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\TITLE
Invariant manifolds associated to non-resonant
spectral subspaces.
\AUTHOR
Rafael de la Llave
\FROM
Department of Mathematics
University of Texas
Austin TX 78712-1082
\ENDTITLE
\vskip 3 em
\noindent
{\titlefont Keywords:} Invariant manifolds, linearizations, partial
linearizations, asymptotic behavior, renormalization group, invariant foliations
\vskip 5 em
\ABSTRACT
We show that, if the linearization of a map at a fixed point leaves invariant
a spectral subspace which satisfies certain non-resonance conditions,
the map leaves invariant a smooth
manifold tangent to this
subspace.
This manifold is as smooth as the map, but is unique
among $C^L$ invariant manifolds, where $L$ depends
only on the spectrum of the linearization.
We show that if the non-resonance conditions
are not satisfied, a
smooth invariant manifold need not exist
and also establish smooth dependence on parameters.
We also discuss some applications
of these invariant manifolds and
briefly survey related work.
\ENDABSTRACT
\SECTION Introduction
Besides their intrinsic appeal, invariant manifold theorems
are interesting in Dynamics because they provide
landmarks
which organize the long--time behavior.
>From this point of view, having more invariant manifolds
is quite desirable, since it means having more tools
for the analysis of the dynamical system. In particular, it is
often the case, that associated to invariant manifolds in
the tangent bundle, one can associate other invariant
structures in the manifold itself. For example one of the
standard constructions of stable and unstable foliations
\cite{HP} includes as one of the steps to
apply the stable manifold theorem to the
operator $f_*$ acting on vector
fields by:
$[f_* v](x) = Df( f^{-1}(x) ) v(f^{-1}(x)$.
The invariant manifolds constructed here,
will also lead to invariant structures on the
manifold. Nevertheless, they are not foliations,
as shown in \cite{JLP}. (These structures seem to
have been considered first in \cite{Pe})
We will discuss some of their uses later.
For example, they play a role in rigidity theory.
Another
motivation for this paper is the study of renormalization
group transformations. Even if a precise
analytical definition of
a renormalization group operator is fraught with technical
difficulties (See e.g \cite{EFS}, \cite{MO}) it is fruitful
to study maps that are well defined as
caricatures of the real situation. (See \cite{Be} Chap. 3
\cite{LMS}, Sec 5., Appendix E)
In that picture, different ways to approach the fixed point
correspond to different properties of finite size scale fluctuations.
An example of physical properties that can
be described by properties of the approach to the
fixed point can be found in \cite{GKT}. See also
\cite{LMS} for the analogy of some of the
manifolds constructed here with the
beta functions or renormalization group theory.
The renormalization group in dynamical systems
-- especially
in the period doubling case --
is much better behaved that the renormalization
group in Statistical Mechanics
and in that case, it is sometimes
possible to write well defined renormalization maps
that are analytic
in an appropriate space and which have
with a compact derivative (See e.g. \cite{CEK}, \cite{CEL}, \cite{VSK})
(These motivations are the reason why we have chosen to prove our results
in the generality of Banach spaces.)
If the system were linear, a very natural invariant set
would be an spectral subspace. For a
non-linear system close to a fixed point
-- and hence approximable by the derivative
at the fixed point
one can ask if there are analogues of the
spectral subspaces of the derivative which
are invariant for the full non-linear system.
The classical theory of invariant manifolds
establishes the existence of
invariant
manifolds associated with spectral subsets which are disks around
the origin or complements of disks around the origin.
Usually the manifold associated to
$\{ z \in \complex \big| |z| \le \rho < 1\}$
is called the strong stable manifold, that associated
to
$\{ z \in \complex \big| |z| < 1\}$
is called the stable manifold,
those associated to sets of the form
$\{ z \in \complex \big| |z| \le 1\}$
are called center stable manifolds
and those associated to sets
$\{ z \in \complex \big| |z| \le \rho > 1\}$
are called pseudostable manifolds.
