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\newtheorem{theo}{Theorem}[section]
\newtheorem{defi}[theo]{Definition}
\newtheorem{prop}[theo]{Proposition}
\newtheorem{exam}[theo]{Example}
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\begin{document}
\title{Decomposition theorems for groups of diffeomorphisms in the
sphere.
}
\author{R. de la Llave\\
Dept. Math. \\
U. Texas at Austin \\
Austin TX 78712\\
USA\\
e-mail llave@math.utexas.edu
\and
{R. Obaya}\\
Dept. Matem\'atica Aplicada a la Ingenier\'{\i}a\\
Escuela Superior de Ingenieros Industriales\\
Univ. de Valladolid\\
47011 Valladolid \\
SPAIN\\
e-mail rafoba@wmatem.eis.uva.es
}
\date{}
\maketitle
\def\sphere{{\bf S}}
\def\torus{{\bf T}}
\def\bfx{{\bf x}}
\def\natural{{\rm I\kern-.18em N}}
\def\integer{{\rm Z\kern-.32em Z}}
\def\real{{\rm I\kern-.2em R}}
\def\complex{\kern.1em{\raise.47ex\hbox{
$\scriptscriptstyle |$}}\kern-.40em{\rm C}}
\def\Diff{\mathop{\rm Diff}\nolimits}
\def\Sym{\mathop{\rm Sym}\nolimits}
\def\OO{{\cal O}}
\begin{abstract}
We study the algebraic structure of several
groups of differentiable diffeomorphism
in $\sphere^n$.
We show that given any sufficiently smooth diffeomorphism
can be written as the composition of
a finite number of
diffeomorphisms
which are symmetric under reflection,
essentially one-dimensional and
about as differentiable as the
given one.
{\em Keywords:} Decomposition theorems, diffeomorphism groups.
{\em AMS 1991 Classification:} 58D05, 57S25, 57S05
\end{abstract}
\section{Introduction}\label{intro}
The goal of this paper is to prove several decomposition theorems for
diffeomorphism groups.
We recall that decomposition theorems
for groups, roughly, state that any
element of the group can be
can be written as a finite product
of elements lying in a
smaller subgroup.
In our case, the groups considered
will be groups of differentiable diffeomorphisms
of the sphere and the subgroups in which we
factor will be groups of diffeomorphisms
that commute with reflections across a
plane and which are essentially one dimensional.
The factors into which we
can decompose a given map will be slightly less differentiable than
the original one.
There are several motivations for the study of theorems of
this type.
For example, in \cite{LS} it is shown that a theorem
of this type for the circle can be used
to solve the inverse problem for scattering of geodesic
fields in surfaces of genus one. Physically, this is the problem
of given the transformation to be effected by a lens,
find the distribution of the refractive index that
produces the desired effect.
The problem is such that it it is solved
for two diffeomorphisms, it is solved for their
composition. Moreover, for symmetric mappings
a simple construction works. Hence, the
decomposition theorem, shows that it can be solved for all
mappings.
For the application above, the loss of
differentiability incurred in the
factorization does not affect essentially
the conclusions.
We also note that
these theorems are analogues of the usual
factorization theorems in Lie algebras and
they could be useful
in the
problem of computing representations of diffeomorphism groups
\cite{Bi}.
If we consider these theorems as infinite dimensional versions of
factorization
theorems for Lie groups, one first difficulty
is that for diffeomorphism groups, the corresponding Lie algebra --
the Lie algebra of vector fields -- is considerably
worse behaved than those of finite dimensional Lie groups.
For example,
the exponential is not surjective
in any neighborhood of the identity. Hence, it
is not possible to
apply the usual implicit function theorem
in Banach spaces to obtain results in a
small enough neighborhood of the origin using the
{\it ``infinitesimal results'' } obtained from the Lie algebra.
Nevertheless, we will show that it is possible to
obtain similar results to those in the finite dimensional case
by using appropriate {\it ``hard''} implicit function theorems of
the Nash--Moser type.
However, one needs to reformulate the
problem so that it has some group structure.
This requires some assumptions on the structure of the
manifold. (Of course, the existence of the reflections
is included in the assumptions.)
In our proof we proceed to prove the theorem
by induction on the dimension. The construction
is somewhat delicate since we want to obtain
a loss of differentiability that is independent
of the dimension, in spite of the fact that
the number of inductive steps has to grow with the
dimension.
\section{Acknowledgments}\label{acknow}
The work of R.L. has been supported by NSF grants.
Also R.L wants to acknowledge the hospitality enjoyed at Valladolid
during several visits.
The work of R.O. has been partially supported by Junta de Castilla y Leon.
Letters from J. Langer and F. Bien discussing
the implications of these theorems were very useful to us
and provided encouragement.
\section{Decomposition theorem for spheres.}\label{dos}
\subsection{Notation and
statement of results}
We start this section by recalling some known results which show
how to endow certain function spaces on compact sets with differentiable
manifold structure, necessarily of infinite dimension.
If $l$ is an integer,
and $\Omega$ is a compact set in $\real^d$
we will say that $f: \Omega \to \real $ is
$C^l$ if it has continuous derivatives
of order $l$. We will denote by $C^l(\Omega)$, the space of $C^l$ functions in
$\Omega$ endowed with the norm ${\|\ \|_{C^l(\Omega)}}$ defined as
supremum of the derivatives of order up to $l$.
It is well known that it is a Banach space.
If $l$ is not an integer,
we define as usual
\begin{equation}\label{Holder}
\| f\|_{C^l(\Omega)} = \max( \sup_{x\ne y} \| D^l f(x) - D^l f(y) \| /
|x-y|^{l-[l]}, \| f\|_{C^{[l]}(\Omega)} )
\end{equation}
To cover Lipschitz conditions as well, we introduce the notation
$l^- = (l-1)+1^-$. In this notation,
$1^-$ will be bigger than any number in $[0,1)$
but smaller than $1$.
These notions of regularity can be lifted to geometric objects that
can be expressed in coordinates by saying that an object is $C^l$ if
we can find coordinate patches that cover the manifold on which the
expression is $C^l$.
Furthermore, if we fix a set of coordinate patches that cover the
manifold we can talk about the $C^l$ distance between two objects by
declaring it to be the maximum of the $C^l$ distance between the
coordinate expressions. A moment's reflection will show that
the statement that a geometric object is $C^l$ is independent of the
coordinate patches chosen (that is, if the coordinate expressions
in one patch are $C^l$ so are in all others).
Unfortunately, the $C^l$ distances do depend on the patch. Even if it is
easy to show that the distances obtained using two patches are equivalent,
they do differ and some statements such as that certain operators acting
on $C^l$ functions are contractions require specific choices of coordinate
patches.
When $l$ is an integer, there is a geometrically natural notion of $C^l$
distance in Riemannian geometry based on the notion of jets.
(See e.g. \cite{Hi} Ch. 2)
For non-integer $l$, since the definition of H\"older distance
(\ref{Holder}) requires comparing the values at different points,
there are no geometrically natural notions of $C^l$ distance.
For our purposes, the dependence on a
coordinate patch is not a shortcoming since many analytical operations
that we will need to perform
require taking coordinates anyway and we will always take a
fixed coordinate system.
We will denote by $\Diff^l (M,N)$, the set of diffeomorphisms
of $M$ into $N$ and, when $M=N$ we will simply write $\Diff^l (N)$.
When $l$ is finite $l \ge 1$, these sets can be given the structure of Banach
manifolds as follows.
Using charts, we can define a norm
on the space of $C^l$ vector fields on $N$
that makes it
a Banach space.
Given a diffeomorphism $f \in \Diff^l(M,N)$
and a $C^l$ vector field $v$, the map ${\rm Exp}(f,v):M \to N$
defined by ${\rm Exp}(f,v)(x) = \exp_{f(x)}v(f(x))$ --
where $\exp$ denotes the Riemannian geometry exponential associated to a
$C^\infty$ metric -- is a $C^l$ mapping.
The implicit function theorem shows that for sufficiently small
$v$, this mapping is also a diffeomorphism
and, moreover, any diffeomorphism in
a $C^l$ neighborhood of $f$ can be written in this way.
If $l<1$, we cannot apply the implicit function theorem
to conclude that ${\rm Exp}(f,v)$ is a diffeomorphism,
nevertheless, it is possible to show that ${\rm Exp}$
covers a neighborhood of $f$ in the space of
$C^l$ mappings.
