda^{-1}a^{(r-1)}_{k-1}
=09=09\right),
$$
from which it follows that:
\begin{equation*}
\begin{split}
f^{(r)}(z) & =3D \sum_{k\in\fZ}z^{k}\Lambda^{k}\left(
=09=09\Lambda a^{(r-1)}_{k+1}-2a^{(r-1)}_{k}+\Lambda^{-1}a^{(r-1)}_{k-1}
=09=09\right)\\
=09& =3D \left(\frac{1}{z}-2+z\right) f^{(r-1)}(\Lambda z)\\
=09& =3D \frac{(z-1)^{2}}{z}f^{(r-1)}(\Lambda z).
\end{split}
\end{equation*}

Now, clearly $f^{(0)}(z)=3D\mu$. Then:
\begin{equation*}
\begin{split}
f^{(q)}(z) & =3D \mu\frac{\left(\prod_{k=3D1}^{q-1}(\Lambda^{k}z-1)\right)^=
{2}}
=09=09{z^{q}\Lambda^{q(q-1)/2}}\\
=09& =3D \mu\Lambda^{-q(q-1)/2}\frac{(1-z^{q})^{2}}{z^{q}}\\
=09& =3D \mu(z^{q}-2+z^{-q}),
\end{split}
\end{equation*}
since $\Lambda$ is a primitive $q$-th root of $1$: this proves=20
(\ref{case:d}).
\end{proof}

Now we proceed with our heuristic argument. We have:
\begin{equation*}
=09D^{2}_{\lambda^{q}}\tilde{\phi}_{\lambda}(z) =3D=20
=09=09\sum_{-q}^{q}a^{(q)}_{k}\tilde{\phi}_{\lambda,k},
\end{equation*}
where:
\begin{align*}
=09\tilde{\phi}_{\lambda,k} & =3D \tilde{\phi}_{\lambda}(z_{k}),\\
\intertext{and}
=09z_{k} & =3D \lambda^{k}z.
\end{align*}
The coefficients $a$ are the one defined in lemma \ref{lem:arith},=20
with $\mu=3D1$. We have choosen our notations so that:
$$
D^{2}_{k}\tilde{\phi}_{\lambda,k} =3D=20
\tilde{\phi}_{\lambda,k+1} - 2\tilde{\phi}_{\lambda,k}
+ \tilde{\phi}_{\lambda,k-1}.
$$

We shall apply $q-1$ times a transformation defined in three steps:
\begin{enumerate}
=09\item  write everything in terms of $\tilde{\phi}_{\lambda,k+1} -=20
=09=092\tilde{\phi}_{\lambda,k} + \tilde{\phi}_{\lambda,k-1}$;

=09\item  apply the functional equation for $\tilde{\phi}_{\lambda}$:
=09$$
=09D^{2}_{\lambda}\tilde{\phi}_{\lambda}(z) =3D=20
=09=09-(\lambda-\Lambda)^{2/q}ze^{\tilde{\phi}_{\lambda}(z)};
=09$$

=09\item  linearize:
=09$$
=09z_{k}e^{\tilde{\phi}_{\lambda}(z_{k})} \approx=20
=09=09\Lambda^{k}z_{0}e^{\tilde{\phi}_{\lambda,0}}
=09=09\left(1+\left(
\tilde{\phi}_{\lambda,k}-\tilde{\phi}_{\lambda,0}-k\log\frac{\lambda}{\Lamb=
da}
=09=09\right)\right).
=09$$
\end{enumerate}

Now we use lemma \ref{lem:arith} and in particular the fact that:
$$
\sum_{k=3D-r}^{r}a^{(r)}_{k}\tilde{\phi}_{\lambda,k} =3D=20
\sum_{k=3D-r+1}^{r-1}b^{(r)}_{k}
(\tilde{\phi}_{\lambda,k+1}-2\tilde{\phi}_{\lambda,k}+\tilde{\phi}_{\lambda=
,k-1}),
$$
where the sequences $a^{(r)}$ and $b^{(r)}$ are those defined in the=20
lemma. Note that conditions (\ref{case:b}) and (\ref{case:c}) of=20
lemma \ref{lem:arith} are crucial for the following to work.