We have used ``is'' or ``are'' on purpose
to indicate whether uniqueness
under local assumptions holds or not. We refer to Section 7.
about other results which also include uniqueness
under global assumptions.
On the other hand, for some finite dimensional systems,
one can apply the Sternberg linearization theorem and
conclude that the system is equivalent to
the linearization
expressed in another smooth system of
coordinates.
Of course, since the spectral subspaces are invariant
under the linearization, their image under the change of variables
that linearizes the map
will be invariant under the full map.
Hence, the non-linear map may leave invariant
some smooth manifolds that correspond
to spectral subspaces.
Even if the above argument indicates that the classical theory of
invariant manifolds is not as general as possible,
the theory based on the the Sternberg
linearization theorem is not very satisfactory either.
For example, for
infinite dimensional systems the non-resonance
conditions become harder to verify -- or even false
if the spectrum includes open sets --.
Even in finite dimensions, the conditions
of Sternberg theorem are non-open.
Moreover, since the linearizing changes of variables
provided by the Sternberg linearization theorem are
highly non-unique, it seems that the
smooth invariant manifolds produced this
way are also not unique.
In this paper we try to obtain a compromise
between the Sternberg linearization theorem
and the classical invariant manifold theory.
The proofs will start as in the Sternberg linearization theorem,
using non-resonance assumptions to eliminate undesired terms,
but we will switch as soon as possible to the -- much easier than
linearization -- invariant manifold theory.
This will allow to prove some invariant manifold theorems for
spectral subspaces satisfying some mild non-resonance conditions
that persist for open sets of problems.
Moreover, there will be local conditions that guarantee
uniqueness. Once that this uniqueness is established,
it makes sense to study the dependence
with respect to parameters. We show that, indeed
these manifolds depend smoothly with respect to parameters
and compute explicit formulas for the derivatives.
These formulas can be used to justify some perturbative
calculations of beta functions in renormalization group.
We will also provide examples that show that if those
non-resonance conditions are
not met, the conclusions are false.
The ideas presented above are closely related to partial
normal forms. That is, showing
certain terms are irrelevant for the dynamics since they can
be eliminated just by switching to appropriate systems of
coordinates. From the renormalization group point of view,
the discussion of when a term cannot be eliminated is
quite interesting.
Note that if we express the dynamics in
a system of coordinates where some terms in the map are not
present,
we can sometimes
obtain uniform estimates about all the iterates of
the map in this system of coordinates. Then,
these estimates can be read off in the original system.
This type of argument is often used
in dynamical systems to obtain very uniform control of iterates.
It is sometimes also used
in renormalization group arguments as a way to
control the approach of surfaces defining an interesting phenomenon
to critical surfaces.
In Section 6 we just quote some results on these problems of
partial linearizations. We point out that, compared to the method of
graph transforms developed in Section 4, they have the advantage
that they can produce invariant manifolds whose spectral
subset straddles the unit circle. Nevertheless the manifolds produced
using this method are much less differentiable than the map.
Finally, in Section 7 we review briefly other
works that study invariant manifolds other than the
classical stable, strong stable etc. and in
Section 8, we sketch some applications of
the results presented here.
\SECTION Acknowledgments.
This paper would not have been written without the friendly prodding of
A. Sokal, who raised the problem in 1987, explained the statistical
mechanics motivation and has kept encouraging me to write the answer.
Independently of that, I was driven to the same problems by an effort
to construct more invariant foliations to be used in rigidity questions.
Y. Yomdin pointed to me the paper \cite{Pe} which uses similar ideas.
(See also \cite{JLP}.) I am also grateful to H. Rosenthal for
bringing to my attention \cite{LeW},\cite{K}. Also to C. Pugh
for encouragement and pointing out that $C^r$ regularity was also possible.
This research has been supported by NSF grants and
a Centennial Fellowship from AMS and a URI from Univ. of Texas.
We also thank an anonymous referee that improved greatly the
exposition.
\SECTION Notation and statement of the main results.