(A Similar argument shows that $\Diff^\infty (M)$ is a Frechet manifold).
In $\Diff^l (M)$ it is natural to define the group operation of composition.
This set is a topological group for integer exponents; this is the
reason we formulate the statements in this paper only for this case.
Unfortunately, when $l$ is not an integer, this operation will be
discontinuous. Moreover, when $l$ is an integer, even if
composition is continuous, it will not be differentiable.
For properties of the composition operator on
H\"older spaces we refer to \cite{LO} where there
is a very systematic treatment of these properties.
We will also consider families of diffeomorphisms depending of a
parameter. We will denote the parameters by subindices. We will say that
$f_\lambda$ is a $C^l$ family of diffeomorphisms when for every fixed
$\lambda \in \Lambda$, where $\Lambda \subset \real^m$ is an open set or a
compact manifold, the map $x \rightarrow f_\lambda(x)$ is a $C^l$
diffeomorphism and, moreover $(\lambda,x) \rightarrow f_\lambda(x)$ is also
$C^l$.
The composition of diffeomorphisms can be extended in a natural way
to families. We denote by $g_\lambda \circ f_\lambda$ the family
$(\lambda,x) \rightarrow g(\lambda,f_\lambda(x))$.
The notation $f_\lambda$ is somewhat ambiguous since
it denotes at the same time the family and the mapping for a concrete value
of $\lambda$. The notation is, however, standard use and it does not lead to
confusion.
\smallskip
\noindent
{\bf Remark.} It is also possible to think of families as mappings from
$\Lambda \times M$ to $\Lambda \times N$ such that the first variable
remains unaltered. This point of view makes it clear that many results
about composition proved for $C^l$ mappings are true for $C^l$
families.
Notice that the map $\Lambda \rightarrow C^l(M,N)$
defined by
$\lambda \rightarrow f_\lambda$ is only $C^0$. There are many $C^0$
mappings $\Lambda \rightarrow C^l(M,N)$ which are not $C^l$ families.
\medskip
From now on, the manifold under consideration will be the
$n$-dimensional sphere $\sphere^n$. We
take its standard embedding in the Euclidean space
\[ \sphere^{n} =
\left\{ x = (x_1,\cdots,x_{n+1}) \in \real^{n+1} \big|
\sum_{i=1}^{n+1} x_i^2 = 1\right\}.
\]
If $x$ is a point in
$\sphere^{n}$, we will denote by $\Pi_x$ the
hyperplane passing through the origin
and consisting of vectors orthogonal to $x$, and by ${\cal M}_{x,y}$
the one-dimensional maximum circle going from $x$ to
$-x$ and passing through the point $y \in \sphere^n-\{x,-x\}$.
We will refer to those sets as {\it meridians\/}.
We will also represent by ${\cal P}_{x,\gamma}$, $-1\leq \gamma\leq 1$,
the subsets of $\sphere^{n}$ whose points have a constant projection
$\gamma$ along the line joining the origin and the point $x$. The sets
of these form will be referred to as {\sl parallels}.
Finally, we denote by $\delta_x$, the reflection across the plane $\Pi_x$.
We call attention to the fact that, with our notation, meridians
are always one-dimensional, but parallels in
$\sphere^{n}$ are copies of $\sphere^{n-1}$.
Outside a neighborhood of the poles $\{x,-x\}$,
meridians ${\cal M}_{x,y}$ and parallels ${\cal P}_{x,\gamma}$
constitute two transversal foliations.
We will denote by $\Diff_0^l (\sphere^{n})$ the connected component
of the identity of $\Diff^l (\sphere^{n})$ and
by $\Sym_{x}^l$ the subset of $\Diff^l (\sphere^{n})$ commuting with
$\delta_x$. The symbol $\Sym_{x,0}^l$ will stand for
the connected component of $\Sym_{x}^l$ containing the identity.
\medskip
Next we study the algebraic structure of the group of diffeomorphisms in
$\sphere^n$. The main theorem in this paper is:
\begin{theo}\label{main}
Let $n\in \natural$, $n\ge 2$. For any
$l \in \natural \cup \{\infty\}$, $l\ge 7$,
we can find points
$y_1,\ldots y_{(n+1)^2}$ in such a way that any
$f\in\,${\rm Diff}$_0^l(\sphere^{n})$ can be written as
$f=f_1\circ f_2\circ \cdots \circ f_N$, where each $f_i \in\,$
{\rm Sym}$^{l-2}_{y_j,0}$ for some $j=1,\ldots,(n+1)^2$.
( The number $N$ may depend on $f$)
\end{theo}
For $n=1$, there exist points $y_1,$ $y_2$
such that any $f\in\,$Diff$_0^l(\sphere^{1})$
can be written as $f=f_1\circ f_2\circ \cdots \circ f_N$ with each $f_i
\in\,$Sym$^{l-2}_{y_j,0}$ for some $j=1,2$.
This theorem was proved in \cite{LS} for $l=\infty$.
A different proof valid in the classes of finitely
differentiable functions was given
in \cite{Ll} where an analogue result for ${\bf T}^n$ was also proved.
We also remark that, as a corollary of the proof
of Theorem \ref{main}, we will obtain that
the diffeomorphisms appearing as factors can be
identified with families of diffeomorphisms of the circle.
\subsection{Proof of Theorem \ref{main}}
First at all, observe that it suffices to prove the result in an
arbitrary small neighborhood of the identity in the $C^l$-topology.
In effect, given any
open neighborhood ${\cal U}$ of the identity,
any diffeomorphism in the connected component of the identity
can be written as the composition of a finite number
of diffeomorphisms in ${\cal U}$.
We will establish the local theorem for families of diffeomorphisms
by induction on the dimension. We introduce in the inductive argument
that the map ${\cal F}$, constructed through the proof, which assigns
to each family of diffeomorphisms
$f_\lambda \in {\cal U}$ the factors $f_{\lambda,i}$, $i=1,\ldots,N$
in a neighborhood of the identity in the $C^{l-2}$ topology,
is continuous and that for each value of the parameter $\lambda \in
\Lambda$ with $f_\lambda=Id$, one obtain factors $f_{\lambda,i}=Id$
for every $i=1,\ldots,N$.
This result for the one dimensional case is stated explicitly in \cite{Ll}
In the statement of the theorem the points $y_1, \ldots,y_{(n+1)^2}$
are left to our choice. We fix a real number $\rho$ and take those
$y= (y_k)$ of $\sphere^n$ whose components are defined
by one of the following conditions:
\begin{itemize}
\item[-]
there is $1\leq i \leq {n+1}$ such that $y_i=1$ and $y_k=0$ for
$k \neq i$, or
\item[-] there are $1\leq i,j \leq {n+1}$ such that $y_i=\cos \rho$,
$y_j=\sin \rho$ and $y_k=0$ for $k \neq i,j$.
\end{itemize}
(However, it will be clear that a more careful selection of these points
would allow us to reduce their number. We have not bothered with this
issue.)
The choices of the angle of separation $\rho$
that are convenient for the method of proof
are those satisfying
the so-called {\it Diophantine} conditions,
\begin{equation}\label{2.4}
\left|\rho- \frac{p}{q} \right|^{-1} \leq C q^{2 + \delta}
\end{equation}
for all pair of integers $p,q$ with $q>0$.
It is well known that for $\delta >0$, the numbers satisfying this kind of
inequalities have full Lebesgue measure. For $\delta=0$, the set of numbers
satisfying these inequalities is called
the set of constant type numbers. Moreover relation
(\ref{2.4}) with $\delta = 0$ is equivalent
to $\rho$ having a bounded continued fraction
expansion. All irrational numbers that satisfy a quadratic equation with
integer coefficients are constant type; in particular, constant type
numbers are dense.
Even if all Diophantine $\rho$'s would lead to a theorem of the type we want
and, for $\delta$ small, with the conditions of differentiability required in
the statement, constant type numbers will lead to the sharpest
differentiability conclusions. From now on we will assume we have chosen
a constant type number. The actual choice will affect the sizes of
the neighborhoods where the local theorem holds,
but not the differentiability properties.
In order to apply the induction process, we will
decompose diffeomorphisms of $\sphere^n$ in a neighborhood of $Id$ into
factors with the same order of differentiability
that preserve lower dimensional spheres.