Then we have:
\begin{equation*}
\begin{split}
\sum_{k=3D-r}^{r} a^{(r)}_{k}\tilde{\phi}_{\lambda,k} & =3D
=09=09\sum_{k=3D-r+1}^{r-1}b^{(r)}_{k}
=09=09(\tilde{\phi}_{\lambda,k+1} -
=09=092\tilde{\phi}_{\lambda,k} +
=09=09\tilde{\phi}_{\lambda,k-1})\\
=09& \approx -(\lambda-\Lambda)^{2/q}ze^{\tilde{\phi}_{\lambda}}
=09=09\sum_{k=3D-r+1}^{r-1}b^{(r)}_{k}\Lambda^{-k}
=09=09\left(1+\left(\tilde{\phi}_{\lambda,k} -=20
=09=09\tilde{\phi}_{\lambda,0} -=20
=09=09k\log\frac{\lambda}{\Lambda}\right)\right)\\
=09& =3D -(\lambda-\Lambda)^{2/q}ze^{\tilde{\phi}_{\lambda}}
=09=09\left(\sum_{k=3D-r+1}^{r-1}a^{(r-1)}_{k} -=09
\log\frac{\lambda}{\Lambda}\sum_{k=3D-r+1}^{r-1}ka^{(r-1)}_{k}\right)\\
=09& \quad -(\lambda-\Lambda)^{2/q}ze^{\tilde{\phi}_{\lambda}}
=09=09\sum_{k=3D-r+1}^{r-1}a^{(r-1)}_{k}
=09=09(\tilde{\phi}_{\lambda,k}-\tilde{\phi}_{\lambda,0})\\
=09& =3D -(\lambda-\Lambda)^{2/q}ze^{\tilde{\phi}_{\lambda}}
=09=09\sum_{k=3D-r+1}^{r-1}a^{(r-1)}_{k}\tilde{\phi}_{\lambda,k},
\end{split}
\end{equation*}
and by keeping iterating this identity $q-1$ times (\ie by setting=20
sequentially $r=3Dq,\ldots,2$) we obtain:
\begin{equation*}
\begin{split}
D^{2}_{\lambda^{q}}\tilde{\phi}_{\lambda} & =3D=20
=09\sum_{k=3D-q}^{q}a^{(q)}_{k}\tilde{\phi}_{\lambda,k}\\
 & \approx=20
=09\left(-(\lambda-\Lambda)^{2/q}ze^{\tilde{\phi}_{\lambda}(z)}\right)^{q-1=
}
=09\left(a^{(1)}_{1}\tilde{\phi}_{\lambda,1}
=09+ a^{(1)}_{0}\tilde{\phi}_{\lambda,0}
=09+ a^{(1)}_{-1}\tilde{\phi}_{\lambda,-1}\right).
\end{split}
\end{equation*}
Since $a^{(1)}_{1}=3Da^{(1)}_{-1}=3D\Lambda$, $a^{(1)}_{0}=3D-2\Lambda$, we=
=20
finally have:
\begin{equation*}
\begin{split}
D^{2}_{\lambda^{q}}\tilde{\phi}_{\lambda} & \approx
=09\left(-(\lambda-\Lambda)^{2/q}ze^{\tilde{\phi}_{\lambda}(z)}\right)^{q-1=
}
=09\Lambda (\tilde{\phi}_{\lambda,1} -=20
=092 \tilde{\phi}_{\lambda,0} + \tilde{\phi}_{\lambda,-1})\\
 & =3D \Lambda=20
=09(-1)^{q}(\lambda-\Lambda)^{2}z^{q}e^{q\tilde{\phi}_{\lambda}(z)}.
\end{split}
\end{equation*}
Taking the limit $\lambda\rightarrow\Lambda$ we have:
$$
\frac{D^{2}_{\lambda^{q}}}{(\lambda-\Lambda)^{2}}\tilde{\phi}_{\lambda}=20
\rightarrow
q^{2}\Lambda^{-2}z\frac{\der\tilde{\phi}_{\Lambda}}{\der z} +
q^{2}\Lambda^{-2}z^{2}\frac{\der^{2}\tilde{\phi}_{\Lambda}}{\der z^{2}},
$$
so that:
\begin{equation}
=09\frac{\der}{\der z}\left(
=09=09z\frac{\der\tilde{\phi}_{\Lambda}}{\der z}\right) =3D
=09=09\frac{(-1)^{q}\Lambda^{3}}{q^{3}}z^{q-1}e^{q\tilde{\phi}_{\Lambda}(z)=
},
=09\label{eq:linssmcalculated}
\end{equation}
with the initial condition $\tilde{\phi}_{\Lambda}(0)=3D0$. Note that=20
the transformation $\psi_{\Lambda}(w)=3Dq\tilde{\phi}_{\Lambda}(z)$ and=20
$w=3D\frac{(-1)^{q}\Lambda^{3}}{q^{3}}z^{q}$ applied to (\ref{eq:linssmcalc=
ulated})
gives the simpler equation:
$$
\frac{\der}{\der w}\left(w\frac{\der\psi_{\Lambda}}{\der w}\right) =3D
e^{\psi_{\Lambda}},
$$
whose solution is $\psi_{\Lambda} =3D -2\log(1-w/2)$. Therefore we=20
finally have:
\begin{equation}
=09\tilde{\phi}_{\Lambda}(z) =3D=20
=09=09-\frac{2}{q}\log\left(1+\frac{(-1)^{q+1}\Lambda^{3}}{2q^{3}}z^{q}\rig=
ht),
=09\label{eq:linssmsolution}
\end{equation}
which agrees with (\ref{eq:solphi1}) for $q=3D1$.