In this paper $X$ will be
Banach space (not necessarily separable),
over the reals or over the complex.
$f$ will be a \ $C^r,\ 1 \le r \le \infty,\omega$,
mapping from $X$ to itself such that it has a fixed point,
which we will place at the origin (i.e. $f(0) = 0$).
We will call $Df(0) = A$ and write $f(x) = Ax+N(x)$.
The question we will address is the existence of
\lq\lq invariant manifolds\rq\rq passing though the
fixed point. That is,
submanifolds $W$ of $X$ such that $f(W) = W,\ 0 \in W$.
In the cases that we will consider, the problem is
equivalent to a local version so that we just need
to assume $f$ is defined in a small neighbor\-hood
of the origin.
The guiding idea behind the results presented here
is that, in small scales, $f$ is a perturbation of
$A$ and the invariant manifolds of $f$ should be
very similar to those invariant under $A$.
The most natural invariant manifolds for a linear
operator (they are not the only ones!) are invariant
subspaces associated to spectral projections. If
$X$ is a complex space, to each closed subset of the
spectrum bounded away from the rest of the spectrum
we can associate a spectral projection whose range
is a closed subspace. (If $X$ is real, the
subspaces are associated to subsets of the spectrum
as above which are also
invariant under complex conjugation). (See \cite{RS} \cite{Ka}.)
The invariant manifolds we will construct are
perturbations of these spaces. As it is often the
case with perturbation theory, it is very
advantageous to try to remove some terms with
an appropriate change of variables. Notice
that if we could eliminate all of them as in
the Sternberg linearization theorem, the result would
follow: the invariant manifolds would be the images
under the linearizing map of the invariant
subspaces.
The problem with applying
the Sternberg linearization
theorem in infinite dimensional Banach spaces,
is that the non-resonance conditions fail in open sets
of maps. (The spectrum of the linearization may
have non-empty interior.)
One can observe that the best known results on
existence of invariant manifolds :
the strong stable, stable, center stable, center,
center unstable pseudostable manifolds (\cite{Ha}, \cite{Ke}, \cite{Fe1},
\cite{Ir1}, \cite{Ir2}, \cite{Ir3}, \cite{HPS},)
amount to considering subsets of the spectrum
obtained by intersecting it with disks, complements of
disks or circles.
We will be able to generalize those results in that
we will be able to consider more general sets.
(At the end of the paper, we will list other results that
go in this direction)
We will not be able to associate invariant manifolds to
all the subsets of the spectrum of $A$,
but will have to impose \lq\lq non-resonance\rq\rq\
conditions so that some eliminations can be performed.
We will, furthermore show that, if these non-resonance
conditions fail, there are counter\-examples to the
theorems considered here (or slightly stronger versions).
The technique we will use is the \lq\lq graph
transform\rq\rq\ coupled with some manipulations
standard in \lq\lq normal form\rq\rq\ theory. These
manipulations, even if they simplify the proof, are not
really necessary and it is possible to construct a proof
without using them, (we will give an sketch of these alternative methods ).
Since the paper \cite{La1} (parts of it reproduced in \cite{MMc})
contains not only an excellent exposition of the graph
transform method for invariant manifolds but also an
exposition of normal forms, we will refer the reader
there for some basic results.
{\bf Notation}
Given a splitting $X = E^S \oplus E^U$ we will denote by
$\Pi_S$, $\Pi_U$ the corresponding projection.
Given a function $N: X \mapsto X$ we will define
$N_S$ (resp. $N_U)$ by
$N_{S,U}(\Pi_Sx,\Pi_Ux)=\Pi_{S,U}N(x)$.
If the splitting is invariant under a linear operator
$A$, we will call $A_S$, $A_U$ the restriction of $A$ to
those subspaces. This is slightly inconsistent with
the previous convention but is also customary and will
not lead to confusion.
Finally, we will call $B^S(\ell)$ the ball of radius
$\ell$ around $0$ in the space $E^S$, and analogously for
all spaces. If some of the indexes are clear from the context,
we will omit them.