We will start by proving a parameterized version of the well known
{\it fragmentation lemma}.
\begin{lemm}\label{frag}
Let $M$ be a compact manifold, and
$\{K_i\}_{i=1}^n$ be a collection of
compact subsets of $M$ such that
\[\bigcap_{i=1}^{n} K_i=\emptyset\;.\]
Then, for each $l\in \natural \cup \{ \infty \}$
we can find a neighborhood ${\cal U}$ of the identity in the space of
$C^l$ families of diffeomorphisms, and a map ${\cal F}_0$ that,
to each family $f_\lambda$ in ${\cal U}$,
associates $C^l$ families $f_\lambda^1,\cdots,f_\lambda^n$, satisfying
\begin{itemize}
\item[i)] $f_\lambda= f_\lambda^1\circ \cdots\circ f_\lambda^n\,$.
\item[ii)] $f_\lambda^i|_{K_i}=Id\,$.
\item[iii)] if for some $x\in M$ and $\lambda \in \Lambda$ one has
$f_\lambda(x)=x$ then $f_\lambda^i(x)=x$ for
every $i=1,\dots,n$.
\end{itemize}
The mapping ${\cal F}_0$ is continuous when we give its domain and its range
the topology of $C^l$ families.
\end{lemm}
\smallskip
\noindent
{\bf Proof.} We will assume that ${\cal U}$ is small enough so that
$f_\lambda(x)$ is within the injectivity radius of the exponential. We
will also need that some intermediate steps in our construction lie in this
domain. Since these are a finite number of conditions, it will be easy to check
that they are satisfied for a non-trivial neighborhood.
Suppose first the case $n=2$; then we can find a $C^\infty$-function
$\phi$ such that
\[\phi|_{K_1}\equiv 1\,,\quad \phi|_{K_2} \equiv 0\;.\]
If $f_\lambda$ can be represented as the exponential of the vector field
$F_\lambda=E^{-1}(f_\lambda)$, then $f_\lambda^2=E(F_\lambda \tau)$
is the identity restricted to $K_2$. Moreover, since $f_\lambda(x)=x\,
\Leftrightarrow E^{-1}(f_\lambda)(x)=0$, the set of fixed points of
$f_\lambda^2$ is bigger than those of $f_\lambda$. Provided that $f_\lambda$
is sufficiently close to the identity --which depends only on the $K$'s and
$\phi$--, we have $f_\lambda^2|_{K_1}=f_\lambda$ and
$f_\lambda^2|_{K_2}=Id\,$.
Hence, setting $f_\lambda^1=f_\lambda\circ(f_\lambda^2)^{-1}$, the claims of the
lemma are satisfied.
Since the choice of $\phi$ depends only on
the compacts $K_1$ and $K_2$ the continuity follows.
Notice that the previous argument works even in the case that
one of the compacts is the empty set.
\smallskip
To prove the general case, we will proceed by induction in $n$,
the number of
compact sets, starting from teh aleady verified case
$n = 2$. We will assume that the conclusion is true for $n\leq n_0$,
$n_0 \geq 2$, and will check it for $n=n_0+1\,$.
If $K_1,\dots, K_{n_0+1}$ are compact sets with empty intersection, we can
find open sets $V_1,\dots, V_{n_0+1}$ such that $K_i\subset V_i$, and
$\cap_{i=1}^{n_0+1} \overline{V}_i=\emptyset\,$.
Set $L_0=\cap_{i=1}^{n_0} \overline{V}_i\,$ and $L_{n_0+1}=K_{n_0+1}$. Since
$L_0 \cap L_{n_0+1}=\emptyset$ we apply the previous result with $n_0=2$ to
obtain families $f_\lambda^0$, $f_\lambda^{n_0+1}$ which are the
identity on $L_0$, $L_{n_0+1}$ respectively, and with not less fixed points than
$f_\lambda$ such that $f_\lambda=f_\lambda^0\circ f_\lambda^{n_0+1}$.
We can now consider the compact sets $L_i=K_i-\cap_{i=1}^{n_0} V_i\,$. They
have empty intersection because $\cap_{i=1}^{n_0} K_i \subset
\cap_{i=1}^{n_0} V_i\,$, so we can apply the induction hypothesis to
$f_\lambda^0$ to write $f_\lambda^0=f_\lambda^1\circ \cdots\circ
f_\lambda^{n_0}\,$. Each one of the $f_\lambda^i$ will be the identity on a
neighborhood of $K_{n_0+1}$. This gives us the representation
$f_\lambda=f_\lambda^1\circ \cdots \circ f_\lambda^{n_0+1}$
satisfying all the conditions of the statement.
The continuity of ${\cal F}_0$ is also a consequence of the induction process.
We emphasize that, since,
properties i)-iii) do not determine ${\cal F}_0$ uniquely
the claim of continuity refers only to the one constructed above.
\qed
In the next lemma, we show how diffeomorphisms can be factored
into simpler maps that conserve the parallels and the meridians,
which are in essence diffeomorphisms of $\sphere^{n-1}$ and
$\sphere^1$ respectively. We will henceforth refer to diffeomorphisms
and families that preserve
the parallels as {\it horizontal} and those that preserve the
meridians as {\it vertical}.
\begin{lemm}\label{vert}
Let $x_0$ be a point in $\sphere^n$, $-x_0$ the antipodal point and
$U$, $U'$ two
open neighborhoods around $x_0$, $-x_0$ respectively. Then, for every
$l\in\,\natural\; \cup \{\infty\}$,
we can find a neighborhood ${\cal U}$ of the
identity in the space of $C^l$ families of diffeomorphisms in such a way
that, for any family $f_\lambda$ in ${\cal U}$, such that $f_\lambda |_U= Id$,
$f_\lambda|_{U'}= Id$, there exist $C^l$ families $f_\lambda^h$, $f_\lambda^v$
satisfying:
\begin{itemize}
\item[i)] $f_\lambda=f_\lambda^h \circ f_\lambda^v\,,$
\item[ii)] $f_\lambda^v\,{\cal M}_{x_0,x}= {\cal M}_{x_0,x}$
for all $x \in \sphere^n-\{x_0, -x_0\},$
\item[iii)] $f_\lambda^h {\cal P}_{x,\gamma}={\cal P}_{x,\gamma}$
for all $\gamma \in [-1,1] \, ,$
\item[iv)] if for some $x\in \sphere^n$ and $\lambda \in \Lambda$
one has $f_\lambda(x)=x$ then
$f_\lambda^h(x)=f_\lambda^v(x)=x$. In particular,
$f_\lambda^h|_{U\cup U'}=f_\lambda^v|_{U\cup U'}=Id$.
\end{itemize}
Moreover, if we restrict $f_\lambda^h$, $f_\lambda^v$ to
be in a neighborhood of the identity, they are unique, and the mapping
${\cal F}_1$ that assigns to each $f_\lambda$ the pair $f_\lambda^h$,
$f_\lambda^v$ is continuous when we give its domain and its range the
topology of $C^l$ families.
\end{lemm}
\noindent {\bf Proof.} For any $x \in \sphere^n-\{x_0,-x_0\}$,
we represent by $\gamma(x)$ the projection of $x$ along
the line joining the origin and the point $x_0$.
Obviously, we should set $f_\lambda^v(x)$ to be the point
in the meridian of $x$ and on the parallel of $f_\lambda(x)$. Then,
$f_\lambda^h$ would move along the
parallels so as to get to $ f_{\lambda} (x) $.
More formally, using the notation for parallels and meridians
introduced in the previous section, we have:
\[ \begin{array}{l}
f_{\lambda} ^v (x) = {\cal M} _{x_0,x} \cap {\cal P} _{x_0,
\gamma(f_\lambda (x))} \\[.2cm]
f^h _{\lambda} (x) = {\cal M} _{x_0,f_\lambda ((f^v _\lambda)^{-1} (x))}
\cap {\cal P} _{x_0,\gamma (x)}.
\end{array} \]
Using the uniform transversality of the meridian and parallel foliations
outside a neighborhood of the poles, the usual implicit function theorem,
establishes that $f^v _\lambda (x)$ $f^h _ \lambda (x) $ depend jointly
$C^l$ on $x$ and on the parameter $ \lambda $, and consequently, they
are $C^l$ families of diffeomorphisms.
\qed
\medskip
The induction on the dimension
to prove Theorem \ref{main} will proceed as follows.
