The body of this paper,
from the line following "BODY" to the line preceding "ENDBODY",
contains a total of 196640 characters.
In the table below, this count is broken down by ASCII code;
following one white space after the code is the corresponding character
(intended for trouble shooting, in case things got garbled).
128087 lowercase letters
6120 uppercase letters
2691 digits
4084 ASCII characters 10
23325 ASCII characters 32
20 ASCII characters 34 "
8 ASCII characters 35 #
6126 ASCII characters 36 $
26 ASCII characters 37 %
30 ASCII characters 38 &
572 ASCII characters 39 '
1398 ASCII characters 40 (
1442 ASCII characters 41 )
40 ASCII characters 42 *
311 ASCII characters 43 +
2227 ASCII characters 44 ,
975 ASCII characters 45 -
1416 ASCII characters 46 .
110 ASCII characters 47 /
159 ASCII characters 58 :
17 ASCII characters 59 ;
153 ASCII characters 60 <
581 ASCII characters 61 =
94 ASCII characters 62 >
35 ASCII characters 64 @
25 ASCII characters 91 [
7646 ASCII characters 92 \
25 ASCII characters 93 ]
1180 ASCII characters 94 ^
2715 ASCII characters 95 _
143 ASCII characters 96 `
1845 ASCII characters 123 {
677 ASCII characters 124 |
1845 ASCII characters 125 }
492 ASCII characters 126 ~
BODY
\input amstex
\documentstyle{amsppt}
\advance\hsize 2cm
\advance\vsize 1cm
\refstyle{C}
% \language=0
\topmatter
\title
Smooth nonautonomous normalizations for contractive mappings
\endtitle
\author
Serge\u\i~A.~Dovbysh
\endauthor
\affil
Institute for Mechanics,
Moscow State University \\
Michurinskij prospekt, d.1,
117192, Moscow, Russia
\endaffil
\email dovbysh\@inmech.msu.su \endemail
\toc
\head 1. Preliminary definitions for number sets\endhead
\head 2. The narrow band spectrum condition\endhead
\head 2. Preliminary definitions for number\endhead
\head 3. Nonautonomous normalization\endhead
\head 4. Proof of Proposition 3 and further discussion\endhead
\head 5. Normalization and semi-centralizer theorems for smooth
contractive extensions\endhead
\endtoc
\keywords Normal forms, normalization, (semi)conjugacy,
hyperbolic fixed point, hyperbolic theory, itinerary scheme
\endkeywords
\abstract
The construction of smooth nonautonomous normalizations for a sequence of
contractive mappings is discussed which generalizes the recent construction
by M.~Gyusinsky and A.~Katok~\cite{20}. The associated Normal Form Theorem and
Semi-Centralizer Theorems for extensions of homeomorphisms on
compact spaces by smooth contraction mappings are considered.
The approach utilized is essentially geometrical and is based on the usual
tools of hyperbolic theory for mappings in Banach spaces.
Some remarks for the general case of formal normal forms without the
contractness assumption are also made.
\endabstract
\endtopmatter
\rightheadtext{Nonautonomous normalizations}
% Some definitions
\define\id{\operatorname{id}}
\define\rang{\operatorname{rang}}
\define\card{\operatorname{card}}
\define\const{\operatorname{const}}
\def\Diff{\operatorname{Diff}}
\let\eps\varepsilon
\let\phi\varphi
\let\kap\varkappa
\redefine\le{\leqslant}
\redefine\ge{\geqslant}
\define\pre{\preccurlyeq}
\define\suc{\succcurlyeq}
\define\supp{\operatorname{supp}}
\define\mD{\overline{\Cal D}}
\catcode`\@=11
% `Cyrillic prime' for use in transliterating Russian soft sign
% in Russian names.
\def\cprime{\begingroup\everymath{}\m@th$'$\endgroup}
\let\mz\cprime % ``myagkii znak'' synonym for \cprime
\define\supsetar{\begingroup\m@th\mkern3mu%
{\ooalign{$\hfil\mkern2mu\supset\mkern0mu\hfil$\crcr%
$\hfil{\lower.48ex\hbox{$\scriptstyle{\leftarrow}$}}\mkern8mu\hfil$\crcr%
$\hfil{\raise1.155ex\hbox to 6pt%
{\vrule height.3pt width6pt depth0pt}}\mkern8mu\hfil$}%
}\mkern1mu\endgroup}
\catcode`\@=\active
\define\tim{\widetilde m}
\define\tPhi{\widetilde\Phi}
\define\tis{\tilde s}
\TagsOnRight
% New definitions
% \define\tJ{{\widetilde J}}
\define\fJ{{\frak J}}
\define\fI{{\frak I}}
\define\cP{{\Cal P}}
\define\cR{{\Cal R}}
\define\cE{{\Cal E}}
\define\cI{{\Cal I}}
\define\cJ{{\Cal J}}
\define\ocR{{\overline\cR}}
\define\St{\operatorname{St}}
\define\tw{\widetilde w}
\define\ttw{\widetilde{\widetilde w}}
\define\Comp{\operatorname{Comp}}
\define\ulambda{\underline\lambda}
\define\olambda{\overline\lambda}
\def\tcR{{\widetilde{\cR}}}
\document
Recently, M.~Gyusinsky and A.~Katok~\cite{20} (see also~\cite{25})
studied the non-autonomous normal forms for smooth
contractions on $\Bbb R^n$. The relevant centralizer theorem has been applied
to establish the differential rigidity of Anosov actions~\cite{25}.
In the most general framework, extensions of topological dynamical systems by
smooth ($C^\infty$) contractions of $\Bbb R^n$ are considered. The usual
considerations based on the contraction argument (see Remark~11 below)
show the presence of a unique invariant
continuous section. The conjugacies of the extensions by a continuous family
of local $C^\infty$-diffeomorphisms of the fibers preserving the invariant
section were dealt with.
Some our considerations are parallel to those in the paper cited.
However, the present work generalizes essentially the paper~\cite{20} because
of the following reasons. Firstly, the authors of~\cite{20} restrict themselves
by studying normal forms that correspond to a rather special stable set $\cR$
(in terms of the present paper).
They denoted these forms as {\it ``sub-resonance normal forms''\/}.
All the consideration were essentially adopted to this particular case,
although the general case can be easily treated by the geometric method
developed below.
(The considerations in~\cite{20} are more analytic.)
We will show that the narrow band spectrum condition introduced by M.~Gyusinsky
and A.~Katok is not only sufficient but also necessary for the possibility to
construct nonautonomous normal form with good properties.
Secondly, the argument used to transfer from the formal normalization
to the actual one (the analytic part of the normalization procedure)
will be not optimal in the case of contraction mappings
of a finite smoothness. In~\cite{25} the authors announced some results by
M.~Gyusinsky for the case of a finite differentiability but the smoothness
assumptions are seen to be not optimal. Thirdly, the authors of~\cite{20}
assumed that the invariant subspaces $M_{\tis}$ are spectrally
separated rather than exponentially separated. This means that conditions~i)
and~ii) of the Proposition~3 below are imposed on the union of all the spectra
of operators $J_i$ rather than on each spectrum. Notice in this connection
that, due to Remark in Appendix~A of~\cite{14}, the case of the exponential
separation is indeed more
typical in applications than the case of the spectral separation%
\footnote{N.~Fenichel~\cite{18, Part~II} has proposed a simple but somewhat
artificial example of bundles
which are exponentially separated but not spectrally separated. In
contrast to it, our example presented in~\cite{14} is very
natural and typical in the smooth dynamical systems.
I.U.~Bronshtein~\cite{10} has noticed that one can construct
an $C^1$-open set of diffeomorphisms on the two-dimensional torus
(originating from an algebraic automorphism) which satisfy the exponential
separation condition but are not spectrally separated.}.
Notice also that, in contrast to~\cite{20} and~\cite{13} (see below),
we prefer to solve appearing functional
equations by the direct contraction arguments rather than via analyzing the
equations obtained by iteratively substituting along the orbit.
We state three Theorems (in fact, the Normal Form Theorem and the
two Semi-Centralizer Theorems) which generalize
the corresponding two results of~\cite{20} related to the existence of
``sub-resonance normal forms'' and to the structure of the centralizers
for these normal forms.
Under the absence of resonances (the case $\cR=\emptyset$),
the nonautonomous normalization constructed below is reduced to
the nonautonomous linearization that has been discussed in detail
in~\cite{14}. A paper by Y.~Yomdin~\cite{38} was cited therein, in which the
construction of the nonautonomous linearization for a bi-infinite sequence of
contractive mappings has appeared.
It happened, however,
that the nonautonomous normalizations (in particular, linearizations) for
systems of differential equations analytic in phase variables have been
considered earlier by Y.~Sibuya~\cite{35} and V.V.~Kostin~\cite{27, 28} (see
also a preprint-like textbook~\cite{29}).
On the other hand, these authors proved the convergence of the formal
normalizing expansions (the analytic part of the normalization procedure) by
the use of the classical analytic tools, in contrast to Y.~Yomdin who applied
the hyperbolic theory geometric in character which can be utilized in the
smooth (non-analytic) case (see Section~3).
So, Y.~Sibuya dealt with the
general case of bounded coefficients and constant linear part and looked for
normalizing transformations bounded over the whole infinite time interval.
The possibility of including parameters was discussed and the convergence of
the normalizing transformation in the case where the spectrum of the linear
part lies in the left complex half-plane was proved.
V.V.~Kostin considered the case where the linear part depends, generally
speaking, on time but it takes the triangular or even Jordan form and the
boundedness conditions are not assumed (in particular, in view of sharp
restrictions for terms left in the normal form). The main attention was
drawn to the convergence problem for nonautonomous normalizing transformations.
The convergence conditions obtained are of an integral form.
Notice also that the papers of the authors cited here contain a transfer of
the classical results~\cite{8} (described in Section~3 below) to the
nonautonomous case. Inheriting
the almost-periodicity of the system by its normal form and by the normalizing
transformation was also shown.
As for the paper by Y.~Yomdin~\cite{38}, we
notice that the simplicity of the spectra and the one-dimensionality
of the associated invariant subspaces that were assumed therein are not
essential.
On the other hand, he considered a slightly more general case where the
sequences of mappings and linearizing transformations
grow not faster than exponentially. This generalized situation can also be
easily described in terms of itinerary schemes as it will be done
in Remark~11 below. Y.~Yomdin dealt also with some special situation
where all the elements of the sequence of differentials
coincide and possess a simple spectrum that lies in the Siegel domain, and,
moreover, all the mappings are analytic (we treat a situation where
the spectra lie inside the unit circle). Then the presence of the
nonautonomous linearization is proved via the usage of the KAM-theory.
The subject of the present paper is, on the one hand, an extension of
the corresponding results of~\cite{14} to the general case of the presence of
resonances, and, on the other hand, it is an immediate generalization and
improvement of the results obtained by M.~Gyusinsky and A.~Katok.
Thus, many definitions and results, including the basic Proposition~3, which
are presented here are proper generalizations of the concepts of~\cite{14} to
the general resonance case.
The nonautonomous normalization constructed was already applied to the
many-dimensional non-integrability problem, see announcement of some results
in~\cite{15}.
Among other nearby results, one should mention the paper by
D.~De~Latte~\cite{13} who proved the non-autonomous variant
of the Moser theorem on the normal forms for hyperbolic area-preserving
two-dimensional maps. Both the analytic and $C^\infty$ cases are considered
and the non-autonomous normal forms are shown to provide a countably many
invariants that form obstructions to smooth conjugacy of symplectic Anosov
diffeomorphisms on the torus. In fact, the general case of extensions of
topological dynamical systems (homeomorphisms on compact spaces) by smooth
hyperbolic area-preserving maps of $\Bbb R^2$ is also considered.
The structure of the paper is as follows. In Section~1 some preliminary
definitions and results for number sets are presented. In the context
of the paper, an important notion of a stable set $\tcR$ is introduced.
Section~2 is devoted to the concept of the narrow band spectrum condition
which is reproduced from~\cite{20} (see also~\cite{25}).
It is established that the validity of this
condition for the spectrum is necessary and sufficient for the existence of
an associated set $\tcR$ which is finite and stable.
In Section~3 some results on normal forms will be recalled and their
nonautonomous analogs will be obtained. For this purpose, the basic
Proposition~3 is stated. Section~4 is devoted to proving this Proposition and
further discussions. The importance of the stability condition for $\tcR$
(achieved provided that the nonautonomous version of the narrow band
spectrum condition is satisfied) is indicated. Namely, it is shown that
the (partial) normal forms associated to the stable set $\tcR$
are the most adequate generalizations of the usual
normal forms. In particular, the corresponding Semi-Centralizer Theorem
generalizing the Centralizer Theorem in~\cite{8} (for the formal case) and
in~\cite{5,20} (for the contractive smooth case) is established.
In Section~5, the extensions of homeomorphisms on compact spaces by smooth
contractive mappings are considered and the corresponding Normal Form Theorem
and two Semi-Centralizer Theorems are stated and proved.
These theorems generalize the results by M.~Gyusinsky and A.~Katok as was
mentioned above. Some modifications of the conditions of these theorems are
considered to avoid an explicit usage of a particular Lyapunov norm.
Three Historical comments are included in the text of the paper, to review
the related papers and to explain the novelty of the results presented.
The author is thankful to A.~Katok and D.~De~Latte for sending offprints
of their papers~\cite{20} and~\cite{13}.
\head
1. Preliminary definitions for number sets
\endhead
Let complex numbers $\lambda_1,\ldots,\lambda_n $ satisfy
the condition $0<|\lambda_j|<1$.
Consider partitions of the set $\{\lambda_j \}$ into classes
$\Lambda_r$.
\definition{Definition 1} A partition is said to be
{\it strongly ordered \/} if the
classes $\Lambda_r $ are sets of numbers, $\lambda_j $, whose moduli lie,
respectively, in some non-intersecting intervals.
\enddefinition
The classes $\Lambda_r$ will be placed in the order of
non-decreasing moduli of the numbers $\lambda\in\Lambda_r $.
Let $\{\lambda_{i,j}: 1\le j\le n\}$
be some number sets consisting of $n$ elements such that
$0<|\lambda_{i,j}|<1$. Suppose that there are strongly
ordered partitions
$$ \xi_i =\bigl\{\Lambda_{i,1},\ldots,\Lambda_{i,p} \bigr\}
$$
of these sets. Moreover, assume that
the quantities of classes, $p$, and the numbers of elements in
each class $\card\Lambda_{i,s}=c_s\, (1\le s\le p)$
do not depend on $i$. Here $\card$ denotes the cardinality (the quantity
of elements) of a set.
\definition{Definition 2} Strongly ordered partitions $\xi_i$ which
satisfy these conditions will be said to be {\it concordant\/}.
\enddefinition
Analogously, one can consider complex numbers $\mu_j$ such that $|\mu_j|>1$
and introduce strongly ordered partitions and concordant partitions.
Obviously,
this case is conjugate with the previous one by the inversion operation
$\mu_j\mapsto \lambda_j=\mu_j^{-1}$. Therefore, elements of partitions for
this case will be placed in the order of non-increasing moduli.
In the sequel, we will write $z^m=\prod_{j=1}^n z_j^{m_j}$ and
$|m|=\sum_j m_j$ for any collection $z=(z_1,\ldots,z_n)$ and any multiindex
$m=(m_1,\ldots,m_n)$ with non-negative integer components $m_j$.
Under discussing the nonautonomous normal forms of contractive mappings,
the usual multiplicative Poincar\'e non-resonance conditions
$$ \lambda_s\ne \lambda^m, \tag 1\mz
$$
where $\lambda=\{\lambda_j\}$ is the spectrum
at the fixed point and the multiindex $m$ is such that $|m|=\sum_j m_j \ge 2$,
will be replaced by that
$$ |\lambda_s|\ne |\lambda^m| \tag 1
$$
(cf.~\cite{38,20,14}).
These conditions are obviously satisfied if $|m|>D$ where
$$ D=\ln \min_j | \lambda_j | /
\ln \max_j | \lambda_j |\ge 1.\tag 2
$$
Let $\cR$ be some finite (possibly, empty) set of couples $(s,m)$ with
multiindex $m$ satisfying $|m|\ge 2$ and index $s$ such that all
inequalities~(1) with $(s,m)\notin\cR$ are valid.
In other words, all the couples $(s,m)$, for which the resonance
conditions $|\lambda_s |=|\lambda^m |$ are met, belong to $\cR$, i.e.
$|\lambda_s |=|\lambda^m |\Rightarrow (s,m)\in\cR$.
Therefore, the set $\cR$ will be conditionally said to
{\it contain resonances\/}. Obviously, there is a smallest set, $\cR_0$,
containing resonances (the set {\it consisting of the resonances\/}).
In Section~4, while discussing the usual normal forms, we will also use
the {\it exact resonances\/} $\lambda_s=\lambda^m$ and
the sets $\cR$ which may be infinite (since the spectrum may lie from both
the sides of the unit circle) and which contain these resonances,
i.e., such that $\lambda_s=\lambda^m\Rightarrow (s,m)\in\cR$.
The corresponding smallest set, $\cR_0^{(ex)}$, consisting of the
exact resonances, will be also utilized therein.
For a given partition
$\xi =\bigl\{\Lambda_1,\ldots,\Lambda_p\bigr\}$,
the sets $\cR$ of couples $(s,m)$ satisfying the following condition have
a great importance: if $(s,m)\in\cR$ where $m=(m_1,\ldots,m_n)$,
the numbers $\lambda_s$, $\lambda_{s'}$ belong to
the same class of $\xi$, and unordered set
$\St(m')=\{\underbrace{1,\ldots,1}_{m_1^\prime},\ldots,
\underbrace{n,\ldots,n}_{m_n^\prime}\}$
is obtained from the set
$\St(m)=\{\underbrace{1,\ldots,1}_{m_1},\ldots,
\underbrace{n,\ldots,n}_{m_n}\}$
via replacement of some numbers $j$ by numbers $j'$ such that $\lambda_j$
and $\lambda_{j'}$ belong,
respectively, to the same classes of $\xi$,
then $(s',m')\in\cR$ where $m'=(m_1',\ldots,m_n')$.
\definition{Definition 3} The set $\cR$ satisfying this condition will
be said to be {\it subordinate to the partition $\xi$\/} or
{\it $\xi$-subordinate\/}.
\enddefinition
Notice that this definition allows some equal numbers $\lambda_j$ with
different indices $j$ to belong to different classes of the partition
(in fact, the partition of the set of indices $\{1,\ldots,n\}$
is considered).
Given an index $s$ and a multiindex
$m=(m_1,\ldots,m_n)$, introduce%
\footnote{Introducing the index $\tis$, we correct a minor
inaccuracy passed in~\cite{14}.}
$\tis\,(1\le\tis\le p)$, where
$\lambda_s\in \Lambda_{\tis}$, and $\tim=(\tim_1,\ldots,\tim_p)$, where
$\tim_t=\sum\limits_{\lambda_j\in \Lambda_t} m_j\ge 0$, and
denote $(\tis,\tim)=\Psi(s,m)$. Obviously,
$|\tim|=|m|$. Then for any set $\tPhi$ of couples $(\tis,\tim)$,
the set $\Phi=\Psi^{-1}(\tPhi)$ is subordinate to
the partition $\xi$, and for any set $\Phi$ of couples $(s,m)$, the
condition $\Phi=\Psi^{-1}\bigl(\Psi(\Phi)\bigr)$ means exactly that $\Phi$
is subordinate to the partition $\xi$.
The operation $\Psi$ is determined by the partition $\xi$ and for concordant
partitions the corresponding $\Psi$'s are obviously coincident.
We will sometimes write $\Psi_\xi$ in order to emphasize the dependence
of $\Psi$ on $\xi$.