Our main theorem is:
\CLAIM Theorem(I.1)
Let $X$ be a real or complex Banach space, $f$ a $C^r$
mapping, $r \in \natural \cup \lbrace \infty,\omega
\rbrace$, $r \ge 1$, $f(0) = 0$, $Df(0) = A$.
Assume that there is a decomposition
$
X = E^S \oplus E^U
$
such that:
\breakline
i)\quad \ The splitting is invariant under $A$ (Use the notation $A_S$, $A_U$
for the restrictions.) \breakline
ii)\quad \ $\sigma (A_S)$ bounded away from $\sigma (A_U)$ \breakline
iii)\quad $\sigma (A_U)$ bounded away from zero \breakline
iv)\quad $\sigma (A_S)$ contained strictly inside the unit disk. \breakline
We will call $L$ a integer big enough so that
$$
(\sup\{ |t| \bigm| t \in \sigma (A_S)\})^L(\sup\{
|t|^{-1} \bigm| t \in \sigma (A_U)\} ) < 1
$$
And we will assume that
\breakline
v)\quad \ $L < r$
\breakline
vi)\quad If $\,i < L$ then
$
[\sigma (A_S)]^i \cap \sigma(A_U) = \emptyset
$
\breakline
where
$[\sigma (A_S)]^i=\{ x_1\cdot
x_2\cdot\dots\cdot x_i \bigm| x_j \in \sigma (A_S) \qquad 1 \le j \le i\}$.
\breakline
vii) The function $N(x)=f(x)-A(x)$ has
sufficiently small $C^{L+1}$ norm when restricted to a ball of
radius $1$ around $0$.
(The smallness conditions depend only on the spectrum of $A$ ).
\breakline
Then, there is a function $w:B^S(1) \longrightarrow U$
such that: \breakline
a)\quad The graph of $w$ is invariant under $f$. \breakline
b)\quad $w$ is $C^{r-1+Lipschitz}$ \breakline
Moreover, $w$ is unique among the $C^L$ functions
satisfying a).
\REMARK
Given any function $f: X \mapsto X$, $f(0) = 0 $ and
$Df(0)$ satisfying the assumptions $i)$--$vii)$, $f_\lambda(x) \equiv
{1 \over \lambda}f(\lambda x) $ will verify the smallness hypothesis
for $\lambda$ small enough.
Indeed, recall that the smallness conditions
only depend on $Df_\lambda(0)$, which does not depend
on $\lambda$. On the other hand, $N_\lambda$ gets smaller
in the $C^r$ sense
as $\lambda$ gets small.
If $w_\lambda$ is the function whose graph is
invariant under $f_\lambda$, $w \equiv \lambda w(x/ \lambda)$ will
have a graph invariant under $f$. Note that $f_\lambda = A + N_\lambda$
and that $N_\lambda$ converges to $0$ in $C^r$ in the ball as
$\lambda$ tends to zero. Therefore, assuming smallness conditions in
$N$ and considering only a small neighborhood are equivalent.
\REMARK
It will be important later that the smallness conditions we impose
to $N$ are only in $C^L$ and not in $C^r$. If we want to consider
$C^\infty$ $f$'s, we will have to do a different proof for every finite $r$.
It will be important that we can choose the same $\lambda$
in all cases so that the function $w$ corresponding
to different $r$'s will be defined in the same domain.
\REMARK
Condition $vi)$ will be referred to as the
\lq\lq non-resonance\rq\rq\ condition. Its interpretation
is obvious when the $X$ is finite dimensional, it
just means that the product of any set of less than
$L$ eigenvalues of $A_S$ are not eigenvalues of $A_U$.
\REMARK
We will derive later stronger uniqueness
properties than those claimed in the theorem.
In particular, Theorem 5.2 implies that the
solution is unique among
$C^{r_0 + \epsilon}$ with
$r_0 = \log\Vert A_U^{-1}\Vert/\log \Vert A_S\Vert$.
Note that $L = [r_0] + 1$.