%End of Paper

\vspace{1.5cm}

\begin{thebibliography}{99}

\bibitem{BM2} Berretti, A. and Marmi, S.,
\textit{Scaling near Resonances and Complex Rotation Numbers for the
Standard Map}, Nonlinearity \textbf{7} 603 (1994)

\bibitem{LT} de la Llave, R., and Tompaidis, S.,
{\textit Computation of domains of analyticity for some perturbative
expansions of mechanics},
University of Texas at Austin Preprint (1992)

\bibitem{BTT} Billi,~L., Todesco,~E., and Turchetti,~G.,=20
\textit{Singularity Analysis by Pad\'{e} Approximants of some=20
Holomorphic Maps}, J. Phys. A: Math. and Gen. \textbf{27} 6215-6229,=20
1994

\bibitem{Yoccoz} Yoccoz,~J.~C., \emph{Th\'{e}or\`{e}me de Siegel, Nombres=
=20
de Brjuno et Pol\^{o}mes Quadratiques}, Ast\'{e}risque \textbf{231}=20
3-38, 1995

\bibitem{GreenePercival} Greene,~J.~M., and Percival,~I.~C.,=20
\textit{Hamiltonian Maps in the Complex Plane}, Physica D \textbf{3}=20
530-548, 1981

\bibitem{Percival} Percival,~I.~C., \textit{Chaotic Boundary of a=20
Hamiltonian Map}, Physica D \textbf{6} 67-74, 1982

\bibitem{Koenigs} K\"onigs,~G., \textit{Recherches sur les integrales=20
de certaines equations fonctionelles}, Ann. Sci. \'{E}c. Norm. Sup.=20
($3^{\text{\`{e}me}}$ s\'{e}r.), 1 Suppl. 1--41, 1884

\bibitem{Poincare} Poincar\'{e},~H., \textit{\OE{}uvres} t. I, p.=20
XXXVI-CXXIX, Gauthier-Villars, Paris 1928-1956

\bibitem{Brjuno}Bjuno,~A.~D., \textit{Analytic Form of Differential=20
Equations}, Trans. Moscow Math. Soc. \textbf{25} 131--288, 1971

\bibitem{MMY} Marmi,~S., Moussa,~P., Yoccoz,~J.~C., \textit{The=20
Brjuno Functions and their Regularity Properties}, preprint Saclay=20
95/028, 1995

\bibitem{Cremer} Cremer,~H., \textit{\"{U}ber die H\"{a}ufigkeit der=20
Nichtzentren}, Math. Ann. \textbf{115}, 573-580, 1938

\bibitem{Ecalle} Ecalle,~J., \textit{Les Fonctions R\'{e}surgentes et leurs=
=20
Applications. Tomes I, II, III}, Publications Math\'{e}matiques=20
d'Orsay \textbf{81-05}, \textbf{81-06}, \textbf{85-05}, 1981-1985

\bibitem{Voronin} Voronin,~S.~M., \textit{Analytical Classification of=20
Germs of Conformal Mappings $(\fC,0)\mapsto(\fC,0)$ with identity=20
linear part}, Funct. Anal. and its Appl. \textbf{15} 1-17, 1981

\bibitem{Beardon} Beardon,~A.~F., \textit{Iteration of Rational=20
Functions}, Graduate Text in Mathematics \textbf{138}, Springer, 1991

\bibitem{DH} Douady,~A., and Hubbard,~J.~H., \textit{On the Dynamics=20
of Polynomial Like Mappings}, Ann.  Sc.  E.N.S., 4\`{e}me S\'{e}rie,=20
\textbf{18} 287-343 (1985); see also Douady,~A., \textit{Syst\`{e}mes=20
Dynamiques Holomorphes}, S\'{e}minaire Bourbaki, Expos\'{e} 599,=20
Ast\'{e}risque \textbf{105-106}, 39-63 (1983)

\bibitem{Marmi} Marmi,~S., \textit{Critical Functions for Complex=20
Analytic Maps}, Journ. of Physics A: Math. and Gen. \textbf{23}=20
3447-3474, 1990

\bibitem{Davie} Davie,~A.~M., \textit{Critical Functions for the=20
Semi--standard and Standard Maps}, Nonlinearity \textbf{7} 219-229,=20
1994

\bibitem{Herman} Herman,~M.~R., \textit{Recent Results and Some Open=20
Questions on Siegel's Linearization Theorem of Germs of Complex=20
Analytic Diffeomorphisms of $\fC^{n}$ near a Fixed Point}, Proc. VIII=20
Int. Conf. Math. Phys., Mebkhout and Seneor Eds., World Scientific, 1986

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