Obviously, the set consisting of the resonances, $\cR_0$, is
$\xi_0$-subordinate and
the set consisting of the exact resonances, $\cR_0^{(ex)}$, is
$\xi_0^{(ex)}$-subordinate, where
$\xi_0$ is the partition into the classes of numbers having the equal moduli
and $\xi_0^{(ex)}$ is the partition into the classes of equal numbers.
\definition{Definition 4}
A partition $\xi$ of the set $\{\lambda_j\}$ is said to be
{\it strongly $\tcR$-right\/} if the set $\cR=\Psi_\xi^{-1}(\tcR)$
contains the resonances of $\{\lambda_j\}$ and the following
condition is satisfied: ($*$)~two numbers $\lambda_{j'}, \lambda_{j''}$
belong to the same class
if and only if in all inequalities~(1) such that $(s,m)\notin\cR$
the signs persist under replacement of all factors $\lambda_{j'}$ by
$\lambda_{j''}$ or all factors
$\lambda_{j''}$ by $\lambda_{j'}$ in the left-hand side and right-hand
side simultaneously.
\enddefinition
\remark{Remark 1} (cf. Remark~1 of~\cite{14}.)
For a given $\cR$, the condition~($*$) indeed determines an
equivalence relation.
However, the existence of a strongly $\tcR$-right partition does not yet follow
from this, since this partition was already used in the auxiliary definition
of the subordinate set $\cR$. Due to the same reason, the strongly $\tcR$-right
partition does exist for not any set $\cR$ containing resonances.
A strongly $\tcR$-right partition, if exists, is unique and is also strongly
ordered.
The definition of the strongly $\tcR$-right partition
admits an equivalent formulation dealing with the replacement
of an arbitrary part of the factors.
This definition admits also a form which will be indicated in the proof
of Proposition~1.
Another fact is that we prefer to say about $\tcR$-right partitions rather
than about $\cR$-right ones. This complicates slightly the notations but
will allow us to consider in the sequel $\tcR$-right partitions for
the sets of different cardinalities%
\footnote{This is the difference from~\cite{14} which is made to remove
an inaccuracy having been passed in condition~ii) of Proposition~5 of
\cite{14} while stating Proposition~3 below.}.
\endremark
According to Definition~4, the attempt to construct the strongly $\tcR$-right
partition can lead to the necessity to modify the set $\cR$ itself. Since the
set $\cR$ has to contain the resonances, one should go ahead to enlarge it
(possibly, choosing originally the smallest set containing resonances).
Generally speaking, the bigger set $\cR$, the less non-resonance conditions
appear in Definition~4 and, accordingly, the rougher is the strongly
$\tcR$-right partition.
In the extreme case, the finally modified set $\cR$ contains
all the couples $(s,m)$ such that $|m|\le N$ where $N>D$ and the corresponding
strongly $\tcR$-right partition consists of the single class. The following
Proposition shows, however, that to any set containing resonances,
one can associate in a natural manner the smallest ambient set $\cR^*$ and,
respectively, the finest strongly $\tcR^*$-right partition.
\proclaim{Proposition 1} Let $\cR$ be a set containing resonances.
Then there is a smallest set $\cR^*\supseteq\cR$, for which the
strongly $\Psi_{\xi^*}(\cR^*)$-right partition $\xi^*$ is defined.
Precisely speaking,
if $\cR'\supseteq\cR$ and $\xi'$ is a strongly $\Psi_{\xi'}(\cR')$-right
partition then
$\cR^*\subseteq\cR'$ and $\xi^*\suc\xi'$ (the partition $\xi'$ is a roughing
of $\xi^*$). If $\cR=\emptyset$ then $\cR^*=\emptyset$.
\endproclaim
\demo{Proof} If $\cR=\emptyset$ then $\cR$ is certainly subordinate to
the partition, and the condition~($*$) gives an equivalence relation that
determines the partition sought for. Therefore, $\cR^*=\emptyset$.
In the general case, the idea is to construct the set $\cR^*$ sought for
as a result of minimal enlargements of the set $\cR$ whose necessity is
stipulated by the subordination condition. We replace the condition~($*$)
by the following condition that determines (after a revision below)
an equivalence relation for any set $\cR$: (${*}{*}$)~two numbers
$\lambda_{j'}$, $\lambda_{j''}$ belong to the same class
if and only if there is a chain of numbers
$\lambda_{j'}=\lambda_{j_0},\lambda_{j_1},\ldots,\lambda_{j_{k-1}},
\lambda_{j_k}=\lambda_{j''}$ such that the numbers $\lambda_{j_s}$,
$\lambda_{j_{s-1}}$ ($0\le s1$.
It is convenient sometimes to consider also couples $(s,m)$ such that $|m|=1$
(these couples will be said to be {\it linear\/})
and to supplement to the set $\cR$, subordinated to the partition $\xi$, all
the couples $(i,{\bold e}_j)$ such that $\lambda_i$ and $\lambda_j$ belong to
the same element of the partition $\xi$
(these linear couples will be said to be $\xi$-{\it trivial\/}).
Here ${\bold e}_i$ denotes the basis
multiindex such that $({\bold e}_i)_j=\delta_{i,j}$, the Kronecker symbol.
Thus, for the given partition $\xi$, one can identify the $\xi$-subordinate
sets $\cR$ and the corresponding ``extended'' sets.
Generally speaking, one can consider ``extended'' sets containing arbitrary
linear couples $(s,m)$. Then the above notation of $\xi$-subordinate set
remains valid. The only restriction in the sequel will be that for every index
$s$ there is a couple $(s,m)\in\cR$ (see also Remark~3 below).
Let $\cR_1$, $\cR_2$ be two such
``extended'' sets of couples $(s,m)$ subordinated to the same partition $\xi$.
Put $(s,m)\ltimes\bigl((s^{(1)},m^{(1)}),\ldots,(s^{(q)},m^{(q)})\bigr)$
to be equal to $(s,m')$ with $m'=\sum_{t=1}^qm^{(t)}$ if $q=|m|$ and every
index $j$ is met exactly $m_j$ times among the numbers
$s^{(1)},\ldots,s^{(q)}$. Otherwise the $\ltimes$-expression will be assumed
to be undefined. In other words, the set $\St(m')$ is obtained from the set
$\St(m)$ via replacement of each element $j$ by the corresponding subset
$\St(m^{(k)})$ such that $s^{(k)}=j$ where different indices $k$ are associated
to different elements $j$. Furthermore, let $\cR_1\ltimes\cR_2$ be the set
of all such the sensible $\ltimes$-expressions where $(s,m)\in\cR_1$ and all
$(s^{(t)},m^{(t)})\in\cR_2$. Obviously, the set $\cR_1\ltimes\cR_2$ is also
subordinate to the partition $\xi$, $\cR_1\ltimes\cR_2\supseteq\cR_1\cup\cR_2$
if $\cR_1$, $\cR_2$ contain linear $\xi$-trivial couples,
and $\Psi(\cR_1\ltimes\cR_2)=\Psi(\cR_1)\ltimes\Psi(\cR_2)$ (the sets in the
right-hand side of the last equality are subordinate to the partitions into
the one-element classes). Moreover, the set operation $\ltimes$ is associative,
but not commutative which is evident from the following Remark~2 explaining
the actual reason for introducing the set operation $\ltimes$.
According to Definition~8 below one can associate in a natural way to each
$\xi$-subordinate (``extended'') set $\cR$ some class of the polynomial
(or, more generally,
formal power series) mappings $g:\Bbb C^n\to\Bbb C^n$, the so-called
$(\{M_{\tis}\},\tcR)$-quasi-normal forms. We explain briefly this notation in
a self-contained and slightly simplified way. Let ${\bold e}_j\,(1\le j\le n)$
be some basis in $\Bbb C^n$ and $M_{\tis}$ be the subspace spanned by
all the ${\bold e}_j$'s such that $\lambda_j\in\Lambda_{\tis}$. Let
$\tcR=\Psi(\cR)$ with the mapping $\Psi$ associated to $\xi$.
A {\it $(\{M_{\tis}\},\tcR)$-quasi-normal form\/} is a polynomial or
formal power series mapping $g:\Bbb C^n\to\Bbb C^n$ which takes in the basis
$\{{\bold e}_j\}$ the form
$$ \bigl(g(x)\bigr)_s=\sum a_{s,m}x^m
$$
such that $a_{s,m}\ne 0$ only if $(s,m)\in\cR$.
\remark{Remark 2}
Denote by $\Cal F_\tcR$ the set of the $(\{M_{\tis}\},\tcR)$-quasi-normal
forms where $\tcR$ is any set of couples $(\tis,\tim)$.
Let $\tcR_1$, $\tcR_2$ be two sets of couples $(\tis,\tim)$. Then
$g_1\circ g_2\in\Cal F_{\tcR_1\ltimes\tcR_2}$ if $g_i\in\Cal F_{\tcR_i}$
($i=1,2$), and this result is unimprovable in the sense that
$\tcR_1\ltimes\tcR_2$ is the minimal set $\tcR$ such that
$g_1\circ g_2\in\Cal F_\tcR$ for all $g_i\in\Cal F_{\tcR_i}$. Moreover, the
linear hull of all such $g_1\circ g_2$ coincides with the whole
$F_{\tcR_1\ltimes\tcR_2}$.
\endremark
\definition{Definition 6} The set $\cR$ (``extended'' or subordinated to a
given partition) is said to be {\it stable\/} if $\cR\ltimes\cR=\cR$.
\enddefinition
Obviously, a set $\cR$ subordinated to a given partition $\xi$ is stable if and
only if the corresponding set $\Psi(\cR)$ is stable. The smallest set
containing resonances $\cR_0$ (containing exact resonances $\cR_0^{(ex)}$,
respectively), is always stable.
The set $\cR$ can be not stable even if the partition $\xi$ is strongly
$\Psi_\xi(\cR)$-right, which is seen in the following simple
\example{Example 1} If $\{\lambda_j\}=\{2,4\}$ then the smallest set containing
resonances, $\cR_0$, has a single element, and for $\cR=\cR_0$, the partition
$\xi^*$ consists of one class, and $\cR^*$ consists of all the six couples
$(s,m)$ such that $|m|=2$.
The set $\underbrace{\cR^*\ltimes\cR^*\ltimes\cdots\ltimes\cR^*}_k$ consists of
all the couples $(s,m)$ such that $|m|\le k+1$.
\endexample
\head
2. The narrow band spectrum condition
\endhead
Recall an important concept introduced originally in~\cite{20}
(see also~\cite{25}).
Let $\{\lambda_j\}$ be a finite collection of numbers such that
$0<|\lambda_j|<1$ and let $\xi=\bigl\{\Lambda_1,\ldots,\Lambda_{p} \bigr\}$
be a strongly ordered partition of the set $\{\lambda_j\}$,
$\olambda=\max\limits_j |\lambda_j|$,
$\ulambda_s=\min\limits_{\lambda_j\in\Lambda_s} |\lambda_j|$,
$\olambda_s=\max\limits_{\lambda_j\in\Lambda_s} |\lambda_j|$
(then $\olambda=\olambda_p$).
\definition{Definition 7}
The strongly ordered partition $\xi$
is said to satisfy the {\it narrow band\/} condition if
$\olambda_s\olambda<\ulambda_s$ (or, equivalently,
$\olambda<\ulambda_s/\olambda_s$) for all $1\le s\le p$.
\enddefinition
This condition means that the partition $\xi$ is not too rough.
In particular, $\xi$ may be the partition into the classes of numbers with
equal moduli.
\proclaim{Proposition 2} Given a partition $\xi$, there is a finite stable
$\xi$-subordinate set $\cR$ containing (exact) resonances and such that $\xi$
is $\Psi_\xi(\cR)$-right, if and only if $\xi$ satisfies the narrow band
condition.
\endproclaim
\remark{Remark 3}
In all the further applications, $\cR$ will be a set containing
(exact) resonances. Thus, all the $\xi$-trivial linear couples belong to $\cR$.
Therefore, without loss of generality, one can account that only $\xi$-trivial
couples are the linear couples presented in $\cR$, since removing the other
linear couples from $\cR$ does not violate the stability of $\cR$.
\endremark
We preface the proof of Proposition~2 by recalling some basic concepts and
by the graphic interpretation as follows.
Firstly, we recall the definition of a
{\it topological Markov chain\/} (TMC) and how it is
determined by a directed graph. The term ``topological Markov chain''
was introduced by V.~M.~Alekseev~\cite{2, Part 1} although the corresponding
mathematical object can be found in earlier papers.
In the English mathematical literature it is usually referred
to as {\it ``subshift of finite type'' \/}
going back to S.~Smale~\cite{36}
(because there is a natural one-to-one topological equivalence
between topological Markov chains and Smale's subshifts
of finite type).
Consider a finite set ({\it ``alphabet''\/})
$\Cal L$ containing $m<\infty$ elements and equipped with the discrete
topology. One can put $\Cal L= \{1,\ldots ,m\}$. Denote by
$\Omega=\prod_{n=-\infty}^{+\infty} \Cal L$ the Tychonoff product of a
countable collection of copies of the space $\Cal L$, i.e., the space of
doubly-infinite sequences
$\omega=\bigl[\omega_n\in \Cal L : -\infty0$ if and only if
$\tim_j\ge 1$ for some $(i,\tim)\in\tcR$. Then, by some permutation of the
indices, the matrix $\Pi$ can be represented in the {\it normal\/}
block-triangle form (see, for instance, F.R.~Gantmacher~\cite{19}).
This means that there are some pairwise non-intersecting subsets
$\Cal L_l\subset \Cal L$ such that
$\pi_{i,j}>0$ only if $i$ and $j$ belong to the same $\Cal L_l$ or if $i0$ and then the TMC as was described above.
The edges whose origins and ends belong to the same
$\Cal L_l$ constitute a {\it connected\/} subgraph $\Gamma_l$, i.e.,
for any two vertices $a, b$ of the graph $\Gamma_l$, there is
a path on $\Gamma_l$ with origin $a$ and end $b$. All the closed circuits
(paths) form exactly the subgraphs $\Gamma_l$.
Introduce the notation
$\ltimes^k\cR=\underbrace{\cR\ltimes\cdots\ltimes\cR}_{\Sb k+1 \\
\text{copies of } \cR\endSb}$.
The graph $\Gamma$ can be used to characterize particularly the set
$\ltimes^k\tcR$ as follows. Given $\tis\in\Cal L$ and $k\ge 0$, one has
$\tim_j>0$ for some $(\tis,\tim)\in\ltimes^k\tcR$ if and only if there
is a path of length $(k+1)$ (i.e., consisting of $(k+1)$ edges) with the
origin at $\tis$ and the end at $j$. In a more general framework, the
correspondence that associates a TMC to a set $\tcR$ possesses the following
contravariance property: if the TMC $T_i$ corresponds to the set $\tcR_i$,
$i=1,2$, then $T_2\circ T_1$ corresponds to $\tcR_1\ltimes\tcR_2$.
One can control also $|\tim|$ using a more detailed description based on
trees as follows. Consider trees whose vertices belong to $\Cal L$ (maybe, with
repetitions) and such that from each tree's vertex $i$, placed not on the top
of the tree, there grows a bunch of tree's edges which consists of exactly
$\tim_j$ edges $\overset\longrightarrow\to{(i,j)}$ for some couple
$(i,\tim)\in\tcR$ (determining uniquely this bunch). Thus, some edges in
one bunch may be identical but their ends, $j$'s, will be considered as
distinct vertices of the tree. Assume that the heights (the quantity of the
successive edges leading from the root to the top) of all the branches of the
tree are equal
exactly to $(k+1)$. In other words, every maximal ordered path embedded into
the tree has length $(k+1)$. Then all the elements of $\ltimes^k\tcR$ are
described by these trees so that $(\tis,\tim)\in\ltimes^k\tcR$ if and only if
the following holds: there is a tree $\Cal T$ such that $\tis$ is the root
vertex of $\Cal T$ and, for any $j$, exactly $\tim_j$ top vertices of $\Cal T$
are placed at a given $j\in\Cal L$.
\demo{Proof of Proposition 2}
Obviously, $\cR_{0,\xi}=\Psi^{-1}\bigl(\Psi(\cR_0)\bigr)$ is the
smallest $\xi$-subordinate set containing resonances and
$\bigcup_{k=1}^\infty \ltimes^k\cR_{0,\xi}$ is the smallest $\xi$-subordinate
stable set containing resonances.
Obviously, these considerations remain valid if only exact
resonances are considered and the set $\cR_{0,\xi}$ is replaced by
$\cR_{0,\xi}^{(ex)}=\Psi^{-1}\bigl(\Psi(\cR_0^{(ex)})\bigr)$.
We will say that a $\xi$-subordinate set $\cR$ (or the associated set
$\tcR=\Psi(\cR)$) {\it contains a circuit\/}
$(\tis^{(1)},\tim^{(1)}),\ldots,%
(\tis^{(t)},\tim^{(t)})=(\tis^{(1)},\tim^{(1)})$ if
$(\tis^{(i)},\tim^{(i)})\in\tcR$ and $\tim^{(i)}_{\tis^{(i+1)}}\ge 1$
for all $1\le iq$ which excludes the stability of
$\ltimes^k\cR$ for any $k\ge 1$.
Now assume that all the circuits are linear and show that $\ltimes^k\cR$ is
stable for some $k$. Every (ordered) path of the graph $\Gamma$ cannot pass
twice
through a vertex in $\Cal L\setminus\cup_l \Cal L_l$ or reenter second time
into some set $\Cal L_l$. Therefore, there is a uniform upper bound for the
quantity of the path edges not belonging to $\cup_l \Gamma_l$ (in fact, this
quantity does not exceed
$t=\card(\Cal L\setminus\cup_l \Cal L_l)+\card\{l\}-1$ with $\card$ denoting
the cardinality of the set). Notice that any tree does not branch at the
origin of its edge provided that this edge lies in some subgraph $\Gamma_l$.
Consequently, we can conditionally cut out all such the edges from the tree,
thus substituting a chain of edges belonging to the same $\Gamma_l$ by a
single vertex. After this surgical operation, the tree's height will admit
the above uniform upper estimate. This allows us to estimate $|\tim|$
uniformly from above for all $(\tis,\tim)\in\ltimes^k\tcR$ and arbitrary $k$
if $\tcR$ is finite. (In
fact, $|\tim|$ does not exceed $M=g^t$ with $g$ being the maximal $|\tim|$ for
$(\tis,\tim)\in\tcR$.) Thus, the set $\ltimes^k\tcR$ ranges a finite
collection of sets, and, consequently, $\ltimes^k\tcR=\ltimes^{k+q}\tcR$ for
some $k=k_0$ and $q>0$. This implies that $\ltimes^k\tcR$ depends
$q$-periodically on $k$ (i.e., $\ltimes^k\tcR=\ltimes^{k+q}\tcR$) for all
$k\ge k_0$. Therefore, $\ltimes^k\tcR$ is stable provided that $k\ge k_0$ and
$k$ is a multiple of $q$.
One could describe $k_0$ and $q$ in a more constructive manner applicable
also to infinite sets $\tcR$. Incidentally, this approach provides the
smaller estimates for $k_0$ and $q$ which depend only on the finite graph
$\Gamma$ (and are independent of the set $\tcR$ itself).
Let $q$ be a natural such that
any vertex in $\cup_l \Cal L_l$ belongs to some closed circuit of length $q$.
Under replacement of the TMC $T$ by its $q$-th power, $T^q$ (which corresponds
to the replacement of $\tcR$ by $\ltimes^{q-1}\tcR$), each set $\Cal L_l$
decomposes, generally speaking, into a collection of the analogous subsets, say
$\Cal L_l^{(j)}$. (In fact, these subsets are closely related to the general
formulation of the Spectral Decomposition Theorem for TMC,
see~\cite{4}.) For some $N$ every two
vertices in each $\Cal L_l^{(j)}$ can be joined by a path of length exactly
equal to $Nq$ (the restriction of $T^q$ to the set of admissible words formed
by the elements of $\Cal L_l^{(j)}$ is a {\it primitive\/} TMC,
see~\cite{4}). Therefore, if the length of a path in $\Gamma_l$
is $Q>q$ and $N'\ge N$ then one can replace an arbitrary subpath of length
$[Q/q]q$ with $[\cdot]$ denoting the entire part, by a subpath of length
$N'q$ with the same origin and end and with the edges lying also in $\Gamma_l$.