\REMARK
Observe that if the spectral subset we consider is given by
$\sigma (A) \cap \{ z \bigm| |z| < \alpha \}$ $\alpha < 1$
\clm(I.1) reduces to the $\alpha$-stable manifold theorem
characterized as the set of points $x$ such that
$\alpha^{-n}f^n(x)$
remains bounded.
In this case, the non-resonance conditions are obviously
satisfied.
\REMARK
Note that by the definition of $L$,
$[\sigma(A_S)]^i \cap \sigma(A_U) = \emptyset$
when $i \ge L$. Hence, with $vi)$,
the intersection is empty for all $i \in \natural$.
>From the presentation in the text, it is clear that the
condition holds in an open set of
mappings.
\REMARK
The conclusion $b)$ of \clm(I.1) can be improved
to $C^r$. See the remarks after the proof for details
There are many equivalent norms we can use in $X$.
Given an operator $A$ we can choose a norm in such a way that
$\Vert A \Vert \le \sup \{ |t| \bigm| t \in \sigma
(A) \} + \varepsilon$ for any $\varepsilon > 0$.
Moreover, we could also chose the norm in such a
way that the splittings
associated to a finite number of closed subsets of the
spectrum have projections of norm~1.
We will, henceforth assume such a norm, with a
sufficiently small $\varepsilon$, has been defined so that
the $L$ introduced in the assumption $iv)$ of the theorem
also satisfies
$$
\Vert A_S \Vert^L\/ \Vert A_U^{-1} \Vert < 1.
$$
Note that all the smallness assumptions etc. are to be understood
in this norm. Since it is equivalent to the original one,
all ``sufficiently small'' requirements in this norm are
implied by ``sufficiently small'' in the original one.
\clm(I.1) will be derived from the following theorem, which we will
prove first:
\CLAIM Theorem(I.2)
In the same set up of \clm(I.1), do not assume iv), but
assume instead
$iv')$\quad $N_U(s,u) = 0\,( \Vert u \Vert^{L+1},\ \Vert s \Vert^2)$ near $0$.
Then, the same conclusions
as in \clm(I.1) hold but we moreover have.
\breakline
c)$|| w(s) || = \OO( || s ||^{L+1})$
\REMARK
Note that, in the conditions of \clm(I.2),
denoting by $(s,u)$ the projections on the
stable and unstable component, we have:
$$
f(s,u) = (A_S s + N_S(s,u), A_U u) + \OO(||u||^{L+1}, ||s||^2)
\EQ(reducedform)
$$
Observe that if we ignore the high order terms, the set
corresponding to points with $u = 0$ is invariant under the
dynamics. In a neighborhood of this set, the
terms ignored are a very small perturbation and,
we will be able to construct the invariant manifold as a
perturbation of the set $\{ (s,0)\}$.
\SECTION Proof of \clm(I.1) and \clm(I.2)
As in \cite{La1}, the proof will consist in
formulating the invariant manifold as the graph of a function
$w: B^S(1) \longrightarrow U$
The fact that the manifold is invariant will be equivalent to
the fact that the function $w$ is a fixed point of an operator
$\Tau $ acting on an appropriate space of functions
First, we will prove the theorem when $r < \infty$.
We will take $\Tau$ to be
$$
\Tau [w](x) =
A_U^{-1}(w(A_Sx+N_S(x,w(x)))-N_U(x,w(x))).
\EQ(I.1)
$$
This operator was used in \cite{La2} -- not in \cite{La1}. It is not exactly the
graph transform, but is somewhat more manageable.
The reason to define the operator is that
if we compute $f$ on a point of the graph of
the $w$, we have:
$$
f(x,w(x)) = \big( A_S x + N_S(x, w(x)),\ A_U w(x) + N_U(x,w(x)) \big)
\EQ(transform)
$$
This transformed point belongs to the graph of $w$ if and only
if the second coordinate is the result of applying $w$ to
the first. That is,
$$
w( A_S x + N_S(x, w(x)) = \ A_U w(x) + N_U(x,w(x))
$$
which, provided that all
the compositions
can be defined, is equivalent to $w$ being a fixed point of $\Tau$.