For $0<\varepsilon<1$, we will consider the sets
$ K^{\varepsilon} _i = \{ x \in \mbox\sphere^n \,/\, |x_i|
\leq \varepsilon \} $.
It is obvious that
$\cap_{i=1}^n K_i^\varepsilon= \emptyset$
when $ \varepsilon $ is sufficiently small.
\smallskip
Lemma~\ref{frag} will allows us to write any
diffeomorphism $f $ as composition
of diffeomorphisms $f_i$, $1 \leq i \leq{n+1}$,
each of which is the identity restricted to $ K^ \varepsilon _i $.
From now on we consider the case $i=1$; it is clear that the same
arguments could be applied to the rest of diffeomorphisms $f_i$, $i=2,
\cdots,{n+1}$.
Furthermore, for $f_1$ we will pick a polar axis going through the points
$x_\pm=(0,\ldots,\pm 1) $ and using Lemma~\ref{vert} we will
write $ f_1 = f^h \circ f^v $, as a composition of an horizontal and a
vertical diffeomorphisms.
If $f_1$ was sufficiently close to the identity then $ f^h$, $f^v $
would be the identity in $ K^\varepsilon _1 $.
We will apply the induction hypothesis to $f^h$ and will show how
to factor $f^v$.
To apply the induction hypothesis, we will think of $f=f^h$ as a family
of maps sending $\sphere^{n-1}$ into $\sphere^{n-1}$,
the parameter being the height of the parallel.
That is the map will be $f^h(p,\gamma)=({\tilde f}_\gamma(p),\gamma)$,
where ${\tilde f}_\gamma$ is a diffeomorphism of $\sphere^{n-1}$
for every $\gamma \in [-1,1]$. Moreover we have ${\tilde f}_\gamma =Id$
on $\sphere^{n-1}$ when $\gamma$ is close enough to $-1$ or $1$.
Notice how we are lead to consider families of diffeomorphisms even
if we start with a single one. With hindsight, this is the reason why
we have formulated the inductive hypothesis and the
statements of the paper in terms of families.
\smallskip
We obtain ${\tilde f}_\gamma={\tilde f}_{\gamma,1} \circ \cdots \circ
{\tilde f}_{\gamma,N'}$ as composition of $C^{l-2}$ families of
diffeomorphisms of $\sphere^{n-1}$. For each $i=1,\ldots,N'$
there exists $j \in \{1, \ldots, n^2\}$ (or $j \in \{1,2 \}$ when
$n=2$) such that
${\tilde f}_{\gamma ,i} \circ \delta_{y_j}= \delta_{y_j} \circ
{\tilde f}_{\gamma ,i}$ on $\sphere^{n-1}$ for all the values of $\gamma$.
Since the induction includes in its conclusions that the
factors ${\tilde f}_{\gamma,i}$ depend smoothly on the height,
and preserve the identity in a neighborhood of the poles,
we conclude that these factors are really diffeomorphisms
in $\sphere^n$ commuting with reflections. Clearly, the above arguments
apply without any change to families of horizontal diffeomorphisms.
\smallskip
We now proceed to establish the factorization of $f^v$ into symmetric
diffeomorphisms. According with the above remarks
we also can take the height as a periodic variable and consider the
parameters in a set $\Lambda={\bf T}^{m-2}$.
\begin{prop}\label{div}
For an $l \in \natural \cup \{ \infty \}$, $l\geq 7$,
we can find two points $y_1, y_2$ in the sphere and a
neighborhood ${\cal U}$ of the identity in the space of $C^l$
families of diffeomorphisms such that, for every family $f_\lambda^v\,\in
{\cal U}$ with
\begin{itemize}
\item[{\rm (h.1)}] $f_\lambda^v\,{\cal M}_{x_+,x}={\cal M}_{x_+,x} \qquad
\forall \; x \in \sphere^n-\{x_{\pm}\} ,$
\item[{\rm (h.2)}] $f_\lambda^v |_{K_1^\varepsilon}=Id$,
\end{itemize}
there exist two $C^{l-2}$ families $f_\lambda^{v,1}$, $f_\lambda^{v,2}$
in a neighborhood of the identity, satisfying
\begin{itemize}
\item[i)] $f_\lambda^{v,1}\,{\cal M}_{x_+,x}=
f_\lambda^{v,2}\,{\cal M}_{x_+,x}={\cal M}_{x_+,x} \qquad
\forall \;x \in \sphere^n-\{x_{\pm}\} ,$
\item[ii)] $f_\lambda^{v,1}\circ \delta_{y_1}=
\delta_{y_1} \circ f_\lambda^{v,1},$
\item[] $f_\lambda^{v,2}\circ \delta_{y_2}= \delta
_{y_2} \circ f_\lambda^{v,2},$
\item[iii)] $f_\lambda^v=f_\lambda^{v,1}\circ f_\lambda^{v,2},$
\item[iv)] if for some $\lambda $ one has $f_\lambda^v=Id$ then
$f_\lambda^{v,1}=f_\lambda^{v,2}=Id$.
\end{itemize}
Besides, the map ${\cal F}_2$ which assigns to $f_\lambda^v$ the factors
$f_\lambda^{v,1}$ and $f_\lambda^{v,2}$ is continuous for the $C^l$-topology
in its domain and the $C^{l-2}$-topology in its range.
\end{prop}
\medskip\noindent
{\bf Proof.} We remark again that, since the pair $f_\lambda^{v,1},
f_\lambda^{v,2}$ is not uniquely determined (even when $y_1, y_2$ are
fixed), the conclusions of continuity in the statement are valid only
for the map ${\cal F}_2$
constructed below.
We will take polar coordinates in $\sphere^n-K_1^\varepsilon$
defined by:
\[
\label{polar}
\begin{array}{rcl}
x_{n+1} &=&\cos \pi\varphi \\
x_n &=& \sin \pi\varphi \; \cos \pi \omega_1 \\
x_{n-1} &=& \sin \pi\varphi \; \sin \pi \omega_1 \; \cos \pi \omega_2 \\
\vdots & & ~~~~~~\vdots \\
x_3 &=& \sin \pi\varphi \; \sin \pi \omega_1 \cdots \cos \pi \omega_{n-2}\\
x_2 &=& \sin \pi\varphi \; \sin \pi \omega_1 \cdots \sin \pi \omega_{n-2}
\; \cos 2\pi \theta \\
x_1 &=& \sin \pi\varphi \; \sin \pi \omega_1 \cdots \sin \pi \omega_{n-2}
\; \sin 2\pi \theta \end{array}\]
for $\varphi$, $\omega_1\,,\dots,$ $\omega_{n-2}$, $\theta \in (0,1)$.
Notice that if $|x_1| \geq \varepsilon$, then
$|x_{n+1}|^2 \leq 1-\varepsilon^2$
so that $\varphi$ is defined univocally, and, since
$|x_n|^2+|x_{n+1}|^2 \leq 1-\varepsilon^2 $, $\omega_1$ is likewise outside
of the critical values of $sin$; this same argument is applicable to
$\omega_2 \ldots \omega_{n-2}$. In others words, this polar coordinate
system defines a smooth transformation $\Psi$ which identifies
$\sphere^n-K^\varepsilon_1$ with an open subset $V$ of ${\bf T}^n$.
Let $\delta$ be a small real positive number and $V_\delta=(\delta,
1-\delta)^{n-1}\times [0,1] \subset {\bf T}^n$.
Then $U_\delta=\Psi^{-1}(V_\delta)$ is
an open subset of $\sphere^n$ which contains $\sphere^n-K_\varepsilon$,
where the above coordinate system still works.
In particular the images by the map
$\Psi^{-1}$
of the points of the plane $\theta =0$ lie in $K_1^\varepsilon$.
It is clear that every diffeomorphism $f$ of
$\sphere^n$ with $f|K^\varepsilon_1=Id$ can be extended by the identity to
a diffeomorphism of ${\bf T}^n$.
We observe that when we express the family $f_\lambda^v$
of vertical diffeomorphisms of $\sphere^n$ in these coordinates,
the value of $\varphi$ changes but the others coordinates remain unaltered.
Therefore, a reflection across the plane $\Pi_{y_1}$ with
$y_1=(1,\ldots,0)$
is expressed as a change in sign for the $\theta$ coordinate while all the
others remain fixed. More generally, a reflection across $\Pi_{y_2}$
with $y_2=(\cos \rho,\sin \rho,\ldots,0)$ can be expressed in
the coordinates (\ref{polar}) by sending
$\theta$ into $2\rho -\theta$ and leaving all the other variables
unchanged.