Thus, if $Q\ge Q_0=(N+1)q$ then one can shorten any path of length $Q$ in
$\Gamma_l$ with the fixed endpoints by exactly $q$ edges.
Notice that among the vertices of each path of length $k+1$ constituting
the original tree, at most $t'$ vertices do not
belong to $\cup_l \Cal L_l$ where $t'\le\card(\Cal L\setminus\cup_l \Cal L_l)$.
Therefore, if $k\ge t'$ then, certainly, each path passes through a vertex
lying in $\cup_l \Cal L_l$. Then one sees that
$\ltimes^{k+q}\tcR\supseteq\ltimes^k\tcR$, due to the possibility to paste
a closed circuit of length $q$ in each of these vertices.
Furthermore, if $k\ge k_1=t'+\card\{l\}\cdot(Q_0-1)$ then at each path,
there is a chain of $Q_0$ successive edges belonging to $\cup_l\Gamma_l$.
Replacing chains of these edges by chains of appropriate $(Q_0-q)$ edges
belonging again to $\cup_l\Gamma_l$, one sees that
$\ltimes^{k-q}\tcR\supseteq\ltimes^k\tcR$. Thus, finally,
$\ltimes^{k+q}\tcR=\ltimes^k\tcR$ as $k\ge k_0=k_1+q$.
The only trouble in these considerations is that every two paths in one tree
have a common initial piece and all the tree paths should be modified
simultaneously in a coherent way so that initial pieces are left identical
if modified. Fortunately, this is easily achieved by the simple rule
as follows: each
path is modified at the nearest to the root place where it is possible.
The sufficiency of this rule is seen from the following fact. Let the initial
pieces of some paths be coincident up to the vertex $\tis\in\Cal L_l$
(i.e., these paths belong to the same branch of the tree up to the vertex
$\tis$) included. Then these paths have coincident maximal chains (maybe,
empty) of successive edges belonging to $\Gamma_l$ and originated from
$\tis$. In other words, the maximal common initial piece is ended at a
vertex which is not the origin of a tree edge belonging to
$\cup_l\Gamma_l$.\qed
\enddemo
The following example generalizes that presented in~\cite{20} and shows
that $\ltimes^k\cR_{0,\xi}$ can ``stabilize'' for $k$ large enough.
\example{Example 2} Let $\Delta_j=[a_j,b_j]$, $1\le j\le p$, be a collection
of $p$ closed intervals in $(0,1)$ such that $a_j\le b_{j-1}$, $1\tis$. Here, the product $\prod_jA_j$ of the sets $A_j$ and the
set $A$ raised to the power $t$ are understood as the collections of the
corresponding numbers $\prod_ja_j$, $a_j\in A_j$, and $a^t$, $a\in A$.
If all $\Delta_j$ are placed in a small enough vicinity of a given point
of $(0,1)$ then~(4) can be satisfied only when $\sum_jf_j=1$ and $f_j=0$ for
$j>\tis$, due to the discreteness of the set of the possible $f_j$. Assume
that this is the case and then choose the length, $b_{j-1}-a_j\ge 0$, of the
intersection $\Delta_{j-1}\cap\Delta_j$ to be small enough for all
$10$ provided that the narrow band spectrum condition is met and have
constructed the associated $(\{M_{\tis}\},\ltimes^k\tcR)$-normal forms.
Incidentally, the desired stability result is easily seen from the fact that
$\tcR$, analogously to $\tcR_{0,\xi}$, takes the
triangular form with the linear diagonal part.
This proof is more simple and explicit than that presented in~\cite{20}.
\endremark
\head
3. Nonautonomous normalization
\endhead
In this Section we recall some results on the normal forms, in particular,
the Sternberg theorem~\cite{37,5} (see also~\cite{23}) on polynomial
resonant normal forms of contraction mappings and describe its
nonautonomous analog.
For this purpose, some quite natural class of particular normalizations will
be introduced and discussed.
We preface a discussion by the following
\remark{Remark 4} It is well-known that
``the problem of finding normal forms for smooth
transformations splits into two parts: one of a purely formal nature and the
other analytic in character''~\cite{37}. The formal part consists of the
normalization of the diffeomorphism jet at the fixed point by means of a formal
transformation, i.e., a transformation represented by a formal Taylor series.
The formal normalizing transformation constructed should be the jet of the
faithful normalizing transformation. The formal part provides also the
normal form. The analytic part consists of proving the existence of a (smooth)
normalizing transformation possessing the prescribed jet and carrying over
the diffeomorphism into the prescribed normal form. The purpose of the formal
normalization is to eliminate the non-resonant terms in the diffeomorphism jet.
It is convenient to do this by the successive steps for increasing powers.
Here, it is convenient to reduce beforehand the linear part to the Jordan
form which requires, generally speaking, the use of complex variables.
Since we consider real case, ``if complex
eigenvalues do occur, they occur in pairs of complex conjugates. In that case,
it is convenient to modify'' the mapping ``by replacing the pair of real
coordinates corresponding to the complex eigenvalues by a pair of conjugate
complex variables and to allow complex formal power series satisfying certain
symmetry (reality) conditions. This procedure is explained in detail
in~\cite{7, p.~60--63}. In that follows, these symmetry conditions are
automatically verified for the series that arise in virtue of the method of
their construction''~\cite{37} (see also~\cite{9}).
Thus, to the complex normal form constructed, there corresponds the so-called
real normal form.
In fact, the normalizing (formal or smooth) transformation is a result of the
following two successive transformations: a real (formal or smooth)
transformation reducing the original mapping to the so-called ``real normal
form''
and the linear transformation which replaces some pairs of real coordinates
by pairs of complex conjugate ones.
We recall the basic result~\cite{8} related to the formal normalization
(for simplicity, S.~Sternberg~\cite{37}, while discussing smooth
contraction mappings, considered the case of the absence of
the multiple elementary divisors of the linear part, which has no importance;
a general case of the smooth contraction mappings was also treated by
B.~Anderson~\cite{5}).
Consider formal power series $g:\Bbb C^n\to\Bbb C^n$ as
$$ \bigl(g(x)\bigr)_s=\sum\Sb s,m\\ |m|\ge1 \endSb g_{s,m}x^m,
$$
where $x=(x_1,\ldots,x_n)\in\Bbb C^n$ and the linear part
$dg{\mid}_0$ takes the (complex) Jordan form,
$\bigl(dg{\mid}_0(x)\bigr)_j=\lambda_jx_j+\sigma_jx_{j+1}$
(where $\sigma_j\in\{0,1\}$ and the eigenvalues $\lambda_j$ are properly
clustered).
Recall that each nonlinear term $g_{s,m}x^m$ is said to be {\it resonant\/}
(with respect to the linear part $dg{\mid}_0$) if $\lambda_s=\lambda^m$, and
otherwise, {\it non-resonant\/}, where
$\lambda=\{\lambda_1,\ldots,\lambda_n\}$ denotes the spectrum.
The {\it (complex) normal form\/} is a mapping $g$ vanishing at the origin
whose linear part $dg\mid_0$ takes the (complex) Jordan form and which
contains only nonlinear resonant terms. Now
let the non-degenerate linear part, $J$, of the formal mapping,
$$ \bigl(T(x)\bigr)_s=\bigl(J(x)\bigr)_s+
\sum\Sb s,m\\ |m|\ge2 \endSb t_{s,m}x^m, \tag 5
$$
is already reduced to a required Jordan form (by a preliminary linear
transformation) with the spectrum $\lambda$.
Then some formal transformation $y\mapsto x$ with the identity linear part,
$$ x_s=y_s+\sum\Sb s,m\\ |m|\ge2 \endSb \phi_{s,m}y^m, \tag 6
$$
transfers the mapping~(5) into the normal form.
This transformation is called a {\it normalization\/} or {\it normalizing
transformation\/}.
Moreover, the coefficients $\phi_{s,m}$ in the resonant terms of the
normalizing transformation~(6) (i.e., ones such that $\lambda_s=\lambda^m$)
can be arbitrarily given. Then the normalizing transformation and the normal
form are uniquely determined.
We will present a simple proof of this result in Remark~15.
For the contractive mappings of a sufficiently high but finite smoothness,
the theorem by S.~Sternberg ensures the
presence of not only formal but also actual polynomial (real) normal forms.
Then the result that the normalization is determined by its resonant terms
is easily carried over to the actual normalizations as will be evident from
Remark~6.
The below-described construction will provide also the proper
reformulations of this result to the situation of some weakened resonance
conditions.
\endremark
Now we recall in detail the Sternberg theorem.
Let $T\:U\to{\Bbb R}^n$ be a map of a vicinity, $U\subset{\Bbb R}^n$,
of the fixed point $0\in{\Bbb R}^n$ and let $J=dT(0)$ be its nondegenerate
linear part, $\lambda=\{\lambda_1,\ldots,\lambda_n\}$ be the spectrum of $J$
such that $0<|\lambda_j|<1$ for all $j$.
Let $D$ be the number determined by the spectrum of $J$ in accordance
with~(2). If $T\in C^N$ where natural $N>D$ then, by some coordinate
$C^N$-transformation called a {\it $C^N$-normalization\/} or
{\it $C^N$-normalizing transformation\/},
the mapping $T$ can be reduced to polynomial (real) normal form~(5).
It is obvious that in this normal form $t_{s,m}=0$ for $|m|>D$
because the corresponding non-resonance conditions $\lambda_s\ne\lambda^m$
are certainly met.
Now we describe the non-autonomous analog of the Sternberg theorem.
Under the non-resonance conditions~(1\mz), the polynomial normal form, $\fJ$,
of a contraction mapping $T$ is reduced to the linear operator $J$.
The corresponding
coordinate transformation is called a {\it linearization\/}.
The nonautonomous counterpart of the linearization (nonautonomous
linearization) for a sequence of contraction mappings has been constructed
by Y.Yomdin~\cite{38} and by the author~\cite{14}.
The basic idea of~\cite{14} was as follows (Y.Yomdin used very similar
considerations).
Sternberg's proof~\cite{37}
can be slightly rephrased in such a way that the linearization
of a contraction mapping $T\:U\to U$, $T(0)=0$, of a neighbourhood
$U\subset \Bbb R^n$ of the origin (under conditions~(1) that are sharper
than conditions~(1\mz) considered in~\cite{37} where $\{\lambda_j\}$ is the
spectrum of the linear part, $J$) will be sought as a fixed point
of a hyperbolic endomorphism $\Cal D_T$ in the Banach space of $C^N$-mappings
$f$. The endomorphism $\Cal D_T$ is defined so that the following
diagram is commutative:
$$\CD
@>T>> \\
@V\Cal D_T(f+\id)VV @VVf+\id V \\
@>>J>
\endCD\tag 7
$$
where $T(0)=0$, $J=dT(0)$, $f\:U\to \Bbb R^n$, and $f(0)=0$, $df(0)=0$.
Then the operator $\Cal D_T$ transforms functions defined in the image
of the mapping $T$ into functions defined in the pre-image, and $l=\id +f$
is the required linearization if $\Cal D_T l=l$. In order to deal with the
linear space of $C^N$-functions $f$ whose 1-jets vanish at the origin,
it is convenient to represent the required $f$ as a fixed point of the
mapping $(\cdot)\mapsto \mD_T(\cdot)=\Cal D_T(\cdot)+d_T$, where
$d_T=\Cal D_T\id-\id=J^{-1}\circ T-\id$. Obviously,
$\mD_{T_1\circ T_2}=\mD_{T_2}\circ \mD_{T_1}$,
$\Cal D_{T_1\circ T_2}=\Cal D_{T_2}\circ \Cal D_{T_1}$,
i.e., the correspondences $T\mapsto \mD_T,\, \Cal D_T$
are {\it contravariant functors\/}.
Now, applying the standard tools of hyperbolic theory for mappings in
Banach spaces, one can construct the desired nonautonomous normalization.
Now we admit the presence of some resonances.
One should consider a modified diagram~(7) where the linear map
$J=dT(0)$ is replaced by the nonlinear normal form $\fJ$. However, the proper
generalization of the above consideration requires some accuracy.
Now we will present in detail the proper modification of the
considerations of~\cite{14} related to the diagram~(7).
Let $\cR$ be a finite set containing resonances of the spectrum,
$\{\lambda_j\}$, of the linear part, $J$, of the mapping $T$, which is
subordinate to some partition $\xi=\{\Lambda_1,\ldots,\Lambda_p\}$ of
$\{\lambda_j\}$.
The only restriction is that all the multiple eigenvalues that correspond to
one Jordan block (``geometrically identical'' eigenvalues) belong to the same
class of the partition $\xi$. In particular, $\cR$ and $\xi$ can be,
respectively, the smallest set containing resonances, $\cR_0$, and the
partition into the classes of numbers with equal moduli, $\xi_0$.
We remove also from the set $\cR$ all the couples $(s,m)$ such that $|m|\ge N$
where natural $N>D$ (recall that these couples are {\it a priori\/}
non-resonant in the strengthened sense~(1)).
In the sequel, the multiindex $m$ will run over the finite set $2\le|m|T>> \\
@V l' VV @VVl V \\
@>>\fJ >
\endCD\tag 8
$$
with $l=f+\id$, $l'=f'+\id$
commutes when
$f'_\ocR=\cE_{T,f_\cR,f'_\cR}(f_\ocR)$,
$\fJ_\cR=\cJ_{T,f_\cR,f'_\cR}(f_\ocR)$, and
$\fJ_\ocR\equiv 0$ (notice that $d\fJ(0)=J$).
Here, in contrast to~\cite{14}, the nonlinearity of the maps introduced is
caused by that of the $(\{M_{\tis}\},\tcR)$-normal form. However, the map
$\cE_{T,f_\cR,f'_\cR}$ possesses good hyperbolicity properties over a given
bounded set in $E_\ocR$.
The correctness of these definitions and the hyperbolicity property of the map
$\cE_{T,f_\cR,f'_\cR}$ will be established in a more general framework while
proving Proposition~3.
Rigorously speaking, in Proposition~3 where a ``uniform'' variant of the above
construction is considered
which is required to build a ``nonautonomous normalization'',
conditions imposed on $\xi$ will be somewhat stronger. However, the proof
will be also valid under the above-mentioned assumptions concerning $\xi$.
In an auxiliary case of the inverted lower arrow of the latter diagram,
we will show the existence of nonlinear maps
$\cE_{T,f_\cR,f'_\cR}^{\leftarrow}:E_\ocR\to E_\ocR$
and $\cI_{T,f_\cR,f'_\cR}:E_\ocR\to E_\cR$
such that the diagram
$$\CD
@>T>> \\
@V l' VV @VVl V \\
@<<\fI <
\endCD
$$
with $l=f+\id$, $l'=f'+\id$
commutes when
$f'_\ocR=\cE_{T,f_\cR,f'_\cR}^{\leftarrow}(f_\ocR)$,
$\fI_\cR=\cI_{T,f_\cR,f'_\cR}(f_\ocR)$, and
$\fI_\ocR\equiv 0$ (notice that $d\fI(0)=I=J^{-1}$).
\remark{Remark 6} (See Remark~14 of~\cite{14}.)
Now we explain the proof of the Sternberg theorem
and establish its locally uniform variant as follows.
The normalizing $C^N$-transformation depends continuously on a
$C^N$-diffeomorphism
$T\:U\to\Bbb R^n$ if all the mappings are treated in the $C^k$-topology for
arbitrary $1\le k\le N$ and $T$ remains uniformly $C^N$-bounded from above.
This result remains valid for the case $k=0$
provided that an additional assumption is met that the perturbation of
the diffeomorphism $T$ is sufficiently small in the $C^1$-topology.
Here $N\ge 2$ is determined by the inequality $N>D$.
Actually, we adopt the original proof~\cite{37}.
Moreover, we will extend these results to the most general case where the
mapping $T$ is conjugated by a $C^N$-transformation $l$ possessing a
prescribed $(N-1)$-jet $j_0^{N-1}l$ with a mapping $\fJ$ possessing a
prescribed $G_\ast$-component, $\fJ-j_0^{N-1}\fJ$. Here $l$ and $\fJ$ will
depend continuously on $T$, $j_0^{N-1}l$, and the $G_\ast$-component of $\fJ$.
A fixed point of the contractive map $T$ depends continuously on $T\in C^1$.
Hence, without loss of generality, one can consider
perturbations of $T$ that leave the fixed point to be located at the origin.
There is a norm in $\Bbb R^n$
that induces an operator norm $\|\cdot\|$ such that $\|J\|<1$ and
$\mu=\bigl\|J^{-1}\bigr\| \|J\|^N<1$ where $J=dT(0)\:\Bbb R^n\to\Bbb R^n$.
Let $U$ be a small enough ball in this norm centered at the origin and let
$G_\ast$ be the space of $C^N$-functions $f\:U\to\Bbb R^n$ with zero
$(N-1)$-jets at the origin. Equip $G_\ast$ with the norm $\|\cdot\|_\ast$
being the exact upper bound for the norm of the $N$-th differential at all
the points of $U$.
Here the $N$-th differential is considered as a polylinear mapping
$\underbrace{\Bbb R^n\times\cdots\times \Bbb R^n}_N\to \Bbb R^n$ symmetrical
in each cofactor. (The desired norm in the space of polylinear mapping is
defined, for instance, in the textbook~\cite{12}.)
Then the affine map $\mD_T$ is well-defined and
the key fact used in~\cite{14}
is that $\bigl\|\Cal D_T\mid G_\ast\bigr\|_\ast\le\tilde\mu$,
$\Cal D_T$ being the linear part of $\mD_T$,
the operator norm
being induced by the norm $\|\cdot\|_\ast$ in $G_\ast$, and $\tilde\mu$ being
close to $\mu$ if the neighbourhood $U$ is chosen to be sufficiently small.
Indeed, elementary estimates show that
$|\tilde\mu-\mu|\le\eps\cdot\bigl\|J^{-1}
\bigr\|P_N\bigl(\|T\|_{C^N},\eps\bigr)$
where $P_N$ is some polynomial depending on $N$ only and $\eps$ is
the radius of the ball $U$.
The formal normalization of the $(N-1)$-jet of $T$ at the fixed point supplies
the normal form $\fJ$ sought for. The corresponding formal normalizing
transformation is the $(N-1)$-jet for $C^N$-functions $R$ with identity linear
part which satisfy the following condition: a $C^N$-transformation
from the old coordinates $x\in U\subset\Bbb R^n$ to the new
ones $y=R(x)$ transfers the mapping $T$ into the mapping
$T'=R\circ T\circ R^{-1}$ such that $T'-\fJ\in G_\ast$.
Then in the new coordinates $y$, the desired normalizing transformation
is sought for as $l'=\id+f$
where $f$ is a fixed point of the map
$(\cdot)\mapsto\widetilde\cE_{T',\fJ}(\cdot)=\fJ^{-1}\circ
\bigl(\id+(\cdot)\bigr)\circ T'-\id$ in the space $G_\ast$.
This map is well-defined and will be shown to be contractive over a
bounded subset of $G_\ast$. In the original coordinates, the normalizing
transformation is written as $l=l'\circ R$.
(In~\cite{14}, the author presented also another version of the above proof
where the preliminary transformation of coordinates is not performed. However,
this consideration is valid only if the normal form is linear.)