The
operator $\Tau$ is well defined if $\Vert A_S\Vert +
\Vert N_S\Vert_{L^{\infty}} < 1$. We will assume that $N$ is small
enough that indeed $\Tau$ is defined
on functions on the unit ball of
$S$.
We will consider $\Tau$ as acting on the following spaces
$$ \eqalign{
{\chi_{\varepsilon_{1}\dots \varepsilon_{r-L}}^{r}} =
\biggl\{ w:B^S(1)&\mapsto U\text{\ such\ that} \cr
&\text{a)}\ \ w \in C^r \cr
&\text{b)}\ \ D^kw(0) = 0,0 \le k \le L \cr
&\text{c)}\ \ \sup\limits_{x\in B^{S}(1)} \Vert D^kw(x)\Vert \le 1,
\ \ 0 \le k \le L \cr
&\text{d)}\ \ \sup\limits_{x\in B^{S}(1)}
\Vert D^{L+i}w(x)\Vert \le \varepsilon_{i},\ \ 1 \le i \le r-L\biggr\}
}
$$
Note that, because of condition $b)$ the $w \in \chi$ is determined uniquely
by $D^L w$. We will therefore consider $\chi$ endowed with the
topology given by the norm $||w|| = ||D^L w||_{L^{\infty( B^S(1))}}$.
We also point out that by condition $b)$ we also
have
$|| D^i w||_{L^{\infty( B^S(1)}} \le
(1/(L-i)!) ||D^L w||_{L^{\infty( B^S(1))}}$.
For the sake of notation we will suppress the $B^S(1)$ from the
spaces, but $L^\infty$ is to be understood always to be on unit ball.
\CLAIM Proposition(findchi)
Given some smallness assumptions in
$\Vert D^kN_S\Vert_{L^\infty}$,
\ $0 \le k \le L$, it is possible
to find $\varepsilon_1,\dots,\varepsilon_{r-L} > 0$ such a
way that
$$
{\Tau} (\chi^r_{\varepsilon_1,\dots,
\varepsilon_{r-L}}) \subset
\chi^r_{\varepsilon{_1},\dots,\varepsilon_{r-L}}.
$$
These $\varepsilon$'s can be found in a recursive
consistent way: that is if \ $\bar r$ $> r$
and we have found $\varepsilon_1\dots \varepsilon_{r-L}$, we
can find $\varepsilon_{r-L+1}\dots \varepsilon_{{\bar
r}-L+1}$ so that
$$
{\Tau}(\chi^{\bar r}_{\varepsilon_1,\dots
,\varepsilon_{r-L},\varepsilon_{r-L+1},\dots
,\varepsilon_{r-L}}) \subset {\chi^{\bar
r}_{\varepsilon_1,\dots,
\varepsilon_{r-L},\varepsilon_{r-L+1},\dots
\varepsilon_{{\bar r}-L}}}. $$
\PROOF
The fact that ${\Tau}[w]$ satisfies a), b)
if $w$ does is quite easy.
Since for functions satisfying a), b) we have for $k L$ and both integers.
(In the finite dimensional case, it is a Gateaux differential)
Its differential is:
$$
\eqalign{
[D\Tau(w)]\eta (x) &= A_U^{-1}[\eta (A_Sx+N_S(x,w(x)))\cr
&+ Dw(A_Sx+N_S(x,w(x)))D_2N_S(x,w(x))\eta (x)\cr
&-\ D_2N_U(x,w(x))\eta (x)].
}
\EQ(I.3)
$$
\PROOF
Since the operator $\Tau$ is obtained
as a repeated application of
the composition
operator, the result
can be proved by the same methods
as those in \cite{Ir1}.