If we take coordinates $(\varphi,\omega,\theta)$, the family $f_\lambda^v$
will be written as
\[ (\varphi,\omega,\theta) \rightarrow
(\varphi+v_\lambda(\varphi,\omega,\theta),\omega,\theta), \]
and we will look for $f_\lambda^{v,1}$ $f_\lambda^{v,2}$ that can be written
as
\[ (\varphi,\omega,\theta) \rightarrow
(\varphi+v_\lambda^i(\varphi,\omega,\theta),\omega,\theta)\]
with
\begin{equation}\label{2.2}
\begin{array}{l}
v_\lambda^1 (\varphi,\omega,\theta)=v_\lambda^1(\varphi,\omega,-\theta),\\[.2cm]
v_\lambda^2 (\varphi,\omega,\theta)=v_\lambda^2(\varphi,\omega,-\theta+2\rho).
\end{array}
\end{equation}
The equation that $v_\lambda^1$ $v_\lambda^2$ have to solve is, as
an straightforward calculation shows
\begin{equation}\label{2.3}
v_\lambda(\varphi,\omega,\theta)
=v_\lambda^1(\varphi+v_\lambda^2(\varphi,\omega,\theta),\omega,\theta)+
v_\lambda^2(\varphi,\omega,\theta).
\end{equation}
Notice that there is an open set of values of $w$ for which all
the points corresponding to $(\varphi,\omega,\theta)$ are
in $K_1^{\varepsilon}$, hence $v_\lambda(\varphi,\omega,\theta)=0$.
For these values of $w$, equation (\ref{2.3})
admits the trivial solution $v_\lambda^1(\varphi,\omega,\theta)= v_\lambda^2
(\varphi,\omega,\theta)=0$. Once established Proposition \ref{div},
its assertion $iv)$ would imply that $f_\lambda^{v,1}=Id$,
$f_\lambda^{v,2}=Id$ for the points in the sphere satisfying this condition.
(We will show a similar behavior in the variable $\varphi$).
Clearly, the solutions of (\ref{2.3})
can be extended by the identity to all the points of the
sphere in $K_1^{\varepsilon}$ even for those for which our coordinate system
does not work. Under conditions of proximity to the identity of
$f_\lambda^v$, $f_\lambda^{v,1}$, $f_\lambda^{v,2}$, these will be solutions
of our original problem.
Since, except for these considerations, $\omega$ does not enter
in either (\ref{2.3}) or in~(\ref{2.2}), we can consider these
variables as parameters in the problem in the same way as $\lambda$.
We will hence assume that $\lambda$ refers both to the parameters in the
family and to $\omega$, and omit $\omega$ from the notation.
We will carry out the proof of the proposition considering the family
$(f_\lambda^v)$ as a vertical diffeomorphism in ${\bf T}^m$
where we will solve the functional equation (\ref{2.3}). Finally, we will
show that the factors obtained are in fact diffeomorphism of
$\sphere^n$.
Proceeding heuristically for the moment,
the importance of the Diophantine condition already imposed on $\rho$
becomes apparent if we consider the {\it linearized equations}
obtained by expanding formally (\ref{2.3}) in the unknowns and keeping
only the linear terms. We obtain
\begin{equation}\label{2.5}
v_\lambda(\varphi,\theta)=v_\lambda^1(\varphi,\theta)+
v_\lambda^2(\varphi,\theta)
\end{equation}
By expanding $v_\lambda,$ $v_\lambda^1,$ and $v_\lambda^2$ in
Fourier coefficients in the variable $\theta$, $v_\lambda(\varphi,\theta)=
\sum_k \hat{v}_{\lambda,k}(\varphi) e^{ 2\pi i k \theta}$ and
$v_\lambda^j(\varphi,\theta)=
\sum_k \hat{v}^j_{\lambda,k}(\varphi) e^{ 2\pi i k \theta}$ for
$j=1,2$, we see that (\ref{2.5})
is equivalent to
\begin{equation}\label{2.6}
\hat{v}_{\lambda,k}(\varphi)=\hat{v}_{\lambda,k}^1(\varphi)
+\hat{v}_{\lambda,k}^2(\varphi).
\end{equation}
Similarly, the symmetry conditions
for $v^1_\lambda$, $v^2_\lambda$ can be expressed as
\begin{equation}\label{2.7}
\begin{array}{c}
\hat{v}^1_{\lambda,-k}(\varphi)=\hat{v}_{\lambda,k}^1(\varphi),\\[.2cm]
\hat{v}_{\lambda,-k}^2(\varphi)e^{4 \pi i\rho}=\hat{v}_{\lambda,k}^2(\varphi).
\end{array}
\end{equation}
When $k\neq 0$, we can group the equations for $k$, $-k$ in~(\ref{2.6})
and, using~(\ref{2.7}) we obtain
\[ \begin{array}{c}
\hat{v}_{\lambda,k}(\varphi)=\hat{v}_{\lambda,k}^1(\varphi)+
\hat{v}_{\lambda,k}^2(\varphi), \\[.2cm]
\hat{v}_{\lambda,-k}(\varphi)=\hat{v}_{\lambda,-k}^1(\varphi)+
\hat{v}_{\lambda,k}^2(\varphi) e^{-4 \pi i\rho}.
\end{array} \]
These equations admit the solutions
\begin{equation}\label{2.8}
\begin{array}{c}
\hat{v}_{\lambda,k}^1(\varphi)=(\hat{v}_{\lambda,k}(\varphi) e^{-4\pi i \rho k}
-\hat{v}_{\lambda,-k}(\varphi))/(e^{-4\pi i\rho k}-1), \\[.2cm]
\hat{v}_{\lambda,k}^2(\varphi)=(\hat{v}_{\lambda,k}(\varphi)
-\hat{v}_{\lambda,-k}(\varphi))/(e^{-4\pi i\rho k}-1). \\[.2cm]
\end{array}
\end{equation}
When $k=0$, there are many solutions. Since $v_\lambda(\varphi,0)=0$
for every $\varphi \in {\bf T}$ we choose
\begin{equation}\label{2.9}
\begin{array}{c}
\hat{v}_{\lambda,0}^1 (\varphi)=
- \sum_{k\neq 0}\hat{v}_{\lambda,k}^1 (\varphi), \\[.2cm]
\hat{v}_{\lambda,0}^2 (\varphi)=
- \sum_{k\neq 0}\hat{v}_{\lambda,k}^2 (\varphi),
\end{array}
\end{equation}
which provides $v^1(\varphi,0)=v^2(\varphi,0)=0$ for every $\varphi \in
{\bf T}$.
In addition, if $\varphi \in {\bf T}$ with $|\sin(\pi \varphi)|
<\varepsilon$, then $v_\lambda(\varphi,\theta)=
0$ for every $\theta \in {\bf T}$, hence their Fourier coefficients are
$ \hat{v}_{\lambda,k} (\varphi)=0$ for every $k$. From (\ref{2.8}) and
(\ref{2.9}) we deduce that both functions $v_\lambda^1(\varphi,\theta)=
v_\lambda^2(\varphi,\theta)=0$ for all $\theta$.
Notice that, if $\rho$ were rational, the denominators
in (\ref{2.8}) would become zero;
that is, there would be no solution for the equation. Even if $\rho$ is
irrational the denominators would become arbitrarily close to zero.
Nevertheless, if $\rho$ is a number of constant type, we have the bounds
\begin{equation}\label{2.10}
|(e^{-2 \pi i k \rho}-1)^{-1}| \leq C'|k|,
\end{equation}
that allow us to control the new Fourier expansions.
If we think of (\ref{2.3}) as an equation in spaces of functions,
expression (\ref{2.5})
would be the derivative --in some sense that will have to be specified--
at zero. The calculation shows that the derivative, even if
invertible, is not bounded;
hence, the usual implicit function theorem in Banach spaces does not apply.
We also emphasize that the above calculation shows only existence of the
inverse of the derivative at zero. If we had started making perturbations
around a non-zero, $v_\lambda^1$, $v_\lambda^2$ we would have been lead to an
equation that cannot be readily analyzed in terms of Fourier components.