One could notice that $\fJ-J=\cJ_{T',0,0}(f)$ for every $f\in G_\ast$ and
$\widetilde\cE_{T',\fJ}=\cE_{T',0,0}\mid G_\ast$ under any choice of the set
$\cR$ containing resonances, since $T'-\fJ\in G_\ast$.
In the sequel, for brevity we will omit the subscript ``prime'' of $T$.
If the eigenvalues $\lambda_j$ are placed in the order of non-increasing moduli
then the normal form takes the lower-triangular form, i.e., the nonlinear part
of $\bigl(\fJ(y)\bigr)_s$ is expressed via coordinates $y_{s'}$ only such that
$s'~~0$ be given and
$B_\ast$ be the (closed) ball in $G_\ast$ of radius $R_\ast$ centered at
the zero.
Then $h_{T,\fJ}\mid B_\ast$ possesses a Lipschitz constant not exceeding
$\eps Q_N\bigl(\|T\|_{C^N},\|\fI\|_{C^N},\eps\bigr)$
where $Q_N$ is some polynomial depending on $N$ and $R_\ast$ only.
Indeed, it suffices to notice that the map $h_{T,\fJ}$ is differentiable
due to below Lemma~1 and to obtain the required upper estimate for its
differential. The differential of the functional map $G_\ast\to G_\ast$ that
determines by the formula
$f\mapsto T_{\tw_\alpha}(\cdot)\bigl(\ttw_\beta\circ f\circ T(\cdot)\bigr)$
the coefficient in front of $\bold e$ in~(9) takes the form
$$ \delta f(\cdot)\mapsto g(\cdot)\cdot \delta f\circ T(\cdot)\tag 10
$$
where $g(\cdot)=T_{\tw_\alpha}(\cdot)\cdot d\ttw_\beta\mid_{f\circ T(\cdot)}$.
The norm $\|g\|_{C^N}$ is estimated from above by a polynomial in
$\|T\|_{C^N}$, $\|\fI\|_{C^N}$, and $\eps$, and the function $g$ certainly
vanishes at the origin due to that $t=\alpha+\beta\ge 2$. The operator~(10) is
the composition of two operators in the space $G_\ast$ which act according to
the formulas
$$ \delta f(\cdot)\mapsto \delta f\circ T(\cdot)\quad\text{and}\quad
v(\cdot)\mapsto g(\cdot)\cdot v(\cdot).\tag 11
$$
It is easily seen that the norm of the first operator~(11) does not exceed
$S'_N(\|T\|_{C^N},\eps)$ and the norm of the second one,
$\eps S''_N(\eps)\cdot\|g\|_{C^N}$ where $S'_N$ and $S''_N$ are analogous to
$P_N$ and $Q_N$. The required estimate for the norm of the differential
follows from these ones.
The multiplier $\eps$ in front of the second estimate is due to the fact
that $g$ vanishes at the origin.
Thus, if $R_\ast>0$ is chosen large enough
($R_\ast>R_\ast\bigl(\|T\|_{C^N},\|\fI\|_{C^N}\bigr)$) and then the radius
of the ball $U$ is set to be small enough
($\eps<\eps\bigl(\|T\|_{C^N},\|\fI\|_{C^N},R_\ast\bigr)$) then
$\widetilde\cE_{T,\fJ}\mid B_\ast$ is a contractive map $B_\ast\to B_\ast$.
Consequently, there is a unique fixed point $f\in B_\ast$.
Thus, {\it a priori\/} upper bound $R_\ast$ for the norm of the normalizing
transformation determines the restriction from above to the radius $\eps$
of the ball $U$ where this normalization is built. However, this does not lead
to losing some possible solutions of the normalization problem and does not
bound a domain where the normalization is defined, due to the following facts.
Firstly, it is obvious that under increasing $R_\ast$ and the corresponding
decreasing $\eps$, one will obtain new normalizing transformations that are
exactly the restrictions of the previous ones to narrower domains.
Secondly, the iterations of the map $T$ and its normal form $\fJ$ will just
determine
the continuation of the normalization onto the whole basin of attraction of
the fixed point of $T$.
Arguments of~\cite{14} show that it suffices to prove the following result
where all the mappings are considered in the $C^k$-topology.
The unique fixed point
$f\in B_\ast$ of the mapping $\widetilde\cE_{T,\fJ}\mid B_\ast$ depends
continuously on $T$ provided that $T$ remains uniformly $C^N$-bounded.
Here, in contrast to~\cite{14}, the mapping $\widetilde\cE_{T,\fJ}$ is not
affine. So, the considerations should be slightly modified as follows.
The sequence $f_{(n)}=\widetilde\cE^n_{T,\fJ}(f)$, where $f\in B_\ast$, is
uniformly convergent in $G_\ast$. This fact is due to the uniformity of the
Lipschitz constant $<1$ for $\widetilde\cE_{T,\fJ}\mid B_\ast$ and the
uniform $C^N$-boundedness of $B_\ast$. Consequently, the sequence is also
uniformly convergent in the $C^k$-norm and the desired result follows
immediately from here because of the continuous dependence of each term on
$T$ in the $C^k$-topology.
Now we consider the most general case mentioned in the beginning of this
Remark. Let $w_N$ be the $G_\ast$-component of $\fI=\fJ^{-1}$. The only
distinction is that now a new summand $w_N\circ(\id+f)\circ T$ will appear
in the expression for $\widetilde\cE_{T,\fJ}(T)$,
in addition to the previous terms~(9). By the Implicit Function Theory
(in Banach spaces), one easily sees from $\fI\circ\fJ=\id$ that the dependence
of $\fI$ on $\fJ$ is continuously differentiable. We have only to prove that
this new term satisfies the above-described Lipschitz estimate. However, this
is easily derived from the above arguments applied to the differential of this
term, since $N\ge 2$.
\endremark
The following important result which is an appropriately generalized
reformulation of Proposition~5 of~\cite{14}
will provide the nonautonomous normalization.
Before stating it, we point out that
if $T=J$ is linear and $f_\cR\equiv 0$, $f'_\cR\equiv 0$ then
$\Cal J_{J,0,0}=0$, so that the ``normal form'' $\fJ$ and the map
$L=\cE_{J,0,0}=\Cal D_J\mid E_\ocR$ are linear, $L(f)=J^{-1}\circ f\circ J$
for $f\in E_\ocR$. One can define a map $L^0:E_\cR\to E_\cR$
using also the latter formula. Introduce also a vector
$d_T=J^{-1}\circ T-\id$ analogous to that used earlier.
\proclaim{Proposition 3} Let the following ``concordance'' and
``uniformity'' conditions be satisfied for
$C^N$-mappings $T_i\:U\to \Bbb R^n$ of a domain $U\subset \Bbb R^n$
that have the common fixed point $0\in U$:
\roster
\item"i)" the sets of eigenvalues for the operators
$J_i=dT_i(0)\:\Bbb R^n\to \Bbb R^n$ possess the $\tcR$-right concordant
partitions $\xi_i=\bigl\{\Lambda_{i,\tis}: \, 1\le \tis\le p\bigr\}$, and,
moreover, invariant subspaces $M_{\tis}$ corresponding to the elements
$\Lambda_{i,\tis}$ of the partitions $\xi_i$ are coincided. The number $N$
satisfies the claim $N>D_i$ where $D_i$ are defined by the equalities of
type~{\rm (2)} via the eigenvalues of the operators $J_i$;
\item"ii)" moreover, some norm in $\Bbb R^n$ and its restrictions
to $M_{\tis}$ induce operator norms $\|\cdot\|$ such that
$\bigl\| J_i\bigr\|<1$ and the partition into classes
$\Bigl\{\bigl\| J_{i,\tis}\bigr\|,\bigl\| J_{i,\tis}^{-1}\bigr\|^{-1}\Bigr\}$,
$J_{i,\tis}=J_i\mid M_{\tis}$, that correspond to $\tcR$-right concordant
partitions $\xi_i$, are also $\tcR$-right%
\footnote{We correct here an inaccuracy passed in Proposition~5 of~\cite{14}.}.
These conditions must be persistent under $\delta$-perturbations of
the operators $J_{i,\tis}$, where $\delta>0$ is some number;
\item"iii)" the $C^N$-norms of $T_i$ % and $T_i^{-1}$
are uniformly bounded.
\endroster
Then in a small enough ball (in the sense of the norm from condition~{\rm ii)})
$U$, centered at the origin $0\in \Bbb R^n$, all the mappings $T_i$ are
contractive. Besides, there exist a number $0<\mu<1$, a
direct sum decomposition
$E_\ocR=E^+\oplus E^-$ of the space $E_\ocR$, and
a norm in $E_\ocR$ (that is equivalent to the original one)
such that the
following statements hold. Each linear operator
$L_i=\cE_{J_i,0,0}=\Cal D_{J_i}\mid E_\ocR:E_\ocR\to E_\ocR$
is hyperbolic with separatrices $E^+$ and $E^-$
and, moreover, $\Bigl\| \bigl(L_i\mid E^\pm\bigr)^{\pm1}\Bigr\|\le\mu$
for the corresponding operator norms. The functional spaces $E^\pm=E^\pm(U)$
do not depend on the choice of $U$ in the sense
that for $U'\subset U''$, restricting the functions $f\in E^\pm(U'')$
onto $U'$ generates the natural projections of $E^\pm(U'')$ onto
$E^\pm(U')$.
The maps $\cE_{T_i,f_\cR,f'_\cR}:E_\ocR\to E_\ocR$ and
$\cJ_{T_i,f_\cR,f'_\cR}:E_\ocR\to E_\cR$ are well-defined and
differentiable (in the sense of Frech\'et). Obviously, to the
decomposition of $E_\ocR$, there correspond the
block forms of the linear operators $d\cE_{T_i,f_\cR,f'_\cR}$:
$$
d\cE_{T_i,f_\cR,f'_\cR}=\left(\matrix A_i & B_i\cr C_i & D_i\endmatrix\right),
$$
where $A_i\:E^+\to E^+$, $B_i\:E^-\to E^+$, $C_i\:E^+\to E^-$, and
$D_i\:E^-\to E^-$ are linear operators.
Next, given bounded sets $B_\cR\subset E_\cR$, $B_\ocR\subset E_\ocR$ and
a number $\eps>0$, the small enough neighbourhood $U$
of the origin and the norm $\|\cdot\|$ in $E_\ocR$ can be chosen in such
a manner that in addition the following conditions will hold:
the set $B_\ocR$ lies in the unit ball, $\|f_\ocR\|<1$, in $E_\ocR$
and for all $i$ the norms of the block components of the differential
$d\cE_{T_i,f_\cR,f'_\cR}$ at points $f_\ocR$ of the unit ball
satisfy estimates $\|A_i\|\le\mu+\eps$, $\|B_i\|\le\eps$,
$\|C_i\|\le\eps$, and $\bigl\|D_i^{-1}\bigr\|\le\mu+\eps$,
and the norms of the vectors $\cE_{T_i,f_\cR,f'_\cR}(0)$ do not exceed $\eps$,
provided that $f_\cR,f'_\cR\in B_\cR$.
\endproclaim
\proclaim{Corollary} If the subscript $i$ ranges over the set $\Bbb N$
and the polynomial mapping $f_i\in E_\cR$ have uniformly bounded
coefficients
then, under the conditions of Proposition~{\rm 3}, there
exists a unique sequence of uniformly $C^N$-bounded mappings $l_i$
possessing at the origin a tangency of (at least) first order with the
identity map $\id$, having the prescribed ``resonant parts'' $(l_i)_\cR=f_i$
and such that the following diagram commutes:
$$ \define \U#1{@>\pretend{T_{#1}}\haswidth{T_{-1}}>>}
\define \UUB#1{@>>\fJ_{#1}>}
\CD \cdots \U{-1} \U{0} \U{1} \U{2} \cdots \cr
&& @Vl_{-1}VV @Vl_0VV @Vl_1VV \cr
\cdots \UUB{-1} \UUB{0} \UUB{1} \UUB{2} \cdots \cr
\endCD \tag 12
$$
where all $\fJ_i$ are ``resonant'' in the sense that $(\fJ_i)_\ocR=0$.
\endproclaim
\remark{Remark 7} The construction of the mappings
$\cE_{T_i,f_\cR,f'_\cR}:E_\ocR\to E_\ocR$,
$\cJ_{T,f_\cR,f'_\cR}:E_\ocR\to E_\cR$ and Proposition~3 can be generalized
to the case where the differentials $dl(0)$, $dl'(0)$ are not necessary the
identity mappings but are non-degenerate and possess $M_{\tis}$ as the
invariant subspaces. The simplest way to show this fact is based on the
following observation. The commutativity of diagram~(8) is equivalent to the
commutativity of the same diagram where the mappings $l$, $l'$, and $\fJ$
are replaced, respectively, by the mappings $\bigl(dl(0)\bigr)^{-1}\circ l$,
$\bigl(dl'(0)\bigr)^{-1}\circ l'$, and
$\bigl(dl(0)\bigr)^{-1}\circ \fJ\circ\bigl(dl'(0)\bigr)$. So, this general
case is reduced to the particular one where $dl(0)=\id$ and $dl'(0)=\id$.
To ensure the uniform choice of $U$ and $\|\cdot\|$ independently of the
differentials $dl(0)$, $dl'(0)$, one should demand their uniform
non-degeneracy in the sense that both $\|L\|$ and $\|L^{-1}\|^{-1}$ are
uniformly bounded for $L=dl(0)$ and $dl'(0)$.
Due to this generalization, the Corollary of Proposition~3 remains valid
if all the prescribed differentials $dl_i(0)$ are uniformly non-degenerate
and possess $M_{\tis}$ as the invariant subspaces.
\endremark
\remark{Remark 8} Assume that there is a commutative diagram
$$ \define \U#1{@>\pretend{J_{#1}}\haswidth{J_{-1}}>>}
\define \UUB#1{@>>J'_{#1}>}
\CD \cdots \U{-1} \U{0} \U{1} \U{2} \cdots \cr
&& @Vt_{-1}VV @Vt_0VV @Vt_1VV \cr
\cdots \UUB{-1} \UUB{0} \UUB{1} \UUB{2} \cdots \cr
\endCD \tag 13
$$
of linear mappings such that $\{t_i\}$ are uniformly non-degenerate
in the sense of the previous Remark, and $\{J_i\}$ and $\{J_i'\}$ satisfy,
respectively, condition~ii) of Proposition~3 with the same collection of
invariant subspaces $\{M_{\tis}\}$.
Then the subspaces $M_{\tis}$ are $t_i$-invariant for all $i$.
Thus, in Remark~7, one can omit the assumption that the subspaces
$\{M_{\tis}\}$
are invariant with respect to the linear parts of the mappings $l_i$
conjugating the original mappings $T_i$ with the normal forms $\fJ_i$.
Indeed, this assumption is automatically guaranteed by the uniform
non-degeneracy of these linear parts and by the commutativity of the
diagram~(13) where $J_i$ and $J_i'$ are the linear parts of $T_i$ and $\fJ_i$,
respectively.
For the proof of this result one uses only the fact that partition into
the classes
$\Bigl\{\bigl\| J_{i,\tis}\bigr\|,\bigl\| J_{i,\tis}^{-1}\bigr\|^{-1}\Bigr\}$
and the analogous partition into the classes
$\Bigl\{\bigl\| J'_{i,\tis}\bigr\|,\bigl\|
(J'_{i,\tis})^{-1}\bigr\|^{-1}\Bigr\}$
are strongly ordered. Indeed, this fact means that the subspaces
$\{M_{\tis}\}$ are {\it exponentially separated\/} in the sense of Section~5.
More precisely speaking, introduce the linear extension
$\bold J=[J_i]:X\to X$, $X=\Bbb Z\times \Bbb R^n$, of the standard shift
$\sigma:\Bbb Z\to \Bbb Z$, $\sigma(i)=i+1$, by the mappings $J_i$ which acts
as $(i,x)\mapsto \bigl(\sigma(i),J_i(x)\bigr)$. Then the trivial bundles
$X_{\tis}=\Bbb Z\times M_{\tis}\to \Bbb Z$ are $\bold J$-invariant and
exponentially separated.
Moreover, they constitute a uniformly non-degenerate decomposition
$X=\oplus_{\tis}X_{\tis}$ is the sense that the angles between vectors
belonging to distinct fibers $M_{i,\tis}=M_{\tis}$ over the same point $i$
of the base $\Bbb Z$ are uniformly bounded away from zero for all $i\in\Bbb Z$.
Note that, in contrast to Section~5, we deal with non-compact base $\Bbb Z$
but implicitly assume the required uniformity conditions.
Obviously, since diagram~(13) is commutative and the linear mappings $t_i$ are
uniformly non-degenerate, they transform $\bold J$-invariant subbundles of $X$,
which are exponentially
separated and constitute a uniformly non-degenerate decomposition of $X$,
into $\bold J'$-invariant subbundles of $X$ which possess the same properties,
where $\bold J'=[J_i']:X\to X$ is analogous to $\bold J$.
But the invariant exponentially separated subbundles, which
constitute a uniformly non-degenerate decomposition of the whole bundle,
are uniquely determined by their dimensions, see Proposition~4 in Section~5.
Thus, the subbundles $X_{\tis}$ are invariant under the linear automorphism
$\bold t=[t_i]$, i.e., $t_i(M_{\tis})=M_{\tis}$.
\endremark
\remark{Remark 9} In the previous Remark, one can omit the condition that
the linear maps $t_i$ are invertible by some sharpening the conditions
imposed on $J$ and $J'$. Assume that for $v\in M_{\tis}$ one has
$$ C^{-1}(g_{\tis}^-)^m\le {\bigl\|J_{i+m}\circ\cdots\circ J_{i+1}(v)\bigr\|
\over \|v\|} \le C(g_{\tis}^+)^m, \quad m\ge 0,
$$
where $g_{\tis}^+0$ is
large. Thus, replacing $T$ and $\fJ$ by
$\vphantom{T}_mT_i=T_{i+m}\circ\cdots\circ T_i$ and
$\vphantom{\fJ}_m\fJ_i=\fJ_{i+m}\circ\cdots\circ \fJ_i$ for large $m>0$, we
obtain the mappings
$\cE_{\vphantom{T}_mT_i,\vphantom{\fJ}_m\fJ_i,t,t',f_\cR,f'_\cR}$
possessing the desired hyperbolicity properties, which proves the result
sought for.
\endremark
\remark{Remark 10}
In Remark~6 and in Proposition~3, the linear operators $\Cal D_{T_i}$ and
$L_i=\Cal D_{J_i}$ in $E_\ast$ are not close, although their norms
admit the close upper estimates. Indeed, let $T\:U\to U$
be a $C^N$-mapping possessing a nondegenerate fixed point and let
$J$ be the corresponding linear part of $T$. Assume that a mapping
$T'\:U\to U$ is $C^N$-close to $T$. Then, using
the Chain Rule, one can represent the $N$-th differential of
$\bigl(\Cal D_{T'}-\Cal D_T \bigr)(f)$ at a point $x\in U$ as a sum
of monomials such that the norm of one of them,
$$ J^{-1}\Bigl(d^Nf\mid_{T'(x)}-d^Nf\mid_{T(x)}\Bigr)
\Bigl(\underbrace{dT\mid_x,\cdots,dT\mid_x}_N\Bigr),
$$
can be estimated from above only in terms of the continuity modulus of $d^Nf$,
while the norms of the other monomials admit a suitable upper estimate
$\|T'-T\|_{C^N}\cdot\bigl\|J^{-1}\bigr\|^2 P_N\bigl(\|T\|_{C^N}\bigr)
\cdot\|f\|_{C^N}$ where $P_N$ is a polynomial depending on $N$ only.