For every point $x$ we have the formula
$$
\eqalign{
\Tau[ w &+ \eta](x) =
\Tau[ w ](x) +
\int_0^1 A_U^{-1}
\, \eta ( A_S x + N_S(x,(w + \lambda \eta)(x)) ) \cr
& + \int_0^1 A_U^{-1} Dw( A_S x + N_S(x,(w + \lambda \eta)(x)) ) \eta(x)
D_2 N_S(x,(w + \lambda \eta)(x))\eta(x) \, d \lambda\cr
& -
\int_0^1 A_U^{-1} D_2 N_U(x,(w + \lambda \eta)(x))\eta(x)
\, d\lambda
}
\EQ(interpolation)
$$
The desired result follows from
interpreting the above formula
as a formula in Banach space of
functions and estimating the continuity of
the remainders.
This is the same method used in
\cite{Ir1} and we refer to this paper for
details.
We note that the composition
operator, considered as an operator in
$C^L$ is differentiable in the Gateaux sense only at functions
which are at least $C^{L+1}$ and with a uniform
continuity of the derivatives of high order.
Other than that, it is only Frechet.
\QED
\CLAIM Proposition(smallness)
Assume that $N$ is $C^r$ and that
$A$ satisfies the assumptions of
\clm(I.2). Denote
by $\chi^r_{\varep_1,\ldots,\varep_{r-L}}$
a set of the type produced in \clm(findchi).
Assume that $\Vert N\Vert_{C^{L}}$,
is sufficiently small
where the smallness assumptions depend only on $\varep_1$
(and, therefore,
only on $\vert D^{L+1} \Vert_{L^\infty}$)
Then, the derivative in \equ(I.3) is a
contraction in $\chi^r_{\varep_1,\cdots,\varep_{r-L}}$
in the norm of the supremum of $D^Lw$.
We emphasize that the smallness
assumptions on $\Vert N \Vert_{C^L}$
needed for the conclusion of the theorem.
are independent on all the
derivatives of order higher than $L+1$.
This is reasonable because the norm only
involves the first $L$ derivatives
and we need one more to
establish that $\Tau$ is Lipschitz.
The reason we chose the wording of the theorem in this
way is that we obtain uniqueness in
$\chi^r_{\varep_1,\ldots,\varep_{r-L}}$, hence it is somewhat interesting to
let the
$\varep_{1},\ldots, \varep_{r-L}$ to be as unrestricted as possible.
We also emphasize that we do not need that $ \Vert D^{L+1} N \Vert$
is small but rather that the smallness assumptions that we
need on $\Vert N\Vert_{C^L}$ depend on
the bounds that we have for it.
In the applications, however, by restricting our attention
to an small enough ball
we can assume, as we have discussed, that $\Vert N\Vert_{C^{L+1}}$ is small
so that for the sake of applications it would have been
enough to prove \clm(smallness) under the assumption that
$\Vert N\Vert_{C^{L+1}}$ is small.
\PROOF
In the formula \equ(I.3) for the differential of $\Tau$
take $L$ derivatives with respect to $x$, expand
using the chain rule and the rules for sums and products
(tensor products)
of derivatives as often as possible.
We get $D_x^L [(D_w \Tau)[w]\eta](x)$
as a sum of terms.
The only term in this sum that does not contain a derivative
of $N$ is:
$$
A_U^{-1}\, D^L\eta (A_Sx+N_S(x+w(x))A_S^{\otimes L}.
\EQ(firstterm)
$$
All the other terms contain a factor which is a
derivative of $N$ of
order not bigger than $L$.
The factors other than derivatives of $N$
of order up to $L$ that appear in the other terms
term are either derivatives of $\eta$ of order
not bigger than $L$ or derivatives of $w$ of order not
bigger than $L+1$.
The only terms that include a derivative of
$w$ of order $L+1$ are those resulting form expanding
$$
\eqalign{
A_U^{-1}& ( D^{L+1}w) ( \ A_S x + N_S(x,w(x))\, ) \cdot\cr
& \cdot [A_S + (D_1 N_S(x,w(x)) + D_2 N_S(x,w(x))Dw(x) ) ]^{\otimes L}
D_2 N_S(x,w(x)) \eta(x))
}
\EQ(secondterm)
$$
Therefore, except from \equ(firstterm) and \equ(secondterm), all the
other terms involve derivatives of $\eta$ of order not more than $L$,
derivatives of $w$ of order not more than $L$ and at least one
factor which is a derivative of $N$ or order not more than $L$.