The above heuristic discussion suggests that, among the several {\it hard}
implicit function theorems which can cope with unbounded inverses, the one that
will be useful for this problem is that of Zehnder \cite{Ze1}, which
requires existence
of inverse only at one point, but which also requires a {\it group}
structure for the equation.
We continue the proof by recalling Zehnder's theorem, which we have taken
from \cite{Ze1} in the version we are going to apply.
\begin{defi}\label{regu}
{\rm We say that $(X_\alpha)_{\alpha\geq0}$ is a} family of regular spaces {\rm when
$X_\alpha$ are real Banach spaces with norm $\| \quad \|_\alpha$ and, whenever
for $\alpha'\leq \alpha$ we have
\[ \begin{array}{l}
X_0\supset X_{\alpha'} \supset X_{\alpha }\supseteq X_\infty =
{\displaystyle \bigcap _{\alpha \geq 0}} X_\alpha, \\[.3cm]
\|x\|_{\alpha'} \leq \|x\|_{\alpha } \quad \forall x\in X_{\alpha }.
\end{array} \]
Furthermore, it is possible to define on them a $C^\infty$-regularization, that
is, a family $(S_t)_{t>0}$ of linear mappings $ S_t\,:\,X_0 \rightarrow
X_\infty$ and constants $C(\alpha,\alpha ')$ satisfying:
\[ \begin{array}{ll}
i) & {\displaystyle \lim_{t\rightarrow \infty}} \|S_t x-x\|_0=0 \quad
\forall x \in X_0, \\[.3cm]
ii) & \| S_t x\|_\alpha \leq t^{\alpha-\alpha '} C(\alpha,\alpha ')
\|x\|_{\alpha '} \quad \forall x \in X_{\alpha '} \quad
0\leq \alpha ' \leq \alpha, \\[.3cm]
iii) & \|S_t x-x\|_{\alpha' }\leq t^{\alpha '-\alpha} C(\alpha,\alpha ')
\|x\|_\alpha \quad \forall x\in X_\alpha \quad
0\leq \alpha' \leq \alpha. \end{array} \]}
\end{defi}
Notice that $X_\infty$ is dense in $X_0$ but not necessarily in $X_\alpha$
for $0<\alpha$; besides, the closure of
$X_\infty$ in $X_\alpha $ contains $X_{\alpha'}$ for every $\alpha
<\alpha'$.
We now consider several families of regular spaces and denote by the same
symbol their norms which are probably different. However, this notation
is standard and clear.
\begin{theo}\label{zehn}
Let $(X_\alpha)_{\alpha \geq 0}$, $(Y_\alpha)_{\alpha \geq 0}$,
$(Z_\alpha)_{\alpha \geq 0}$ be families of regular spaces
and $(x_0,y_0) \in X_\infty \times Y_\infty$.
Take $F\,:\, X_0 \times Y_0 \rightarrow Z_0$ with $F(x_0,y_0)=0$
and continuous restriction on the set $B_0$.
(For any $\alpha$, we will write $B_\alpha
= \{ (x,y) \in X_\alpha \times Y_\alpha \; /\; \|x-x_0\|_\alpha < 1/2\,, \;
\|y-y_0\|_\alpha < 1/2\}$.)
Assume:
\begin{description}
\item[{\rm (h.1)}] $F(x,\cdot)\,:\; Y_0\rightarrow Z_0$ is twice differentiable and
\[ \|d_2 F(x,y)\|\leq M_0\,, \quad \|d_2^2F(x,y)\|\leq M_0 \quad
\forall \;x,y \in B_0 \;.\]
\item[{\rm (h.2)}] $F$ is uniformly Lipschitz in $X_0$. That is, for
all $(x,y) \in B_0$, $(x',y)\in B_0$
\[ \|F(x,y)-F(x',y)\|_{0} \leq M_0 \|x-x'\|_{0}\;.\]
\item[{\rm (h.3)}] The triple $(F,x_0,y_0)$ is of order $\infty$. That is
\begin{itemize}
\item[{\rm (h.3.1)}] $F(B_0 \cap X_\alpha\times Y_\alpha) \subset Z_\alpha
\quad 0\leq \alpha$.
\item[{\rm (h.3.2)}] there exist constants $M_\alpha$, $1\leq \alpha$
such that if $(x,y)\in X_\alpha\times Y_\alpha\cap B_1$ satisfies
$\|x-x_0\|_\alpha \leq K,$ $\|y-y_0\|_\alpha \leq K$,
then $\|F(x,y)\|_\alpha \leq K\,M_\alpha$.
\end{itemize}
\item[{\rm (h.4)}] There exist an approximate right-inverse of loss $\sigma$,
$1\leq \sigma$. That is, for every $\alpha \geq \sigma$ and
$(x,y) \in B_\sigma \cap (X_\alpha \times Y_\alpha)$ there exist
a linear mapping $\eta (x,y) \in {\cal L}(Z_\alpha,Y_{\alpha-\sigma})$
such that:
\begin{itemize}
\item[{\rm (h.4.1)}] $\|\eta(x,y)z\|_0\leq M_0 \|z\|_\sigma$
for all $z \in Z_\sigma $ .
\item[{\rm (h.4.2)}] $\|d_2F(x,y)\circ \eta(x,y)-1)z\|_0 \leq M_0
\|F(x,y)\|_\sigma\,\|z\|_\sigma$ for all $z \in Z_\sigma $.
\item[{\rm (h.4.3)}] if $\|x-x_0\|_\alpha \leq K,$
$\|y-y_0\|_\alpha \leq K$, then
$\| \eta(x,y)(F(x,y))\|_{\alpha-\sigma} \leq K\,M_\alpha$.
\end{itemize}
\end{description}
Then, if $\varepsilon$ is a small positive number and
$\lambda=2\sigma+\varepsilon$, we can find an open neighborhood
$D_\lambda=\{ x\in X_\lambda \;/\; \|x-x_0\|_\lambda < C_\lambda \}$
and a mapping $\psi\,:\, D_\lambda \rightarrow Y_\sigma$ such that
\begin{itemize}
\item[i)] $F(x,\psi(x))=0$.
\item[ii)] $\|\psi(x)-y_0\|_\sigma \leq C_\lambda ^{-1} \|x -x_0\|_\lambda$.
\item[iii)] there exist $0<\beta <1$ and constants $C_\tau$ for
$\lambda< \tau$, such that for every $x \in D_\lambda \cap X_\tau$
one has $\psi(x) \in Y_{\beta (\tau-1)}$ and
$\|\psi(x)-y_0\|_{\beta (\tau-1)} \leq C_\tau \|x -x_0\|_\tau^\beta $.
\end{itemize}
\noindent In particular, $\psi(D_\lambda \cap X_\infty) \subset
Y_\infty$. Moreover, if $\eta (x,y)$ depends continuously on $(x,y)$,
then the mapping
$\psi\,:\, D_\lambda \rightarrow Y_\sigma$ is continuous.
\end{theo}
We next precise the spaces and functional to which the implicit function
theorem will be applied to yield the main result. For each $\mu \in {\bf T}$, we denote by $C^r_\mu({\bf T}^m)$
the set of functions $v$ of $ C^r(${\bf T}$^m)$ satisfying
\[v_\lambda(\varphi,\theta)=v_\lambda(\varphi,-\theta+2\mu), \quad
\forall \; (\lambda,\varphi,\theta)\in {\mbox{\rm \bf T}}^m\,.\]
Let us fix $l \in \natural$, $l \geq 7$, we take
\[X_\alpha=Z_\alpha=\left\{ v\in C^{l-7+\alpha}({\bf T}^m)\,/\,
v_\lambda(\varphi,0)=0 \quad \forall \;(\lambda,\varphi)
\in {\bf T}^{m-1}\right\}\]
and
\begin{eqnarray*}
\lefteqn{
Y_\alpha=\left\{ (v^1,v^2) \in C^{l-5+\alpha}_0({\bf T}^m)
\times C_\rho^{l-5+\alpha} ({\bf T}^m)\,/\,\right.} \\[.2 cm]
& & \hspace{2cm} \left. v^i_\lambda(\varphi,0))=0, \;
\forall \; (\lambda,\varphi) \in {\bf T}^{m-1},\;i=1,2 \right\}
\end{eqnarray*}
for any $\alpha \geq 0$, with the corresponding H\"{o}lder
norms and define the operators
\[ \begin{array}{lcl}
F\;: & X_\alpha\times Y_\alpha &\longrightarrow Z_\alpha\\
& (v,v^1, v^2) & \longrightarrow
v_\lambda^1(\varphi+v_\lambda^2(\varphi,\theta),\theta)+
v_\lambda^2(\varphi,\theta)-v_\lambda(\varphi,\theta).