Therefore, Remark~6 is unimprovable in the above-mentioned sense if one
imposes no restrictions on the continuity modulus of the $N$-th
differential of $f$. Due to the same reason, the operator $\Cal D_T$
and the mappings $\cE_{T,f_\cR,f'_\cR}$, $\cJ_{T,f_\cR,f'_\cR}$ in the
topology of the uniform convergence over bounded sets do not
depend continuously on $T\in C^N$ although $\Cal D_Tf$,
$\cE_{T,f_\cR,f'_\cR}(f_\ocR)$, $\cJ_{T,f_\cR,f'_\cR}(f_\ocR)$ do for each
$f\in E$ or $f_\ocR\in E_\ocR$.
\endremark
\remark{Remark 11} The previous Corollary is easily established
by standard means of hyperbolic theory
(the finite-dimensional case with invertible mappings $S_i$ is usually
considered. We deal here with the {\it infinite-dimensional Banach case with
non-invertible\/} $S_i$). We sketch the underlying geometrical ideas
that are very natural and can be found in many works (see also a relevant
discussion in~\cite{6} and~\cite{24}). These ideas are ``nonautonomous''
and many-dimensional variant of a
geometrical approach used by J.~Hadamard~\cite{21} to prove the
presence of separatrices (invariant curves) of a hyperbolic fixed point of
a two-dimensional diffeomorphism
(see also an exposition in~\cite{33}; J.~Hadamard considered only the
two-dimensional case, but this is of no importance).
This approach was denoted as {\it ``graph transform''\/} in~\cite{24}.
In essence, we will construct a mathematical object that was called an
{\it itinerary scheme\/} by V.~M.~Alekseev~\cite{2, Part~1}. Different names
were attached to the elements of the below-defined spaces
$\Sigma^-,\;\Sigma^+$.
For example, they can be called {\it unstable\/} and {\it stable slices\/}.
Let $R>0$ be large enough and $D=D^+\times D^-$ be the ``rectangle'',
where $D^\pm$ is an $R$-ball in $E^\pm$ (centered at zero).
Consider spaces
$\Sigma^\pm$ of continuous mappings $f^\pm\: D^\pm\to D^\mp$ whose
Lipschitz constants are $\le \rho^\pm$, where the positive numbers $\rho^-$
and $\rho^+$ are such that $\rho^-\rho^+<1$.
We will identify functions $f^\pm$ and their graphs. Let
$S=S_i=\cE_{T,f_\cR,f'_\cR}$.
Then the correspondences $\alpha\mapsto S^{\mp1}(\alpha)\cap D$,
for $\alpha\in\Sigma^\pm$, will determine well-defined contracting (in the
usual $C^0$-norm) mappings
$\frak S_i^\pm=\frak S^\pm\:\Sigma^\pm\to\Sigma^\pm$
(although the mappings $S_i$ are non-invertible) if small enough $\eps>0$ was
chosen depending on $\mu$ and $\rho^\pm$. The Lipschitz constant $\lambda<1$
of these contracting mappings and a lower bound for the radius $R$ depend
only on $\mu$, $\rho^\pm$, and $\eps$. In our case $R=1/2$ is enough.
Thus, to~(12) there corresponds a double-infinite chain
$$ \cdots @< \cE_{T_{-1},f_{-1},f_{-2}} << @< \cE_{T_0,f_0,f_{-1}} <<
@< \cE_{T_1,f_1,f_0} << @< \cE_{T_2,f_2,f_1} << \cdots \; . \tag 14
$$
Then the sets
$\frak S_{i+1}^-\circ\frak S_{i+2}^-\circ\cdots\circ\frak S_m^-(\Sigma^-)$
and
$\frak S_i^+\circ\frak S_{i-1}^+\circ\cdots\circ\frak S_{-m}^+(\Sigma^+)$
shrink, respectively, exponentially fast to some elements
$\alpha_i^-\in \Sigma^-$ and $\alpha_i^+\in \Sigma^+$ as $m\to+\infty$.
The intersection $\alpha_i^-\cap\alpha_i^+$ consists of a single point due
to the inequality $\rho^-\rho^+<1$ and
coincides with the mapping $l_i-\id$.
We notice that the mapping $\frak S^+$ of the stable slices is well-defined
and contractive although $S$ may be non-invertible. A simple construction of
$\frak S^+$, that is based on the invertibility of the restrictions of the
mapping $S$ onto unstable slices, can be found in~\cite{32}
(somewhat other constructions can be also found
in~\cite{24, Proof of Theorem 5.1; 22})%
\footnote{In essence, the itinerary scheme was
constructed in~\cite{22, 32} to establish an analog of the well-known
V.M.~Alekseev's theorem~\cite{3}
for a homoclinic structure of a smooth, possibly non-invertible map of a
Banach manifold (the simplest case of a single homoclinic trajectory was
discussed in~\cite{22, 32}, but the result obtained is immediately
transferable to the general case).}
and will be shortly described here. A stable slice
$\alpha\in \Sigma^+$ will be transformed into the set
$\frak S^+(\alpha)=S^{-1}(\alpha)\cap D\in \Sigma^+$ which is the
graph of a function $g\: D^+\to D^-$ to be constructed as follows. The mapping
$S$ transfers an unstable slice $\beta_\xi=\{\xi\}\times D^-\in \Sigma^-$,
$\xi\in D^+$, into its image, whose piece which is located inside
$D=D^+\times D^-$ is an unstable slice
$\widetilde{\beta_\xi}=\frak S^-(\beta_\xi)$. Then
$\bigl(\xi,g(\xi)\bigr)\in D^+\times D^-$ is the pre-image of a unique point
of intersection, $\alpha\cap\widetilde{\beta_\xi}$, under the invertible map
(diffeomorphism)
$S\mid \beta_\xi^\prime\: \beta_\xi^\prime\to\widetilde{\beta_\xi}$, where
$\beta_\xi^\prime=S^{-1}(\widetilde{\beta_\xi})\cap\beta_\xi$.
Incidentally, one could notice that Proposition~3 would be applicable even
if the differentiability was treated in G\^ateaux sense only. Indeed, the
above estimates for the block components of the G\^ateaux differential
imply all the required properties of the induced mappings of stable and
unstable slices.
\endremark
In a slightly more general case%
\footnote{We omit the most general case~\cite{2, Part~1}
where the itinerary scheme is associated with an arbitrary TMC.},
the construction of the itinerary scheme includes the sequence of identical
copies $E_i$ of a Banach space $E=E^+\oplus E^-$, a sequence of
``rectangles'' $D_i=D_i^+\times D_i^-\subset E_i$ equipped with the
spaces $\Sigma^-$ and $\Sigma^+$ of unstable and stable slices with some
positive maximal ``slopes'' $\rho^-,\rho^+$ such that $\rho^-\rho^+<1$,
and a sequence of mappings $S_i\:D_i\to E_{i+1}$,
$$ \cdots @> S_{-1} >> @> S_0 >> @> S_1 >> \cdots
$$
(in contrast to~(14) the order is assumed to be direct) such that the
deformed rectangle $S(D_i)$ and the rectangle $D_j$ have ``good''
intersection~\cite{30}, i.e., they cross.
Then the corresponding mappings $\frak S_i^-\:\Sigma_i^-\to\Sigma_{i+1}^-$
and $\frak S_i^+\:\Sigma_{i+1}^+\to\Sigma_i^+$
are well-defined and contractive with some Lipschitz constant
$\lambda=\lambda(\rho^+,\rho^-)<1$ under a proper choice of
$\rho^-$ and $\rho^+$. The precise formulations and strict estimates
for these geometrically visual results can be found in~\cite{2, Part~1}.
\remark{Remark 12}
Let us consider the itinerary scheme just described above. Then the stable
$\alpha_i^+\in\Sigma_i^+$ and unstable $\alpha_i^-\in\Sigma_i^-$ slices,
defined as above by the itinerary scheme, and their intersection,
$r_i=\alpha_i^+\cap\alpha_i^-$, will depend continuously on the
sequence of mappings $S_j$. To attach the precise meaning to the last
statement, one should consider $[S_i:i\in\Bbb Z]$ as an element of
the corresponding Tychonoff product (over $i\in\Bbb Z$) of the spaces
$V_i$, where $V_i$ is the space of mappings $S_i$ for which $S_i(D_i)$ and
$D_{i+1}$ cross and define the contractive maps
$\frak S_i^-\:\Sigma_i^-\to\Sigma_{i+1}^-$ and
$\frak S_i^+\:\Sigma_{i+1}^+\to\Sigma_i^+$ with Lipschitz constant $\lambda$.
Equip each $V_i$ with the usual $C^0$-norm and recall that $\Sigma_i^\pm$ are
equipped with the analogous norms.
Then, firstly, $\frak S_i^-\beta^-\in\Sigma_{i+1}^-$
($\frak S_i^+\beta^+\in\Sigma_i^+$, respectively) depends continuously on
the map $S_i\in V_i$ and the slice $\beta^-\in\Sigma_i^-$
($\beta^+\in\Sigma_{i+1}^+$). Next, recall that, secondly, the $C^0$-diameter
of $\frak S_{i-1}^-\circ\cdots\circ\frak S_{-m}^-\bigl(\Sigma_{-m}^-\bigr)$
($\frak S_i^+\circ\cdots\circ\frak S_{m-1}^+\bigl(\Sigma_{m}^+\bigr)$,
respectively) tends uniformly to zero as $m\to+\infty$ for all the mappings
$S_j\in V_j$, and, thirdly, the intersection map
$\kap_i\:\Sigma_i^+\times\Sigma_i^-\to D_i$ is continuous. It is easily
seen from these results that $\alpha_i^\pm$ and $r_i=\alpha_i^+\cap\alpha_i^-$
depend continuously on $[S_i]$. (Moreover, the mapping $\kap_i$ and the
dependence of $\frak S_i^\pm\beta^\pm$ on $S_i\in V_i$ are Lipschitz with some
constants determined by $\rho^+$ and $\rho^-$ only. Therefore, the dependencies
of $\alpha_i^\pm$ and $r_i$ on $S_j\in V_j$ are also Lipschitz with constant
$C\lambda^{|i-j|}$ where $C=C(\rho^+,\rho^-)>0$.) For further purposes, we
notice that the analogous results can be easily proven if the spaces
$\Sigma_i^\pm$ and $V_i$ are equipped with the pointwise convergence
topologies.
We emphasize that these topologies are finer than the $C^0$-topologies
if $E^\pm$ or $E$ are infinite-dimensional, respectively. This is a direct
consequence of the analogous result that in the space of linear operators
$E^\pm\to \Bbb R$, whose norms do not exceed a given positive constant, the
topology defined by the operator norm (the strong topology) is coarser than
that of the pointwise convergence (the weak topology).
\endremark
\remark{Remark 13} One can establish completely analogous to Remark~19
of~\cite{14}
that the $C^N$-mappings $l_i$ depend continuously on the sequence of the
$C^N$-mappings $T_i$ and on the sequence of the polynomial mappings $(l_i)_\cR$
if all the mappings are treated in the $C^k$-topology, $1\le k\le N$, and the
sequences are treated in the Tychonoff topology. This result remains valid in
the case $k=0$ provided that the perturbations of $T_i$ are small enough in the
$C^1$-sense. We omit the explanation of these facts since it consists of
repeating the arguments described in Remark~19 of~\cite{14} and in Remark~6
above. Recall only that it is based on Remark~12 above.
\endremark
\remark{Remark 14}
The nonautonomous normalization $l_i$ constructed above admits an invariant
coordinate-free description. The explanation of this fact repeats almost
literally the arguments presented in item~{\bf c)} of Remark~20 in~\cite{14}.
Indeed, consider a sequence
$\bigl\{T_i\bigr\}$ that satisfies the conditions of Proposition~3.
Let $U_i=U$ be the definition domain of the mapping
$T_i\:U_i\to U_{i+1}$. There
exists a unique sequence $r_i=0\in U$ such that $T_i(r_i)=r_{i+1}$. The
nonautonomous normalization is given by the mappings $l_i\:U_i\to \Bbb R^n$
such that $l_i(r_i)=0$, $dl_i(r_i)=\id\mid \Bbb R^n$. Using the
``natural'' coordinates in $U_i\subset\Bbb R^n$, one can identify the spaces
$\Bbb R^n$ containing the images $\operatorname{im}l_i$ with the tangent spaces
$T_{r_i}U_i$. Then the nonautonomous $(\{M_{\tis}\},\tcR)$-normalization is
just the sequence of $C^N$-mappings $l_i\:U_i\to T_{r_i}U_i$ uniformly
$C^N$-bounded and such that
$l_i(r_i)=0\in T_{r_i}U_i$, $dl_i(r_i)=\id\mid T_{r_i}U_i\equiv \id\:
T_{r_i}U_i\supsetar$ (the space $T_rU$ is equipped with the natural linear
structure and, therefore, for each $\xi\in T_rU_i$ there is the canonical
identification $T_rU_i\equiv T_\xi\bigl(T_rU_i\bigr)$ used here for
$\xi=(r_i,0)$, $0\in T_{r_i}U_i$) and diagram~(12) commutes, where
$\fJ_i\:T_{r_i}U_i\to T_{r_{i+1}}U_{i+1}$ are
the $(\{M_{\tis}\},\tcR)$-quasi-normal forms.
Nevertheless, the Corollary of Proposition~3 does not admit a completely
coordinate-free description since the ``resonant parts'' $(l_i)_\cR$ that
should be prescribed depends on the coordinates used in the domain $U_i$.
Indeed, recall from item~{\bf c)} of Remark~20 in~\cite{14} the following fact.
If $l_i\:\Bbb R^n\supset U\to \Bbb R^n=T_0U$ is an arbitrary mapping written
in the natural coordinates then in the coordinates $y=g_i(u), u\in U_i$,
where $g_i(0)=0$, this mapping possesses the form $l_i'=\Cal D_{g_i^{-1}}l_i$.
Thus, generally speaking, the new ``resonant part'' $(l_i')_\cR$ will depend
on the old ``non-resonant part'' $(l_i)_\ocR$.
\endremark
\head
4. Proof of Proposition 3 and further discussion
\endhead
The proof of Proposition~5 in~\cite{14} requires
some modifications only which will be described in all detail.
At the end of the Section, we will indicate the importance of the
stability condition for the set $\tcR$ associated with the
$(\{M_{\tis}\},\tcR)$-normalizations.
In particular, a result generalizing the Centralizer Theorem
of~\cite{8,20} will be established.
First of all, we will state an important Lemma which is related to the
smooth properties of the composition map $\Comp$ acting by the formula
$(f,g)\mapsto f\circ g$ in the functional spaces.
For any two domains $U,\,V$ in Banach spaces we denote by $C_b^r(U,V)$
the linear space of $C^r$-functions $f:U\to V$ with the finite $C^r$-norm
$\|f\|_{C^r}=\sup_{x\in U}\sum\limits_{t=0}^r\bigl\|d^tf\mid_x\bigr\|$.
Here and below the differentiability is understood in the sense of Frech\'et.
\proclaim{Lemma 1} Let $U,\,V,\,W$ be some domains in Banach spaces,
$r\ge 0$, $s\ge 1$. The composition map
$\Comp:C_b^{r+s}(V,W)\times C_b^r(U,V)\to C_b^r(U,W)$ is linear in the
first argument and $s$ times differentiable in the collection of the
arguments at the points $(f,g)$ such that the $(r+s)$-th differential
of $f$ is uniformly continuous in $V$. The associated differential is
calculated by the formula
$$ d\Comp\mid_{(f,g)}(\delta f,\delta g)(\cdot)=
\delta f\circ g(\cdot)+\nabla f\mid_{g(\cdot)}\cdot\delta g(\cdot)
\tag 15
$$
where $\nabla$ denotes the differential in the phase space (i.e., the gradient)
and $\delta$, the differential in the functional spaces.
\endproclaim
The corresponding statements on the linearity in the first argument and
the $s$-times differentiability in the second argument are local
variants of the so-called Alpha- and Omega-Theorems. The global variant
of these Theorems~\cite{1, 16} (see also~\cite{34})
are related to functional spaces of the mappings of compact manifolds
(but not charts). (Notice also
that we treat with infinite-dimensional charts, in contrast to
compact manifolds that are covered by finite-dimensional charts;
the compactness is needed only to provide the finiteness of the $C^r$-norms
of smooth functions and the uniform continuity of continuous functions.)
An independent proof which is less formal than that in~\cite{1} will
be now presented.
\demo{Proof of Lemma 1} Let $r=0$ and $s=1$.
Using the Chain Rule, the Mean Value Theorem, and the Lagrange Theorem, one
can estimate from above the modulus of the difference of
$\Comp(f+\delta f,g+\delta g)-\Comp(f,g)$ and the right-hand side of~(15) as
$\bigl\|\nabla(\delta f)\bigr\|\|\delta g\|+
\omega_{\nabla f}\bigl(\|\delta g\|\bigr)\|\delta g\|$ where $\omega_h$
denotes the continuity modulus of $h$ and $\|\cdot\|$, the $C^0$-norm.
This estimate is seen to be
$o(\eps)$ uniformly as $\eps=\|\delta f\|_{C^1}+\|\delta g\|_{C^0}\to 0$.
This verifies the differentiability and gives the equality~(15) in the case
$r=0$, $s=1$. Obviously, (15) remains valid in the general case provided that
the differentiability of $\Comp$ is established. This is an elementary
corollary of the fact that the convergence in the $\|\cdot\|_{C^l}$-norm
implies that in the $\|\cdot\|_{C^t}$-norm where $t|\lambda^m|$, respectively) in inequality~(1), where
$\{\lambda_j\}$ is the spectrum of $J_i$. Then let $\Phi$ be
constituted by the non-intersecting subsets $\Phi^-$ and $\Phi^+$.
Introduce $\tPhi^0=\Psi(\Phi^0)$, $\tPhi^\pm=\Psi(\Phi^\pm)$, and
$\tPhi=\Psi(\Phi)=\tPhi^+\cup\tPhi^-$
and notice that the correspondence $\Psi$ does not depend on the number $i$
due to condition~i). Let the non-intersecting sets $F^-$, $F^+$ constituting
$F$ be defined as
$$ F^\pm=\bigoplus_{(\tis,\tim)\in \tPhi^\pm} G_{\tis,\tim}.
$$
Then the outgoing separatrix $E^-$ is $F^-$ and the incoming separatrix $E^+$
is the space of functions that have at the origin $0\in \Bbb R^n$
$(N-1)$-jets lying in $F^+$, i.e., $E^+=F^+\oplus G_\ast$. Indeed, in
the spaces $G_\ast\subset E$ and $G_{\tis,\tim}$ invariant under $L_i$,
the following norms $\|\cdot\|_\ast$ and $\|\cdot\|_{\tis,\tim}$ will be
induced. The norm $\|\cdot\|_\ast$ was defined above as the exact upper bound
for the norm of the $N$-th differential, and $\|\cdot\|_{\tis,\tim}$ is
the norm in the
space of polynomials $P$ that are considered as polylinear mappings
$P\: M_1^{\tim_1}\times\cdots\times M_p^{\tim_p}\to M_{\tis}$
symmetrical in each cofactor $M_t^{\tim_t}$
(the case $p=1$ being discussed in~\cite{12} but the results being
transferable immediately to the case of arbitrary $p\ge 1$).
The corresponding operator norms (denoted by the same symbols)
satisfy inequalities
$$ \align
\biggl\| \Bigl( L_i \bigm| G_{\tis,\tim} \Bigr)^{-1} &\biggr\|
_{\tis,\tim}
\le\mu\quad\text{for}\quad (\tis,\tim)\in \tPhi^-, \\
\Bigl\| L_i \bigm| G_{\tis,\tim} &\Bigr\|
_{\tis,\tim}
\le\mu\quad\text{for}\quad (\tis,\tim)\in \tPhi^+, \qquad
\text{and } \\
\Bigl\| L_i \bigm| G_\ast &\Bigr\|_\ast \le\mu
\endalign
$$
for some $0<\mu<1$. In our case
$$\aligned
\mu=\max &\left\{ \max_{(\tis,\tim)\in \tPhi^-}
\bigl\| J_{i,\tis} \bigr\|
\prod_{j=1}^p \bigl\| J_{i,j}^{-1} \bigr\|^{\tim_j};\right. \\
&\hskip7pt\left. \max_{(\tis,\tim)\in \tPhi^+}
\bigl\| J_{i,\tis}^{-1} \bigr\|
\prod_{j=1}^p \bigl\| J_{i,j} \bigr\|^{\tim_j}
\right\}.
\endaligned \tag 16
$$
It follows from condition~ii) that the number $\mu$ can be chosen
independently of $i$. Define a norm in
$$E_\ocR=E^+\oplus E^-=\bigoplus_{(\tis,\tim)\in\tPhi}
G_{\tis,\tim}\oplus G_\ast
$$
as the direct sum (or maximum) of the norms in $G_{\tis,\tim}$, $G_\ast$.
Then $\Bigl\| \bigl(L_i\mid E^\pm\bigr)^{\pm1}\Bigr\|\le\mu$.
The required norm in $E_\ocR$ is built of the norms in
$G_{\tis,\tim}$ and $G_\ast$ with corresponding weight coefficients
$\kap_{\tim}$ and $\kap_\ast$ that decrease quickly as
$|\tim|$ rises (one supposes $|\tim|=N$ for $G_\ast$),
$$ \left\| \bigoplus_{(\tis,\tim)\in\tPhi}
u_{\tis,\tim}\oplus u_\ast \right\| =
\kap_\ast\bigl\| u_\ast\bigr\|_\ast+\sum
_{(\tis,\tim)\in\tPhi}
\kap_{\tim}\bigl\| u_{\tis,\tim}\bigr\|_{\tis,\tim}
$$
(the sums in the right-hand side can be replaced by the maxima).
The main trouble now is to construct the nonlinear mappings
$\cE_{T,f_\cR,f'_\cR}$ and $\cJ_{T,f_\cR,f'_\cR}$ and to prove
the required properties of $\cE_{T,f_\cR,f'_\cR}$.
Let $G_t=\bigoplus\limits\Sb (\tis,\tim)\in\tPhi \\
|\tim|=t\endSb G_{\tis,\tim}$
and $G_t^0=\bigoplus\limits\Sb (\tis,\tim)\in\tPhi^0 \\
|\tim|=t\endSb G_{\tis,\tim}$
for $2\le t|\tim''|$.
Then the differentials of the mappings $\cE_{T,f_\cR,f'_\cR}^{\leftarrow}$ and
$\cI_{T,f_\cR,f'_\cR}$ can be evaluated with an error uniformly small
in the induced operator norm via the use of formulas~(20), (21) where the
right-hand sides are simplified as follows. The quantities
$\overline\Theta_t,\,\overline\Theta_t^0\,(t\alpha$ in the sense that it has
zero $(\alpha'-1)$-jet at the origin and depends polynomially on
$\overline u$ and $\overline v$.
Therefore, these partial derivatives of terms~(9) under discussion can be
made to be arbitrarily small by choosing a small enough ball $U$ for a given
$\kap_\ast$. Indeed, the terms constituting the partial derivatives under
consideration are built with the use of the linear operators~(11) in the space
$G_\ast$ and the nonlinear map $(\cdot)\mapsto f\circ(\cdot)$ in $G_\ast$
where $f\in C^N$ in a vicinity of the origin, $f(0)=0$ (in fact,
$f=\ttw_\beta$). It is easily seen that $\|f\circ v\|_\ast$ admits a uniform
upper estimate $\|f\|_{C^N}S_N\bigl(\|v\|_\ast,\eps\bigr)$ which can be
multiplied by $\eps$ if, in addition, $df(0)=0$. Here, as above, $S_N$ is a
polynomial depending on $N$ only, $\eps$ is the radius of the ball $U$.
The result sought for is implied immediately by these estimates and the
above-written estimates for the norms of operators~(11), taking into account
the fact that certainly $\alpha'\ge\alpha\ge 1$ or $\beta\ge 2$. Later, the
above-described arguments show that the norm of $\Cal D_T\mid G_\ast$ will
admit an upper estimate close to $\mu$ and the partial derivative
$(\delta/\delta u_N)\overline\Theta_N(\overline u,\overline v^0,u_N\circ T)$
will be small provided that the size of the neighbourhood $U$ is chosen small
enough. Finally, if all the coefficients $\kap_{\tim},\,\kap_\ast$ were
chosen small enough then the given bounded set $B_\ocR$ lies inside the
unit circle of the norm constructed, $\|\cdot\|$, and the conditions
$\bigl\|\cE_{T,f_\cR,f'_\cR}^{\leftarrow}(0)\bigr\|\le\eps$
are satisfied for all $f_\cR,f'_\cR\in B_\cR$.
The analog of Proposition~3 for the mappings
$\cE_{T,f_\cR,f'_\cR}^{\leftarrow}$ and $\cI_{T,f_\cR,f'_\cR}$
is proven.
Now we will show that the structure of the original required mappings
$\cE_{T,f_\cR,f'_\cR}$ and $\cJ_{T,f_\cR,f'_\cR}$ is analogous and
this fact will enable us to extend the results just obtained to the
latter mappings.
Let $\fI_\cR=\oplus_t w_t^0$, $\fI_\ocR=\oplus_t w_t$, where $w_t\in G_t$
and $\fI=\fJ^{-1}$.
Completely analogously to the considerations above, we deduce from the
relation $\fI\circ\fJ=\id$ that
$$ \aligned
0&= w_t\circ J+\Xi_t\bigl(\bigoplus_{t'0$ is some constant. (Recall that the $t$-th derivative of a mapping
$\Bbb R^n\to\Bbb R^n$ is a polylinear symmetrical mapping
$\bigl(\Bbb R^n\bigr)^t\to\Bbb R^n$~\cite{12, 34} which allows one to define
the $C^N$-norm in a coordinate-free way~\cite{34}.)
Moreover, one can take in the right-hand side
of~(24) a slightly more general expression
$C\eps^{-1}\sum\limits_{\tis=1}^p \alpha_{\tis} \|x_{\tis}\|_{\tis}$,
where $\alpha_{\tis}\ge 1$.
\endremark
\remark{Remark 17}
The polylinear mappings
$P:M_1^{\tim_1}\times\cdots\times M_p^{\tim_p}\to M_{\tis}$
symmetric in each cofactor $M_t^{\tim_t}$ are identified with the linear
mappings $P':O^{\tim_1}M_1\otimes\cdots\otimes O^{\tim_p}M_p\to M_{\tis}$.
One can show that the induced norm, $\|\cdot\|_{\tis,\tim}$, in the
space of the polylinear mappings coincides with the induced norm in the space
of the above mentioned linear mappings if the linear space
$O^{\tim_1}M_1\otimes\cdots\otimes O^{\tim_p}M_p$ is equipped with a natural
norm. The latter norm can be characterized as follows. Let $U$ and $V$ be two
linear finite-dimensional spaces equipped with some norms both denoted as
$\|\cdot\|$. Set $\|u\otimes v\|_1=\|u\|\cdot\|v\|$ for arbitrary $u\in U$,
$v\in V$, and
$$ \|w\|=\inf\left\{\sum_{i=1}^g\|u_i\otimes v_i\|_1: \,\,\,
w=\sum_{i=1}^gu_i\otimes v_i,\,\,\, u_i\in U,\,v_i\in V \right\}
$$
for arbitrary $w\in U\otimes V$. Then $\|\cdot\|$ is a norm in $U\otimes V$.
Incidentally, $\|u\otimes v\|=\|u\otimes v\|_1$ for all $u\in U$, $v\in V$
if the norms in both $U$ and $V$ are symmetric, i.e., the involutions
$x\mapsto -x$ are metric isomorphisms.
Indeed, let $B\subset U$ be the ball in $U$ centered at the origin with the
radius $\|u\|>0$ where $u\in U$. Due to the convexity of $B$, there is an
affine hyperplane
$u+\Pi_u$ such that $B$ lies entirely from one side of it, $\Pi_u$ being a
linear subspace of codimension one. Choose an arbitrary basis
$\overline u_1,\ldots,\overline u_p$ in $\Pi_u$ and denote $\overline u_0=u$.
Let $\Pi_v$ be the analogous subspace in $V$ associated to $v\in V$ with
$\|v\|>0$ and let $\overline v_1,\ldots,\overline v_q$ be a basis in $\Pi_v$,
and $\overline v_0=v$. Assume that
$$ w=\sum_{i=1}^gu_i\otimes v_i,\,\,\, u_i\in U,\,v_i\in V \tag 25
$$
and denote by $u_{i,0}$ (by $v_{i,0}$, respectively) the
$\overline u_0$-coordinate of $u_i$ in the basis $\{\overline u_j\}$
(the $\overline v_0$-coordinate of $v_i$ in the basis $\{\overline v_j\}$).
Then
$$ \|u_i\|\ge\|\overline u_0\|\cdot |u_{i,0}|,\quad
\|v_i\|\ge\|\overline v_0\|\cdot |v_{i,0}|, \tag 26
$$
in view of the symmetry assumptions and, consequently,
$ \|u_i\otimes v_i\|_1\ge\|u\otimes v\|_1\cdot \bigl(|u_{i,0}|\cdot
|v_{i,0}| \bigr)$. Assuming that $w=u\otimes v$ and expanding the
both sides of~(25) in the basis
$\{\overline u_j\otimes \overline v_k\}$ of $U\otimes V$, one sees that
$\sum\limits_{i=1}^g u_{i,0}\cdot u_{i,0}=1$. Therefore,
$\|u\otimes v\|\ge \|u\otimes v\|_1$ and, consequently,
$\|u\otimes v\|=\|u\otimes v\|_1$. Now prove that in the general case
$\|\cdot\|$ is a norm in $U\otimes V$. Let $\{\overline u_j\}$ and
$\{\overline v_j\}$ be arbitrary two basis in $U$ and $V$, respectively.
Then the inequalities~(26) remain valid for all the coordinates with only
some multiplier $c>0$ in the right-hand sides:
$\|u\|\ge c\|\overline u_j\|\cdot |u_j|$,
$\|v\|\ge c\|\overline v_j\|\cdot |v_j|$, where $u_j$ and $v_j$ are the
$\overline u_j$- and $\overline v_j$-coordinates of arbitrary $u\in U$ and
$v\in V$, respectively. So, for any $w\in U\otimes V$ one obtains that
$\|w\|\ge c^2 \|\overline u_j\|\cdot \|\overline v_k\|\cdot |w_{j,k}|$ where
$w_{j,k}$ is the $\overline u_j\otimes \overline v_k$-coordinate of $w$ in the
basis $\{\overline u_j\otimes \overline v_k\}$. Thus, $\|w\|>0$ for $w\ne 0$
which guarantees that $\|\cdot\|$ is a norm in $U\otimes V$.
Furthermore, for any three finite-dimensional linear normed spaces $U$, $V$,
and $W$, the norm of a bilinear mapping $P:U\times V\to W$ coincides with the
norm of the associated linear map $P':U\otimes V\to W$. The inequality
$\|P\|\le \|P'\|$ is evident from that $P(u,v)=P'(u\otimes v)$ and
$\|u\otimes v\|\le\|u\otimes v\|_1$. For every
$w\in U\otimes V$ and $\eps>0$ there is a representation~(25)
such that $\sum_i\|u_i\otimes v_i\|_1$ is $\eps$-close to $\|w\|$. Then
$\bigl\|P'(w)\bigr\|=\bigl\|\sum_i P(u_i,v_i)\bigr\|\le
\sum_i \bigl\|P(u_i,v_i)\bigr\|\le \|P\|\sum_i\|u_i\otimes v_i\|_1\le
\|P\|\cdot\bigl(\|w\|+\eps\bigr)$ which implies that $\|P'\|\le \|P\|$.
Thus, $\|P\|=\|P'\|$.
Now we are going to describe the norm looked for.
The following results provide immediately this norm that is constructed
inductively.
For arbitrary linear finite-dimensional spaces $U$, $V$, and $W$ the natural
isomorphisms of the linear spaces $U\otimes V\to V\otimes U$ and
$(U\otimes V)\otimes W\to U\otimes (V\otimes W)$ are also the metric
isomorphisms. This fact for the latter isomorphism is valid since
$$ \|z\|=\inf\left\{\sum_{i=1}^g\|u_i\|\cdot \|v_i\|\cdot \|w_i\|: \,\,\,
z=\sum_{i=1}^gu_i\otimes v_i\otimes w_i,\,\,\, u_i\in U,\,v_i\in V,
\,w_i\in W \right\}
$$
for arbitrary $z\in U\otimes V\otimes W$.
The norm in $O^mU$ is induced by the natural factorization
$\otimes^mU\to \otimes^mU/L_m=O^mU$~\cite{31} so that the distance between
the subsets $x+L_m$, $y+L_m$ of $\otimes^mU$, which are just the elements of
$O^mU$, is defined as the minimal distance between their points. Then
the norm of a polylinear symmetric mapping $\otimes^mU\to V$ coincides with
the norm of the associated linear map $O^mU\to V$.
\endremark
\remark{Remark 18} Obviously, the mappings
$(f_\cR,f'_\cR,f_\ocR)\mapsto\cE_{T,f_\cR,f'_\cR}(f_\ocR),\,
\cJ_{T,f_\cR,f'_\cR}(f_\ocR)$ are even of class $C^\infty$ in the collection
of the variables. Moreover, they are polynomial in the sense that
they can be represented as the finite Taylor series.
Indeed, the elements $\fI_\cR=\bigoplus\limits_{tD$, the number $D$ being
determined by~(2) via the spectrum of the linear part of these two
$(\{M_{\tis}\},\tcR)$-normal forms. B.~Anderson~\cite{5} claimed this result
for the particular case where self-conjugacies of the usual normal form are
considered. However, his proof contains some inaccuracies and mistakes as
will be discussed in Historical comment~2.
The key observation to prove these results is that $\cR$ takes a
``triangular form with the linear diagonal part'' described in the proof
of fact~{\bf 2)} while proving Proposition~2.
Indeed, the stability of $\cR$ implies the absence of non-linear circuits
which, in turn, implies that $\cR$ takes this desired form (see a remark
in the proof of fact~{\bf 2)} above).
Then, firstly, completely analogous to the usual normal forms,
this form of $\cR$ implies that
$f^{-1}\in\Cal F_\tcR$ when $f\in\Cal F_\tcR$ and $df(0)$ is nondegenerate.
Secondly, if $F\in\Cal F_\tcR$ and $dF(0)$ is non-degenerate then $F$
conjugates any $(\{M_{\tis}\},\tcR)$-normal form with a
$(\{M_{\tis}\},\tcR)$-normal
form due to the stability of $\tcR$. Inversely, the mapping $F$ that
conjugates two $(\{M_{\tis}\},\tcR)$-normal forms when being considered
as a $(\{M_{\tis}\},\tcR)$-normalization (reducing a given
$(\{M_{\tis}\},\tcR)$-normal form to an arbitrary
$(\{M_{\tis}\},\tcR)$-normal form) is uniquely determined by its
$\cR$-component, $F_\cR$, that can be arbitrary. Therefore, $F_\ocR=0$ since
this case is suitable.
\endremark
Now we prove that the second fact can be generalized to the semi-conjugacies
and, thus, the following result holds.
\proclaim{Theorem 1}
Let $\fJ_1$ and $\fJ_2$ be two $(\{M_{\tis}\},\tcR)$-normal
forms with coincident spectra of their linear parts and let the set $\tcR$
(containing exact resonances) be stable. Then any mapping (maybe,
non-invertible) $f$ such that $f\circ \fJ_1=\fJ_2\circ f$ is a
$(\{M_{\tis}\},\tcR)$-quasi-normal form provided that $f$ is treated
as a formal
power series, in the general formal case, or as a $C^N$-mapping, in the
smooth contractive case (where $\fJ_1$ and $\fJ_2$ are contractive, $N>D$
with $D$ being determined by~(2) via the coincident spectra of $d\fJ_i(0)$,
$i=1,2$, and $|m|D$ where $D$ is
determined by formula~(2) via the spectrum of the linear part of $f$.
According to B.~Anderson~\cite{5},
the $C^N$-centralizer $Z^N(f)$, i.e., the group of the $C^N$-diffeomorphisms
commuting with $f$, contains only polynomials of degrees $0$ being any number less than the moduli of the eigenvalues
of $F$ at the origin. For an analogous reason, inequality~(1) in that Lemma
seems to be far from optimal for large $k$. Fortunately, the desired result is
immediately provided by the contraction property of the map
$\widetilde\cE_{f,f}$ discussed in Remark~6. Recall also in this connection
that the proof by N.~Kopell is almost identical to that for Sternberg's
theorem in the non-resonant case which uses the contractive affine map
$\mD_f=\widetilde\cE_{f,f}$ (here, the first point of the above proof
is superfluous, while the second point is evident).
Incidentally, the result under discussion is immediately carried out to
the case of the $C^N$-mappings which commute with $f$ but whose linear parts
at the origin may be degenerate. In order to show this, it suffices to
apply a map
$(\cdot)\mapsto f^{-1}\circ\bigl(g_{N-1}+(\cdot)\bigr)\circ f-g_{N-1}$
in $G_\ast$ whose contraction property is established in a completely
analogous way
(see the last formula~(17) and the consequent analysis of the nonlinear
term $\Theta_N$ in the proof of Proposition~3).
Thus, the proof by B.~Anderson provides that any element of $Z^N(f)$ (i.e., a
coordinate $C^N$-transformation conjugating the normal form $f$ with itself)
is a quasi-normal form with respect to $df(0)$. The same remains valid for the
transformations $g$ conjugating two normal forms $f_1$ and $f_2$, i.e., such
that $g\circ f_1=f_2\circ g$, if $df_1(0)=df_2(0)$ (or, more generally,
$df_1(0)$ and $df_2(0)$ possess the coincident spectra and the coincident
generalized eigensubspaces associated to the equal eigenvalues).
The formal part of this result
(i.e., one related to the formal Maclaurin series) is provided by the first
point in B.~Anderson's proof without any contraction assumptions on $df(0)$.
This is just the general classic result (see~\cite{8}) where, in addition,
one sees that $df_1(0)=df_2(0)$ or $dg(0)$ may be degenerate
(thus, $g$ may be a semi-conjugacy).
We notice that the proof presented here is purely geometric in contrast to
the analytic approach utilized in~\cite{8}.
Returning to the above-discussed result on the (semi)conjugacy, notice that
its ``analytic'' part is provided by the second point in B.~Anderson's proof
and by the third point via considering a map
$(\cdot)\mapsto f_2^{-1}\circ\bigl(g_{N-1}+(\cdot)\bigr)\circ f_1-g_{N-1}$
in $G_\ast$. The necessary minor modifications will be explained in the proof
of Theorem~1. We emphasize that the linear part of $g$ is not assumed to
be non-degenerate, i.e., $g$ may be a semi-conjugacy.
\endremark
\demo{Proof of Theorem~1}
Applying the basic Anderson's observation to the linear parts of $\fJ_1$,
$\fJ_2$, and $f$, we see that all the subspaces $M_{\tis}$ are
$df(0)$-invariant. Now, given $f_\cR$, we will seek for $f_\ocR$ as
a unique fixed point of the functional map
$\cE:(\cdot)\mapsto \fJ_2^{-1}\circ\bigl(f_\cR+(\cdot)\bigr)\circ \fJ_1-f_\cR$
in $E_\ocR$. The arguments are completely analogous to those presented in
Remark~6. Similarly to~(17), the map $\cE$ takes the triangular form
$$ v_t=\eta_t+L u_t+\Theta_t\bigl(\bigoplus_{t'0$.