\end{array} \]
>From Lemma 2.5 of \cite{Ze1} it is immediate to show that
$(X_\alpha)_{\alpha \geq 0}$, $(Y_\alpha)_{\alpha \geq 0}$ and
$(Z_\alpha)_{\alpha \geq 0}$ are families of regular spaces.
Observe that we only use those diffeomorphisms in ${\bf T}^m$ which are the
identity on the plane $\theta=0$ in order to guarantee the uniqueness
of the decomposition into symmetric factors.
\smallskip
{\bf Remark.} Consider a $C^r$ vertical diffeomorphism of
${\bf T}^m$ in a small neighborhood of the identity,
$f_\lambda(\varphi,\theta)=(\varphi +v_\lambda(\varphi,
\theta),\theta)$ with $r\geq 1$.
Then $f_\lambda^{-1}$ is also a $C^r$ vertical
diffeomorphism $f_\lambda^{-1}(\varphi,\theta)=(\varphi +u_\lambda(\varphi,
\theta),\theta)$. Besides, taking $B_r=\{v \in C^r/ ||v||_{
C^r}<1/2 \}$ the map $B_r \subset C^r({\bf T}^m) \rightarrow
C^{r'}({\bf T}^m) $, $v \rightarrow u$ is continuous for
every $r'0$, depending only on $\rho$, such that
\begin{equation}\label{2.11}
|h(z)|\leq \frac{\kappa\,M}{\tau-\tau'} \qquad {\mbox {\rm for}} \quad
|{\mbox {\rm Im}} z| < \tau'.
\end{equation}
\end{lemm}
We take a periodic real analytic function $v_{\lambda}(\varphi,\theta)$
defined in the strip $U_\tau^m$ with $v_\lambda(\varphi,0)=0$ for
every $(\lambda, \varphi) \in {\bf T}^{m-1}$ and consider the
pair $h_{\lambda}^1,h_{\lambda}^2$ whose Fourier coefficients
in the variable $\theta$ ,$({\hat h}_{\lambda,k}(\varphi))_{k\in Z}$,
are given by the relations (\ref{2.8}) when $k \neq 0$ and by
${\hat h}_{\lambda,0}(\varphi)=0$ for every $\lambda, \varphi$. Define
\[v_\lambda^1(\varphi,\theta)=h_\lambda^1(\varphi,\theta)-
h_\lambda^1(\varphi,0)\]
\[v_\lambda^2(\varphi,\theta)=h_\lambda^2(\varphi,\theta)-
h_\lambda^2(\varphi,0).\]
Observe that the Fourier coefficients in the variable $\theta$ of
$v_\lambda^1, v_\lambda^2$
are given by the equations (\ref{2.8}) and (\ref{2.9}). The functions
$h_\lambda^1, h_\lambda^2, v_\lambda^1, v_\lambda^2$
are analytic in the strip $U_{\tau'}^m$ for every
$\tau'<\tau$ and satisfy a
inequality similar to (\ref{2.11}) with maybe some other constants.
(Notice that if $\rho$ is a constant type number, so is
$-2 \rho$, $e^{-4 \pi i \rho k }/( e^{-4 \pi i \rho k } - 1) =
1+1/(e^{-4 \pi i \rho k }-1)$ and that
$\sup_{ \Im z \le \tau} |\sum_k {\hat v}_k e^{ 2 \pi i k z }| =
\sup_{ \Im z \le \tau} |\sum_k {\hat v}_{-k} e^{ 2 \pi i k z }|$).
Besides $v_\lambda(\varphi,\theta)=v_\lambda^1(\varphi,\theta)+
v_\lambda^2(\varphi,\theta)$.
>From Lemma \ref{rus} it is also possible to
deduce regularity properties of the above decomposition
in the classes of finitely differentiable functions.
The key is that the differentiability properties of functions
can be read of from the size of analytic approximations
on thin strips --see \cite{Mo}--.
The natural spaces on which to consider such characterizations are
the $\Lambda_r$ spaces endowed with the Zygmund norms.
We just remark that,
when $r \notin \natural$, $\Lambda_r = C^r$,
whereas $r \in \natural$, $C^r \subset \Lambda_r $
and the inclusion is strict.
Notice, however that the definition of $C^r$ functions
makes sense on Banach spaces (for $0 \le r \le 1^-$
it makes sense in metric spaces)
whereas the characterization by analytic approximations is
useful only for functions on $\real^n$.
A detailed argument along these lines can be found in
\cite{Mo} p. 528, \cite{Ze1}, \cite{Kr} which allows us
to state the following:
\begin{lemm}\label{mos}
Let $v \in C^r({\bf T}^m)$, $r \notin \natural$ $r>1$,
with $v_\lambda(\varphi,0)=0$ for every $(\lambda,\varphi) \in {\bf T}^{m-1}$.
There are functions $v^1 \in C^{r-1}_0({\bf T}^m)$,
$v^2 \in C^{r-1}_\rho({\bf T}^m)$ and a constant $\kappa$ depending only on
$\rho$ and $r$ such that
\begin{itemize}
\item[i)] $v=v^1+v^2$.
\item[ii)] $v^1_\lambda(\varphi,0)=v^2_\lambda(\varphi,0)=0$ for all
$(\lambda,\varphi) \in {\bf T}^{m-1}$.
\item[iii)] $ || v^1||_{C^{r-1} }, || v^2||_{C^{r-1} } \le \kappa ||v||_{C^r}$ .
\end{itemize}
\end{lemm}
Notice that $F:X_{\alpha} \times Y_{\alpha} \rightarrow
Z_{\alpha}$ is differentiable with respect to the $Y's$ components and
$d_2F(0,0,0)(v^1,v^2)=v^1 +v^2$ in all the cases.
Consequently, the conclusions of Lemma (\ref{mos})
provides a direct estimation of the exact inverse at $(0,0,0)$
of $d_2 F$. We obtain
\begin{lemm}\label{pde}
Let $\epsilon >0$ be a small parameter and $\sigma=3+\epsilon$ .
For every $\alpha \geq \sigma$ and
$v\in Z_{\alpha}$, there is an unique element
$(v^1,v^2) \in Y_{\alpha-\sigma}$ such that $v=v^1+v^2$. Moreover, the map
\[ \begin{array}{rclc}
\eta\;:&Z_{\alpha}&\longrightarrow & Y_{\alpha-\sigma}\\
&v &\longrightarrow &(v^1,v^2)
\end{array} \]
is continuous.
\end{lemm}
(More precisely, the map $ \eta\;:Z_{\alpha-\epsilon/2}
\rightarrow Y_{\alpha-\sigma}$ is also continuous).
Now an approximate right-inverse in a complete neighborhood of $(0,0,0)$ is
required. An important observation is that one exact right-inverse
can be obtained thanks to the
group action. See Hamilton \cite{Ham}, pg 198 and Zehnder \cite{Ze1}, pg. 133.
Let us consider $F_2:Y_\alpha \rightarrow Z_\alpha$,
$(v^1,v^2) \rightarrow F(0,v^1,v^2)$. It is
obvious that $d_2F(v',v^1,v^2)=dF_2(v^1, v^2)$.
We write $B_\alpha^2=\{(v^1,v^2) \in Y_\alpha
\;/\; ||v^1||_\alpha <1/2, ||v^2||_\alpha < 1/2\}$.
\begin{lemm}
For each $\alpha \geq \sigma$, there exists a continuous map
$\eta: \;B_\sigma^2 \cap Y_\alpha \rightarrow {\cal L}(Z_\alpha,
Y_{\alpha-\sigma})$ such that
\[dF_2(v^1,v^2) \circ \eta(v^1,v^2)
(v)=v\] for every $(v^1,v^2) \in B_\sigma^2 \cap Y_\alpha$
and $v \in Z_{\alpha}$.
\end{lemm}
\medskip
{\bf Proof.}
First of all we solve this question in the spaces of infinitely
differentiable functions.
The sets $X_\infty, Y_\infty, Z_\infty$ can be understood as
tangent spaces at the identity of groups of
$C^\infty$ diffeomorphisms in
${\bf T}^m$.