Let $J:X\to X$ be a linear extension of homeomorphism $f:B\to B$ by
non-degenerate linear mappings $J(b)$ (i.e., $J:X\to X$ is a linear
automorphism) and let $X=X_1\oplus X_2$ be the
decomposition of the bundle $X$ into the Whitney
sum of $J$-invariant subbundles. The subbundles $X_1,\,X_2$ (up to their
permutation) are said to be {\it exponentially separated\/} if there are
numbers $C>0$ and $\nu\in(0,1)$ such that
$$ {\bigl\|J^n(x_1)\bigr\| \over \|x_1\|} :
{\bigl\|J^n(x_2)\bigr\| \over \|x_2\|}\le C\nu^n\tag 28
$$
for any $n\ge 0$ and $x_1\in X_1$, $x_2\in X_2$, $\pi(x_1)=\pi(x_2)$. This
means that the vectors in the fibers of $X_2$ are ``stretched'' under
iterations of $J$ faster, than those in the fibers of $X_1$. In this case
we will write $X_1\prec X_2$.
The subbundles $X_1,\,X_2$ are said to be {\it spectrally separated\/} if
there are positive numbers $C$, $g_1^+$, $g_2^-$ such that $g_1^+0$
and $\lambda\in(0,1)$ such that
$$
\|F^n(x)-F^n(y)\|\le d\lambda^n \|x-y\| \tag 30
$$
for any $n\ge 0$ and $x,y\in X$, $\pi(x)=\pi(y)$.
Obviously, this definition is independent of the choice of the (quasi)norm.
Analogously, one can consider extensions $F$ which are contractive over some
$F$-invariant closed subset $Y\subset X$ covering the whole base $B$,
i.e., such that $F(Y)\subset Y$, $\pi(Y)=B$, and (30) holds for $x,y\in Y$.
The contractive extension possesses a single invariant section and this
section is continuous. This result is an immediate corollary of the ``graph
transform'' method, since the contractive extension induces a contractive map
in the space of the continuous sections.
One can account in the sequel that the invariant section coincides with
the zero section. Moreover, we will consider extensions which are smooth
enough along the fibers.
Precisely speaking, the mapping $F\mid X(b):X(b)\to X\bigl(f(b)\bigr)$
of the fibers is of class $C^N$ and depends continuously in the
$C^N$-topology on the point $b$ of the base where $N$ is some natural.
Given a homeomorphism $f:B\to B$, a vector bundle $X\to B$, and a natural $N$,
we denote by $E$ the space of all the extensions $F:X\to X$ of $f$ with the
latter property and whose 1-jets in the fiber directions at the zero
section vanishes. This is a natural generalization of the space $E$ of
the $C^N$-function introduced above (for simplicity we do not change the
notation). The linear structure in $E$ is naturally induced by the linear
structure in the fibers.
For any contractive extension $F$, the linear extension $DF$
being the differential of $F$ in the fiber direction at the invariant zero
section, is also contractive. From the other hand, given a contractive linear
extension $J=DF$, there is a norm such that $d=1$ in formula~(30) for $DF$
(with an arbitrary bigger $\lambda$). This norm is called {\it Lyapunov\/}
if $\lambda<1$.
Then~(30) is valid for $F$ with $d=1$ (and a bigger $\lambda$)
with respect to this norm in some $F$-invariant neighbourhood of
the zero section constituted by small balls in the fibers.
In particular, the extension $F$ is contractive in this neighbourhood.
Let $X=X_1\oplus\cdots\oplus X_p$ where all the (continuous)
vector bundles $X_{\tis}$ are invariant with respect to the linear extension
$DF$ being the differential of $F$ in the fiber direction at the zero section.
Denote $J=DF$, $J(b)=J\mid X(b)$, $J_{\tis}=J\mid X_{\tis}$,
$J_{\tis}(b)=J_{\tis}\mid X_{\tis}(b)$,
$1\le \tis\le p$, where $X_{\tis}(b)=X(b)\cap X_{\tis}$ are the fibers
of $X_{\tis}$.
Now we transfer the basic concepts of Definition~8 to the extensions,
assuming in the sequel that the linear extension $J$ is an automorphism.
\definition{Definition 9}
An extension $F$ will be called a {\it $(\{X_{\tis}\},\tcR)$-quasi-normal
form\/} if all the fiber mapping $F\mid X(b)$ become
$(\{M_{\tis}\},\tcR)$-quasi-normal
forms when the linear spaces $X_{\tis}(b)$ are
identified with $M_{\tis}$, respectively. This identification provides also
the decomposition $E=E_\cR\oplus E_\ocR$ where $E_\cR$ and $E_\ocR$ are the
spaces of all the extensions of $f$ whose fiber mappings are ``resonant''
and ``non-resonant'', respectively.
A $(\{X_{\tis}\},\tcR)$-quasi-normal form $F$ will be called a
{\it $(\{X_{\tis}\},\tcR)$-normal form with respect to the norm\/}
in the bundle $X$ if, in addition, the differential $J=DF$ at the zero
section satisfies the following property: the partition $\xi(b)$ into classes
$\biggl\{\bigl\| J_{\tis}(b)\bigr\|,
\Bigl\|\bigl(J_{\tis}(b)\bigr)^{-1}\Bigr\|^{-1}\biggr\}$
are $\tcR$-right concordant for all $b\in B$ where $\|\cdot\|$ denotes the
induced operator norms.
Any continuous transformation of the bundle $X$ preserving the fibers and
reducing the extension to a $(\{X_{\tis}\},\tcR)$-normal form with respect
to some norm in $X$ will be called a
{\it $(\{X_{\tis}\},\tcR)$-normalization\/}
or {\it $(\{X_{\tis}\},\tcR)$-normalizing transformation\/}.
\enddefinition
Obviously, a linear extension $F$ is a $(\{X_{\tis}\},\tcR)$-quasi-normal
form if and only if all the subbundles $X_{\tis}$ are $F$-invariant.
\remark{Remark 21}
Obviously, these definitions do not depend on the choice of the identifying
isomorphisms and can be easily reformulated to avoid the usage of $M_{\tis}$.
The idea is to use $X_{\tis}(b)$ immediately instead of $M_{\tis}$.
Thus, in the present context,
$G_{\tis,\tim}$ will be the linear space of the continuous fiber mapping
$X_1\times\cdots\times X_p\to X_{\tis}$ whose restriction to each fiber,
$X(b)=X_1(b)\times\cdots\times X_p(b)$, is homogeneous polynomial in $p$
variables that has degree $\tim_t$ in the $t$-th argument ranging over
$X_t(b)$. Analogously, $G_\ast$ will be the space of all the extensions
of $f$ whose restrictions to the fibers are $C^N$-mappings depending
continuously in the $C^N$-topology on the fiber and possessing the zero
$(N-1)$-jets at the origin.
Notice that the condition imposed on the partition $\xi(b)$ in the definition
of a $(\{X_{\tis}\},\tcR)$-normal form with respect to the norm
is an appropriate counterpart of condition~ii) in Proposition~3.
Here, the uniformity assumptions are automatically satisfied due to the
compactness of the base $B$.
\endremark
\proclaim{Theorem 2 (Normal Form Theorem)}
Let the differential $J=DF$ of a contractive extension $F$ be a
$(\{X_{\tis}\},\tcR)$-normal form with respect to some Lyapunov norm
$\|\cdot\|$ and let $N>D(b)$ for all $b\in B$ where
$D(b)=\ln\bigl(\|J^{-1}(b)\|^{-1}\bigr)/\ln\|J(b)\|$.
Then, in a small enough neighbourhood of the zero section, there is a
$(\{X_{\tis}\},\tcR)$-normalization $H$ of $F$ which is $C^N$ along the fibers.
This normalization is uniquely determined by its ``resonant part''
$H_\cR$ that can be arbitrarily chosen.
\endproclaim
\demo{Proof}
The result can be immediately obtained via applying the Corollary of
Proposition~3 to the fiber mappings over a particular orbit of $f:B\to B$.
Here $T_i$, $l_i$, and $\fJ_i$ are, respectively, the restrictions of $F$,
$H$, and the
normal form sought for to the fiber $X(b_i)$ where $\{b_i=f^i(b)\}$ is
a trajectory of $f$. Then the mappings $l_i$ depend continuously on the
sequences $[T_i]$ and $\bigl[(f_i)_\cR\bigr]$, according to Remark~13,
and, in turn, the latter sequences depend continuously on the initial
point $b$ of the orbit.
Another possibility is to reformulate the proof of Proposition~3 in order
to consider the whole cascade $f$ rather than its particular trajectories.
Thus, given $H_\cR$, we consider an extension $\cE_{H_\cR}:E_\ocR\to E_\ocR$
of the homeomorphism $f:B\to B$ by the hyperbolic mappings
$\cE_{H_\cR}(b)=\cE_{F\mid X(b),H_\cR\mid X\bigl(f(b)\bigr),H_\cR\mid X(b)}:
E_\ocR\bigl(f(b)\bigr)\to E_\ocR(b)$. One could call this extension to be
{\it hyperbolic\/} since its fiber mappings, $\cE_{H_\cR}(b)$, possess the
uniform hyperbolic properties for all $b\in B$. Notice also that the fiber
mapping depends continuously in the pointwise convergence topology on
the point of the base. One can see that these properties
guarantee the required result.
Recall that the uniformity conditions of Proposition~3 (in each of these
two approaches) are automatically satisfied due to the compactness of
the base $B$.\qed
\enddemo
\remark{Remark 22} If the linear extension $DF$ is contractive then the
invariant zero section of $F$ possesses some open basin of attraction. Then,
if $DF$ satisfies the conditions of Theorem~2, the
$(\{X_{\tis}\},\tcR)$-normalization described therein exists and is
extended over the whole basin of attraction.
\endremark
\remark{Remark 23} In the present context, the given linear structure of the
bundle $X\to B$ is not natural, since it is used only to determine the
$(\{X_{\tis}\},\tcR)$-normal form. Thus, in the spirit of Remark~14, one can
describe the $(\{X_{\tis}\},\tcR)$-normalization $H$ in a natural
coordinate-free way as a
fiber mapping $X\to T_{\bold 0}X$ with the identity differential
$DH=\id\mid T_{\bold 0}X$ at the zero section $\bold 0:B\to X$ where
$T_{\bold 0}X$ denotes the linear bundle constituted by the fibers
$T_{\bold 0(b)}X(b)$, $b\in B$. However, Theorem~2 cannot be reformulated
in a completely coordinate-free way since the ``resonant part'' $H_\cR$
does not admit such a description.
\endremark
Now we state two Semi-Centralizer Theorems. The difference is that in
the first Theorem, semi-conjugacies are considered which are extensions by
diffeomorphisms of the fibers, while the second Theorem is related to
semi-conjugacies which are arbitrary smooth extensions.
\proclaim{Theorem 3}
Let a contractive extension $F$ of a homeomorphism $f:B\to B$ be a
$(\{X_{\tis}\},\tcR)$-normal form with respect to some Lyapunov norm
$\|\cdot\|$
and let the number $N$ be as above. Furthermore, let $g:B\to B$ be a
continuous map commuting with $f$ and $G$ be an extension of $g$ by
$C^N$-diffeomorphisms of the fibers such that $G$ commutes with $F$.
Then $G$ is a $(\{X_{\tis}\},\tcR)$-quasi-normal form if the set $\tcR$
is stable.
\endproclaim
\proclaim{Theorem 3\mz}
Let the extension $F$, the number $N$, the homeomorphisms $f$, the
continuous map $g$, and
the set $\tcR$ be as in Theorem~3, and let $G$ be an extension of $g$ by
(maybe, non-invertible) $C^N$-mappings of the fibers such that $G$ commutes
with $F$.
Then the conclusion of Theorem~3 remains valid provided that
(exponentially separated) subbundles $X_{\tis}$ are spectrally separated
with the spectral intervals $[g_{\tis}^-,g_{\tis}^+]$ being the elements
of a $\tcR$-right partition.
\endproclaim
Notice that the assumption imposed on the subbundles $X_{\tis}$ in
Theorem~3\mz\ imply that the spectral intervals satisfy the narrow
band condition
$$ g_{\tis}^+g_p^+0$.
Accordingly to item~c) of Proposition~4, we obtain the desired result that
$x\in X_{\tis}\Rightarrow DG(x_{\tis})\in X_{\tis}$ (cf. Remark~9).
Now we complete the proof analogously to the proof of Theorem~3.\qed
\enddemo
Now we are going to remove the choice of a particular Lyapunov norm from the
conditions of Theorems~2, 3, and~3\mz.
The properly modified conditions will be
``asymptotic counterparts'' of the original ones and indeed do not depend on
the choice of a particular norm. Notice that
a $(\{X_{\tis}\},\tcR)$-quasi-normal form $F$ is a $(\{X_{\tis}\},\tcR)$-normal
form with respect to the norm in the bundle $X$ if and only if given any
$(\tis,\tim)\notin\tcR$ one of the two conditions
$$ \bigl\|J_{\tis}^n(b)\bigr\| \cdot \prod_{i=1}^p
\Bigl\|\bigl(J_i^n(b)\bigr)^{-1}\Bigr\|^{\tim_i}<1
\qquad\text{or}\qquad
\Bigl\|\bigl(J_{\tis}^n(b)\bigr)^{-1}\Bigr\| \cdot \prod_{i=1}^p
\bigl\|J_i^n(b)\bigr\|^{\tim_i}<1
$$
is satisfied for all the points $b\in B$.
Here $J_i^n(b)=J_i\bigl(f^{n-1}(b)\bigr)\circ\cdots\circ J_i\bigl(f(b)\bigr)
\circ J_i(b)$ denotes the fiber mapping $X(b)\to X\bigl(f^n(b)\bigr)$ of the
linear extension $J^n$, $n\ge 0$. Consequently, $\bigl(J_i^n(b)\bigr)^{-1}$
is the fiber mapping $X\bigl(f^n(b)\bigr)\to X(b)$ of the linear extension
$J^{-n}$. Notice that here and in the sequel, $J_i^n(b)$ for $n\ge 0$ can be
also understood as the fiber mapping $X\bigl(f^{-n}(b)\bigr)\to X(b)$.
This replacement requires only a minor modification of a long formula in the
proof of Lemma~4.
\proclaim{Proposition 5}
Let, as above, $F:X\to X$ be an extension of a homeomorphism $f:B\to B$
constituted by the fiber $C^N$-diffeomorphisms depending continuously in
the $C^N$-topology of the point of
the base and let $X=X_1\oplus\cdots\oplus X_p$ be the decomposition into
$J$-invariant (continuous) bundles where $J=DF$ is the differential of $F$
at the invariant zero section. (Then $J$ is a linear automorphism.)
Assume that the extension $F$ and the natural $N$ satisfy the following
conditions for some (and, consequently, for any) norm (or quasinorm)
in $X$ where $\|\cdot\|$ will denote the induced operator norms:
\roster
\item"1)" given any
$(\tis,\tim)\notin\tcR$ one of the two conditions
$$
\lim_{n\to+\infty}\bigl\|J_{\tis}^n(b)\bigr\| \cdot \prod_{i=1}^p
\Bigl\|\bigl(J_i^n(b)\bigr)^{-1}\Bigr\|^{\tim_i}=0
\qquad\text{or}\qquad
\lim_{n\to+\infty}\Bigl\|\bigl(J_{\tis}^n(b)\bigr)^{-1}\Bigr\| \cdot
\prod_{i=1}^p \bigl\|J_i^n(b)\bigr\|^{\tim_i}=0
$$
is valid for all the points $b\in B$,
\item"2)" $ N>\lim\limits_{n\to+\infty}\sup
\dfrac {\ln\biggl( \Bigl\| \bigl(J^n(b)\bigr)^{-1}\Bigr\|^{-1}\biggr)}
{\ln\bigl\|J^n(b)\bigr\|}$
for all the points $b\in B$.
\endroster
Then $J$ and $N$ satisfy the conditions of Theorem~{\rm 2\/} under some
choice of the norm.
The desired norm can be chosen to be
Euclidean (in the real case) or Hermitian (in the complex case).
In particular, the subbundles $X_{\tis}$ are exponentially separated so that
$X_i\prec X_j$ if and only if
$$ \lim_{n\to+\infty} \Bigl\|\bigl(J_j^n(b)\bigr)^{-1}\Bigr\|
\bigl\|(J_i^n(b)\bigr\|=0 \quad\text{for all} \quad b\in B.
$$
Moreover, the condition~{\rm 2)} is equivalent to that
$$ \lim_{n\to+\infty} \bigl\|J^n(b)\bigr\|^N \cdot
\Bigl\|\bigl(J^n(b)\bigr)^{-1}\Bigr\| =0
\quad\text{for all the points}\quad b\in B.
$$
Finally, the subbundles $X_{\tis}$ are spectrally separated so that
$X_{\tis}\prec X_{\tis+1}$ if and only if there are numbers $g_{\tis}^\pm$,
$1\le \tis\le p$, such that $g_{\tis}^+0$ and $\lambda\in(0,1)$ and any $n\ge 0$ is an immediate
generalization of the exponential separation condition that
corresponds to the case $\|\alpha^+\|=1$ and $\|\alpha^-\|=1$.
We will write $\alpha^+\prec\alpha^-$ if (32) is valid.
This definition does not depend on the choice of the norm or quasinorm
$\|\cdot\|$ because this choice affects $C$ only. The relation
$\prec$ is a {\it partial order relation\/} on the set of all the
collections $\alpha=\{\alpha_i:1\le i\le p\}$.
The following Lemma~3 is identical to Lemma~A2 in the Appendix~A of~\cite{14}.
The Proposition~6 and Lemma~2 are also immediate generalizations of
Proposition~A2 and Lemma~A1 in that Appendix. Their proofs
are almost identical to those presented therein.
\proclaim{Proposition 6} Let a finite collection of conditions~{\rm (32)}
be met with some constants $\lambda=\lambda_j\in(0,1)$. Furthermore, let
numbers $\mu_j$ lie in the intervals $(\lambda_j,1)$.
Then there is a norm $\|\cdot\|$
in $X$ such that all of these conditions~{\rm (32)}
are valid with constants $C=1$ and $\lambda=\mu_j$, respectively.
The desired norm $\|\cdot\|$ can be chosen to be Euclidean
(in the real case) or Hermitian (in the complex case).
\endproclaim
So, by slightly increasing $\lambda_j$, one is able to put all the constants
$C$'s to be equal to 1.
\proclaim{Lemma 2} Let the conditions of Proposition~{\rm 6} be satisfied,
so that a finite collection of conditions
$h^{(n)}(\beta^{j,+},\beta^{j,-},b)\le C\lambda_j^n$ is valid where
$\beta^{j,\pm}$ denote some $p$-dimensional vectors
$\alpha=\{\alpha_i:1\le i\le p\}$
with non-negative entries.
Then there are cocycles $G_{\tis}^\pm$, $1\le \tis\le p$, such that
the following two properties hold:
\roster
\item"a)" $g_{\beta^{j,+}}^+(b):g_{\beta^{j,-}}^-(b)\le\mu_j$,
$b\in B$, where $g_\alpha^\pm$ are determined via
$g_{\tis}^\pm(\cdot)=G_{\tis}^\pm(\cdot,1)$.
\item"b)" $g_{\tis}^-(b)\le g_{\tis}^+(b)$ and
$C_1^{-1}G_\alpha^-(b,n)\le \Phi_\alpha^{(n)}(x)\le
C_1G_\alpha^+(b,n)$,
$n\ge 0$, $x\in X(b)$, $b\in B$, where constant $C_1>0$
depends on the choice of a quasinorm $\|\cdot\|$.
\endroster
\endproclaim
\demo{Proof} Set
$\|x\|_\ast=\left(\prod\limits_{n=0}^{N-1}\bigl\|J^n(x)\bigr\|_1\right)^{1/N}$,
$\|\cdot\|_1$ being some (quasi)norm and $N$, a sufficiently large positive
integer. Obviously, $\|\cdot\|_\ast$ is a quasinorm. However, $\|\cdot\|_\ast$
can be not a norm even if $\|\cdot\|_1$ is a norm (an appropriate example is
easily constructed).
Let $\Phi_{\ast,\alpha}$, and the desired functions $g_{\tis}^-$ and
$g_{\tis}^+$ be the functions
$\Phi_\alpha$, $g_{\tis}^-$, and $g_{\tis}^+$ determined
via the constructed quasinorm $\|\cdot\|_\ast$. The functions
$g_{\tis}^\pm$ determine the required cocycles $G_{\tis}^\pm$. The item~b) of
the Lemma is obviously satisfied. In order to prove item~a), we recall that
$$ \Phi^{(n)}_{\beta^{j,+}}(x): \Phi^{(n)}_{\beta^{j,-}}(x')\le C\lambda_j^n,
\quad
n\ge 0,\ \pi(x)=\pi(x'),
$$
where $C>0$. One can easily check that
$$ \Phi_{\ast,\alpha}(x)=\biggl(\Phi_\alpha^{(N)}(x)\biggr)^{1/N},
$$
where $\Phi_\alpha^{(N)}$ corresponds to the original (quasi)norm
(for which the assumed conditions~(32) are met).
Therefore
$$ \Phi_{\ast,\beta^{j,+}}(x): \Phi_{\ast,\beta^{j,-}}(x')=
\biggl(\Phi^{(N)}_{\beta^{j,+}}(x):
\Phi^{(N)}_{\beta^{j,-}}(x')\biggr)^{1/N}\le
C^{1/N}\lambda_j\le \mu_j
$$
if $N>0$ is chosen to be large enough. The item~a) of the Lemma follows
immediately from this inequality.\qed
\enddemo
\proclaim{Lemma 3} Let $J:X\to X$ be a linear automorphism and
let $G^\pm$ be cocycles such that
$g^-(b)\le g^+(b)$, $b\in B$, where $g^\pm(\cdot)=G^\pm(\cdot,1)$, and
$$ C^{-1}G^-(b,n)\le \Phi^{(n)}(x)\equiv {\bigl\|J^n(x)\bigr\|\over \|x\|}
\le CG^+(b,n),\quad
n\ge 0,\ x\in X(b),
$$
$C>0$ being some constant and $\|\cdot\|$, a quasinorm in $X$.
Let $\nu>1$. Then there exists a norm $\|\cdot\|_0$ in $X$ such that
$$\nu^{-n}G^-(b,n)\le{\bigl\|J^n(x)\bigr\|_0\over \|x\|_0}\le \nu^nG^+(b,n),
\quad n\ge 0,\ x\in X(b).
$$
The desired norm $\|\cdot\|_0$ can be chosen to be
Euclidean (in the real case) or Hermitian (in the complex case).
\endproclaim
\demo{Proof} Without loss of generality, one can assume that $\|\cdot\|$
is a norm. Set
$$\|x\|_0=\sum_{n\ge 0}\bigl\|J^n(x)\bigr\| \bigl(G^+(b,n)\bigr)^{-1}\nu^{-n}+
\sum_{n<0}\bigl\|J^n(x)\bigr\| \bigl(G^-(b,n)\bigr)^{-1}\nu^n.
$$
Obviously, these two series converge uniformly and $\|\cdot\|_0$ is a norm.
Then it is easily seen that
$$\nu^{-1}g^-(b)\le{\bigl\|J(x)\bigr\|_0\over \|x\|_0}\le \nu g^+(b),\quad
x\in X(b),
$$
which implies that the norm $\|\cdot\|_0$ is the desired one. If the norm
$\|\cdot\|$ was chosen to be Euclidean or Hermitian then the norm
$\|\cdot\|_0$ possesses the same property.\qed
\enddemo
\demo{Proof of Proposition~6} It follows easily from
Lemmas~2 and~3. Apply Lemma~2, using some numbers $\mu_j^\prime$
instead of numbers $\mu_j$. Then we apply Lemma~3 to each invariant
subbundle $X_{\tis}$, assuming that
$G^\pm=G_{\tis}^\pm$, $g^\pm=g_{\tis}^\pm$, $\nu=\nu_{\tis}$. As a result,
some norms $\|\cdot\|_{\tis}$ in subbundles $X_{\tis}$ will be obtained.
Finally, we require inequalities
$\mu_j^\prime\nu^{\beta_j^++\beta_j^-}\le\mu_j$ where the notation
$\nu^{\{\alpha_{\tis}\}}=\prod\limits_{\tis}\nu_{\tis}^{\alpha_{\tis}}$
is used and construct the
desired norm $\|\cdot\|$ in $X=X_1\oplus\cdots\oplus X_p$ from the norms
$\|\cdot\|_{\tis}$ in $X_{\tis}$. For instance, one can set
$$ \|x_1\oplus\cdots\oplus x_p\|=\left(\sum_{\tis=1}^p
\|x_{\tis}\|_{\tis}^2\right)^{1/2},
\quad x_{\tis}\in X_{\tis},\ \pi(x_{\tis})=b.\qed
$$
\enddemo
Finally, the following Lemma show that in the asymptotic conditions of
Proposition~5, the decaying quantity possess an exponential upper estimate.
\proclaim{Lemma 4}
If $h^{(n)}(\alpha^+,\alpha^-,b)\to 0$ as $n\to+\infty$ for
all $b\in B$ then there are $C>0$ and $\nu\in(0,1)$ such that
$h^{(n)}(\alpha^+,\alpha^-,b)\le C\nu^n$ for all $n\ge 0$
and $b\in B$.
\endproclaim
\demo{Proof} For any $b\in B$ there is a natural $T(b)$ such that
$h^{(n)}(\alpha^+,\alpha^-,b)<1$ for $n=T(b)$. Then there is a
neighbourhood $U(b)$ of $b$ such that
$h^{(n)}(\alpha^+,\alpha^-,b')<1$ for $n=T(b)$ and all
$b'\in \overline{U(b)}$. Since $B$ is compact, we may choose finitely many
points $b_1,\ldots,b_k$ such that $\bigl\{U(b_j):1\le j\le k\bigr\}$ is a
covering of $B$. Then there is $\nu\in(0,1)$ such that
$h^{(n)}(\alpha^+,\alpha^-,b')<\nu^n$ for $n=T(b_j)$ and all
$b'\in U(b_j)$, $1\le j\le k$.
Now let $b$ be given. Define a sequence of integers $\{i(k):k>0\}$ as
follows. Choose $i(1)$ such that $b\in U(b_{i(1)})$. If $i(1),\ldots,i(k)$
have been chosen, let $\tau(k)=T(b_{i(1)})+\cdots+T(b_{i(k)})$.
Choose $i(k+1)$ such that $f^{\tau(k)}\in U(b_{i(k+1)})$.
Now let $n>0$ be given. Then there is $k$ such
that $\tau(k)\le n<\tau(k+1)$ and thus $n=\tau(k)+r$ where
$0\le r1$ or $g^+\equiv\const<1$. Note that any cocycle $G$ and
linear extension $\{F^n\}$ determine a linear extension $\{F_G^n\}$ via the
formula $F_G^n(x)=G\bigl(n,\pi(x)\bigr)F^n(x)$. The possibility of constructing
a norm satisfying the one-sided estimate follows immediately from the existence
of the Lyapunov norm for the corresponding linear extension $\{F_G^n\}$. This
idea was utilized in~\cite{10, 11} for proving Proposition~6 in the simplest
case mentioned above.
The case of the two-sided estimates with $g^-=\const$, $g^+=\const$ is
considered in~\cite{24, points~2.8--2.9; 20}. The corresponding proof
generalizes the well-known construction of the Lyapunov norm. In turn, our
proof is an immediate generalization of the latter proof.
A simplified version of Lemma~2 which is related to the exponentially separated
subbundles has been considered in~\cite{14}. The proof described therein was
reproduced from~\cite{10}. For the present general case the proof remains
almost identical.
The Lemma~4 is a slightly generalized version of
the Uniformity Lemma by N.~Fenichel~\cite{17; 18, Part~I} (see
also~\cite{10}). The proof is reproduced from the sources cited.
\endremark
\Refs
\ref\no1
\by Abraham R.
\book Lectures of Smale on Differential Topology
\bookinfo Mimeographed notes
\publaddr Columbia
\yr 1962
\endref
\ref\no2
\by Alekseev V.M.
\paper Quasi-random dynamical systems.~{\rm I}.
Quasi-random diffeomorphisms
\jour Mat. Sb.
\vol 76
\issue 1
\yr 1968
\pages 72--134
\transl\nofrills English transl. in:
\jour Math. USSR-Sb.
\vol 5
\issue 1
\yr 1968
\pages 73--128
\moreref
\paper {\rm II}. One-dimensional non-linear oscillations in a field
with periodic perturbations
\jour Mat. Sb.
\vol 77
\issue 4
\yr 1968
\pages 545--601
\transl\nofrills English transl. in:
\jour Math. USSR-Sb.
\vol 6
\issue 4
\yr 1968
\pages 505--560
\moreref
\paper {\rm III}. Quasi-random oscillations of one-dimensional
oscillators
\jour Mat. Sb.
\vol 78
\issue 1
\yr 1969
\pages 3--50
\transl\nofrills English transl. in:
\jour Math. USSR-Sb.
\vol 7
\issue 1
\yr 1969
\pages 1--43
\endref
\ref\no3
\by Alekseev V.M.
\paper Final motions in the three-body problem and
symbolic dynamics
\jour Uspekhi Mat. Nauk
\vol 36
\issue 4
\yr 1981
\pages 161--176
\transl\nofrills English transl. in:
\jour Russian Math. Surveys
\vol 36
\issue 4
\yr 1981
\pages 181--200
\endref
\ref\no4
\by Alekseev V.M., Yakobson M.V.
\paper Symbolic dynamics and hyperbolic dynamical systems
\paperinfo\nofrills Appendix to
R. Bowen,
Methods of Symbolic Dynamics,
Collected papers,
\publ ``Mir'', Moscow, 1979, pp. 196--240
\transl\nofrills English transl. in:
\jour Phys. Reports
\vol 75
\issue 5
\yr 1981
\pages 287--325
\endref
\ref\no5
\by Anderson B.
\paper Centralisers of $C^\infty$-diffeomorphisms
\jour Trans. Amer. Math. Soc.
\vol 222
\yr 1978
\pages 97--106
\endref
\ref\no6
\by Anosov D.V.
\book Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature
\bookinfo Trudy Matem. Inst. Steklova AN SSSR, Vol. 90
\yr 1967
\transl\nofrills English transl. in:
\bookinfo Proc. Steklov Inst. Math., Vol. 90
\yr 1967
\endref
\ref\no7
\by Birkhoff G.D.
\book Dynamical Systems
\bookinfo rev.~ed., Amer. Math. Soc. Colloq. Publ., Vol.~9
\publ Amer. Math. Soc.
\publaddr Providence
\yr 1966
\endref
\ref\no8
\by Brjuno A.D.
\paper Analytic form of differential equations
\jour Trudy MMO
\vol 25
\yr 1971
\pages 119--262
\transl\nofrills English transl. in:
\jour Trans. Moscow Math. Soc.
\vol 25
\yr 1971
\pages 131--288
\moreref
\paper The analytic form of differential equations.~{\rm II}
\jour Trudy MMO
\vol 26
\yr 1972
\pages 199--238
\transl\nofrills English transl. in:
\jour Trans. Moscow Math. Soc.
\vol 26
\yr 1972
\pages 199--239
\endref
\ref\no9
\by Brjuno A.D.
\paper Normal form of real differential equations
\jour Matem. Zametki
\vol 18
\issue 2
\yr 1975
\pages 227--241
\transl\nofrills English transl. in:
\jour Math.Notes
\vol 18
\issue 1--2
\yr 1975
\pages 722--731
\endref
\ref\no10
\by Bronshtein I.U.
\book Nonautonomous Dynamical Systems
\publ ``Shtiintsa''
\publaddr Kishin\"ev
\yr 1984
\lang Russian
\endref
\ref\no11
\by Bronshtein I.U. % (Bron\v ste\u\i n)
\paper Transversality implies structural stability
\jour Dokl. AN~SSSR
\vol 257
\issue 2
\yr 1981
\pages 265--268
\transl\nofrills English transl. in:
\jour Sov. Math. Dokl.
\vol 23
\issue 2
\yr 1981
\pages 251--254
\endref
\ref\no12
\by Cartan H.
\book Calcul Diff\'erentiel
\publ Hermann
\publaddr Paris
\yr 1967
\endref
\ref\no13
\by De Latte D.
\paper Nonstationary normal forms and cocycle invariants
\jour Random Comput. Dynamics
\vol 1
\issue 2
\yr 1992/93
\pages 229--259
\endref
\ref\no14
\by Dovbysh S.A.
\paper Transversal intersection of separatrices and
branching of solutions as obstructions to the
existence of an analytic integral in
many-dimensional systems.~{\rm I}.
Basic result: Separatrices of hyperbolic
periodic points
\jour Collectanea Mathematicae
\vol 50
\issue 2
\yr 1999
\pages 119--197
\endref
\ref\no15
\by Dovbysh S.A.
\paper Intersection of separatrices and the non-integrability of
multidimensional systems
\jour Uspekhi Mat. Nauk
\vol 55
\issue 3
\yr 2000
\pages 179--180
\endref
\ref\no16
\by Eells J.
\paper A setting for global analysis
\jour Bull. Amer. Math. Soc.
\vol 72
\issue 5
\yr 1966
\pages 751--807
\endref
\ref\no17
\by Fenichel N.
\paper Persistence and smoothness of invariant manifolds for flows
\jour Indiana Univ. Math. J.
\vol 21
\issue 3
\yr 1971
\pages 193--226
\endref
\ref\no18
\by Fenichel N.
\paper Asymptotic stability with rate conditions
\jour Indiana Univ. Math. J.
\vol 23
\issue 12
\yr 1974
\pages 1109--1137
\moreref
\paper {\rm II}
\jour Indiana Univ. Math. J.
\vol 26
\issue 1
\yr 1977
\pages 81--93
\endref
\ref\no19
\by Gantmacher F.R.
\book The Theory of Matrices
\publ Gostekhizdat
\publaddr Moscow
\yr 1953
\finalinfo 4th ed. by ``Nauka'', Moscow, 1988
\transl\nofrills English transl. in:
\publaddr New York
\publ Chelsea
\yr 1959
\endref
\ref\no20
\by Guysinsky M., Katok A.
\paper Normal forms and invariant geometric structures for dynamical
systems with invariant contracting foliations
\jour Math. Res. Letters
\vol 5
\issue 1--2
\yr 1998
\pages 149--163
\endref
\ref\no21
\by Hadamard J.
\paper Sur l'it\'eration et les solutions asymptotiques des
\'equations diff\'erentielles
\jour Bull. Soc. Math. France
\vol 29
\yr 1901
\pages 224--228
\endref
\ref\no22
\by Hale J. K., Lin Xiao-Biao
\paper Symbolic dynamics and nonlinear semiflows
\jour Ann. Math. Pura Appl.
\vol 144
\yr 1986
\pages 229--259
\endref
\ref\no23
\by Hartman P.
\book Ordinary Differential Equations
\publ Wiley
\publaddr New York--London--Sydney
\yr 1964
\finalinfo reprint 2nd ed. by Birkh\"auser, Boston--Basel--Stuttgart, 1982
\endref
\ref\no24
\by Hirsch M.W., Pugh C.C., Shub M.
\book Invariant Manifolds
\bookinfo Lecture Notes in Math.,
Vol. 583
\publaddr Berlin--Heidelberg--New York
\publ Springer
\yr 1977
\endref
\ref\no25
\by Katok A., Spatzier R.J.
\paper Differential rigidity of Anosov actions of higher rank Abelian
groups and Abelian lattice actions
\inbook Trudy Matem. Inst. Steklova RAN, Vol. 216
\pages 292--319
\yr 1997
\transl\nofrills English transl. in:
\bookinfo Proc. Steklov Inst. Math., Vol. 216
\pages 287--314
\yr 1997
\endref
\ref\no26
\by Kopell N.
\paper Commuting diffeomorphisms
\inbook Global Analysis
\bookinfo Proc. Symp. in Pure Math., Vol. 14
\eds S.-S. Chern, S. Smale
\publ Amer. Math. Soc.
\publaddr Providence
\yr 1970
\pages 165--184
\endref
\ref\no27
\by Kostin V.V.
\paper Normal form of non-autonomous systems
\jour Dokl. AN~USSR. Ser.~A.
\issue 8
\yr 1973
\pages 693--696
\lang Russian
\endref
\ref\no28
\by Kostin V.V., Le Dinh Thuy
\paper Some tests for convergence of normalizing transformation
\jour Dopovidi AN~URSR (Dokl. AN~USSR). Ser.~A.
\issue 11
\yr 1975
\pages 982--985
\lang Ukrainian
\endref
\ref\no29
\by Kostin V.V.
\book Normalizing Transformations of Nonautonomous Systems
\bookinfo Preprint
\publ Odessa Univ.
% \publaddr Odessa
\yr 1975
\lang Russian
\endref
\ref\no30
\by Ku\v snirenko A.G., Katok A.B., Alekseev V.M.
\paper Smooth dynamical systems
\inbook Ninth Summer Mathematical School
\publaddr Kiev
\publ ``Naukova Dumka''
\yr 1976
\pages 50--341
\transl\nofrills English transl.:
\book Three Papers on Dynamical Systems
\bookinfo AMS Transl., Ser. 2. Vol. 116
%\publaddr
%\publ
\yr 1981
\endref
\ref\no31
\by Lang S.
\book Algebra
\publ Addison--Wesley
\publaddr Reading (Mass.)
\bookinfo Addison--Wesley Series in Math.
\yr 1965
\endref
\ref\no32
\by Lerman L.M., Shil\mz nikov L.P.
\paper Homoclinic structures in infinite-dimensional systems
\jour Sibir. Matem. Zh.
\vol 29
\issue 3
\yr 1988
\pages 92--103
\transl\nofrills English transl. in:
\jour Siberian Math. J.
\vol 29
\issue 3
\yr 1989
\pages 408--417
\endref
\ref\no33
\by Nemycki\u\i\ V.V., Stepanov V.V.
\book Qualitative Theory of Differential Equations
\publ Gostekhizdat
\publaddr Moscow--Leningrad
\yr 1947
\finalinfo 2nd ed., 1949
\transl\nofrills English transl. in:
\bookinfo Princeton Math. Series, no. 22
\publaddr Princeton
\publ Princeton Univ. Press
\yr 1960
\finalinfo reprint Dover, New York, 1989
\endref
\ref\no34
\by Nitecki Z.
\book Differentiable Dynamics@. An Introduction to the
Orbit Structure of Diffeomorphisms
\publ MIT Press
\publaddr Cambridge (Mass.)--London
\yr 1971
\endref
\ref\no35
\by Sibuya Y.
\paper Non-linear ordinary differential equations with periodic coefficients
\jour Funkcialaj ekvacioj
\vol 1
\issue 2
\yr 1958
\pages 121--204
\endref
\ref\no36
\by Smale S.
\paper Differentiable dynamical systems
\jour Bull. Amer. Math. Soc.
\vol 73
\issue 6
\yr 1967
\pages 747--817
\endref
\ref\no37
\by Sternberg S.
\paper Local contractions and a theorem of Poincar\'e
\jour Amer. J. Math.
\vol 79
\issue 4
\yr 1957
\pages 809--824
\endref
\ref\no38
\by Yomdin Y.
\paper Nonautonomous linearization
\inbook Dynamical Systems
\bookinfo Lecture Notes in Math.,
Vol. 1342
\ed J. C. Alexander
\publaddr Berlin--Heidelberg--New York
\publ Springer
\yr 1988
\pages 718--726
\endref
\endRefs
\enddocument
ENDBODY
~~