Let us fix $(v^1,v^2)\in B_0^2 \cap Y_\infty$
(that is $||v^i||_{C^{l-5 }}<1/2, i=1,2$),
with which we associate
the following differentiable functions
\[ \begin{array}{clcl}
\Phi_{(v^1,v^2)}:\;&Z_\infty &\longrightarrow &Z_\infty\\
&w &\longrightarrow & v_\lambda^1(\varphi+
w_\lambda(\varphi+{ v}_\lambda^2(\varphi,\theta),\theta)
+{ v}_\lambda^2(\varphi,\theta),\theta)\\
&&& \qquad + w_\lambda(\varphi+{ v}_\lambda^2(\varphi,\theta),\theta)
+{ v}_\lambda^2(\varphi,\theta)
\end{array} \]
and
\[ \begin{array}{clcl}
{\cal C}_{(v^1,v^2)}:\;&\:Y_{\infty}&\longrightarrow& Y_{\infty}\\
&(\omega^1,\omega^2)& \longrightarrow &
( v_\lambda^1(\varphi+\omega_\lambda^1(\varphi,\theta),\theta)
+ \omega_\lambda^1(\varphi,\theta),\\
&&& \qquad \omega_\lambda^2(\varphi+v_\lambda^2(\varphi,\theta),\theta)
+ v_\lambda^2(\varphi,\theta)).
\end{array} \]
Take the vertical diffeomorphisms $f_\lambda^{i}
(\varphi, \theta)=(\varphi+v^{i}_\lambda(\varphi,\theta)),\theta)$
with inverses $(f_\lambda^{i})^{-1}
(\varphi, \theta)=(\varphi+u^{i}_\lambda(\varphi,\theta)),\theta)$,
$i=1,2$.
Notice that $F_2(v^1,v^2)= \Phi_({v^1,v^2})(0)$;
besides we find the following relations
\begin{equation}\label{2.12}
\Phi_{(v^1,v^2)} \circ
\Phi_{((u^1),(u^2))}=Id.
\end{equation}
\begin{equation}\label{2.13}
\Phi_{(v^1,v^2)} \circ F_2=F_2
\circ {\cal C}_{(v^1,v^2)}.
\end{equation}
By differentiating (\ref{2.13}) we obtain
\[ d\Phi_{(v^1,v^2)}(0) \circ dF_2(0,0)=
dF_2{(v^1,v^2)} \circ d{\cal C}_{(v^1,v^2)}(0,0);\]
therefore, from relation (\ref{2.12}) and Lemma (\ref{pde}),
we can deduce that $\eta(v^1,v^2)=
d{\cal C}_{(v^1,v^2)}(0,0) \circ \eta \circ
d\Phi_{(u^1,u^2)}(F_2(v^1,v^2))$
is a linear continuous map from $Z_\infty$ to $Y_\infty$ with
\begin{equation}\label{2.14}
dF_2(v^1,v^2) \circ \eta(v^1,v^2)(v)=v
\end{equation}
for every $v \in Z_{\infty}$.
We now focus our attention to the classes of finitely differentiable
functions where we consider defined $\Phi_{(v^1,v^2)}$
and ${\cal C}_{(v^1,v^2)}$.
It follows from Propositions 6.1, 6.2 and Theorem 6.10 in \cite{LO}
that if $(v^1,v^2) \in B^2_\sigma \cap Y_\alpha$
then $\eta(v^1,v^2) \in {\cal L}(Z_\alpha,Y_{\alpha-\sigma})$ and
besides $\eta :\;B_\sigma^2 \cap Y_\alpha \rightarrow
{\cal L}(Z_\alpha, Y_{\alpha-\sigma})$, $(v^1,v^2) \rightarrow
\eta(v^1,v^2)$ is continuous.
Actually, a more precise analysis of the continuity of $\eta$
allows us to state that,
if $(v^1,v^2) \in B^2_0
\cap Y_{\alpha-\epsilon/2}$, then
$\eta(v^1,v^2) \in {\cal L}(Z_{\alpha-\epsilon/2}
,Y_{\alpha-\sigma})$ and the map $\eta :\;B_{0}^2
\cap Y_{\alpha-\epsilon/2} \rightarrow
{\cal L}(Z_{\alpha-\epsilon/2}, Y_{\alpha-\sigma})$
still remains continuous.
Taking into account that the closure of
$B_0^2 \cap Y_\infty$ and $Z_\infty$
in the natural topologies of $Y_{\alpha-\epsilon/2}$ and
$Z_{\alpha -\epsilon/2}$ contain $B^2_{\sigma} \cap Y_{\alpha}$ and
$Z_\alpha$ respectively, we deduce from (\ref{2.14}) that
\[dF_2(v^1,v^2) \circ \eta(v^1,v^2)(v)=v\]
for every $(v^1,v^2) \in B_\sigma^2 \cap Y_\alpha$ and
$v \in Z_\alpha$. This proves that $\eta$ is an exact
right-inverse of loss $\sigma$ of $d_2F$.
\qed
We have verified all the hypotheses of Theorem (\ref{zehn}).
Its conclusions provide the factorization of every $C^l$ vertical
diffeomorphism of ${\bf T}^m$ in a neighborhood of $Id$ into $C^{l-2}$
factors commuting with reflections across the planes $\theta =0$ and
$\theta =\rho$ respectively. In these conditions we can return
to our original problem in the sphere.
Let us assume that one of the above diffeomorphisms on the torus
becomes from $f_\lambda^v$, a $C^l$ family of vertical
diffeomorphisms of $\sphere^n$ with $f_\lambda^v|_{K_\varepsilon^1}=Id$,
and consider its decomposition $f_\lambda^v=f_\lambda^{v,1} \circ
f_\lambda^{v,2}$ according to the thesis of Theorem (\ref{zehn}).
We denote, as usual, $f_\lambda^{v}(\varphi,\theta)= (\varphi+v_\lambda
(\varphi,\theta),\theta)$, the factors
$f_\lambda^{v,i}(\varphi,\theta)= (\varphi+v_\lambda^{i}(\varphi,
\theta),\theta)$ and their inverses $(f_\lambda^{v,i})^{-1}(\varphi,\theta)=
(\varphi+u_\lambda^{i}(\varphi,\theta),\theta)$, $i=1,2$.
Let us fix $\lambda \in {\bf T}^{m-2}$ and take $ \varphi \in {\bf T}$ with
$|\sin (\pi \varphi)| \leq \varepsilon$. Then $v_\lambda(\varphi,\theta)=0$,
which implies that $(f_\lambda^{v,1})^
{-1}(\varphi,\theta)=(f_\lambda^{v,2})(\varphi,\theta)$ and hence
$u_\lambda^1(\varphi,\theta)=v_\lambda^2(\varphi,\theta)$ for every
$\theta \in {\bf T}$. An analysis of this equality based on the
Fourier coefficients, given in (\ref{2.7}) and
(\ref{2.9}), shows that $u_\lambda^1(\varphi,\theta)=v_\lambda^2(\varphi,\theta)
=0$ and hence $f_\lambda^{v,1}(\varphi,\theta)=f_\lambda^{v,2}(\varphi,\theta)
=(\varphi,\theta)$ for every $\theta \in {\bf T}$.
This same argument proves that if for some $\lambda$ one has
$f_\lambda^v= Id$ then also $f_\lambda^{v,1}=f_\lambda^{v,2}= Id$.
Thus, it is clear that $f_\lambda^{v,1}, f_\lambda^{v,2}$ can be extended
by the identity on the open subset of $\sphere^n$, where our coordinate
system does not work, and define
$C^{l-2}$ families of vertical diffeomorphisms of
$\sphere^n$ satisfying all the conditions
$(i)-(iv)$ of Proposition (\ref{div}). This
finishes the induction process and completes the proof
of Theorem (\ref{main}).
\qed
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% This are to make the latex
%bibliography compatible with other
%set of macros that I had for other bibliography data bases
%
%
\def\by#1{ #1 }
\def\jour#1{{\sl #1} }
\def\vol#1{{\,\bf #1}}
\def\yr#1{\quad(#1)}
\def\paper#1{{\em #1}}
\def\book#1{\,{\em #1}\,}
\def\publisher#1{ #1 }
\def\pages#1{ #1 }
\def\endref{}
\def\ref{}
\def\no{\bibitem}
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\end{document}