The term \equ(firstterm) can be bounded by $\lambda \Vert
D^L\eta \Vert _{L^\infty}$ where
$0 < \lambda \equiv \Vert A_S\Vert^L \/\Vert A_U^{-1}\Vert < 1$
by
assumption $iv)$. Hence, the linear operator
defined in \equ(firstterm) has a norm
which is strictly smaller than $1$.
To bound \equ(secondterm), we note that
the $L+1$ derivative of $w$ is bounded by $\varepsilon_{1}$
and all the other derivatives of $w$ are bounded by 1.
Recalling that, since $D^i \eta(0) = 0$ for $0 \le i \le L-1$
we can bound the supremum of $\eta$ by
$1/L! \Vert D^L \eta ||_{L^\infty}$.
Therefore, we can bound the norm of the linear
operator in \equ(secondterm) by
$$(1/L!)\varepsilon_{1} || A_U^{-1}|| ( || A_S|| + 2\vert DN\Vert_{L^\infty} )
\Vert D N\Vert_{L^\infty}.$$
This, clearly, can be made arbitrarily small by assuming that
$ \Vert DN\Vert_{L^\infty}$ is sufficiently small.
Since all the other terms include at least a factor
which is a derivative of
$N$ of order not bigger than $L$ and
derivatives of $w$ of order not larger than $L$
and derivatives of $\eta$ up to order $L$
the norm of all the other
terms can be bounded
by
$\rho \Vert \eta \Vert_{C^L}$ where $\rho$ can be made
as close to zero as desired by making $\Vert N\Vert_{C^L}$
sufficiently small.
That is, we can estimate the norm of
the derivative by a number, which is
strictly smaller than $1$ and a finite
number of other terms that can be made arbitrarily small
by assuming that $|| N||_{C^L}$ is
sufficiently small.
Hence, by the contraction mapping theorem, there is a
unique fixed point in the sequence closure of $\chi^r$. This
finishes the proof of the \clm(I.2) except
for the $C^\infty$, $C^\omega$ cases.
(Notice that this method automatically produces the
uniqueness statement claimed in the theorem ).
To prove the $C^\infty$ result,
the key observation (standard in the theory, see \cite{La1},
\cite{HPS}) is that all the fixed points in different
$\chi^r$ have to agree.
Using only smallness assumptions in $||N||_{C^L}$,
for any $r$ we have the choices of $\varepsilon_1, \dots, \varepsilon_{r-L}$
in such a way that
$\chi^r_{\varepsilon_1,\dots,\varepsilon_{r-L}}$
is mapped into
itself and $\Tau$ is a contraction.
We have that if $r' > r$, then
$\chi^{r'}_{\varepsilon_1,\dots,\varepsilon_{r'-L}} \subset
\chi^r_{\varepsilon_1,\dots,\varepsilon_{r-L}}$
The fixed points produced for every $r$ have to be the same.
We emphasize
that here we use essentially
that $S$ is contained in the unit circle. Indeed, there are examples
in which the spectrum of $A_S$ contains the unit circle
and in which there is no $C^\infty$ invariant manifold.
The proof of $C^\omega$ regularity is simpler. We have
to consider $\Tau$ acting on a space of analytic
functions vanishing up to order $L$ at the origin with
the $C^L$ norm in a complex neighborhood of the space.
The same argument used here shows it is a contraction
(the properties of the absolute value are the same be it
real or complex) and the uniform limit of analytic
functions is analytic (\cite{Ka} ch.~7).
\QED
\CLAIM Proposition (subset)
If $S$, $S'$ are spectral subsets both of which satisfy
the assumption of the theorem and $S\subset S'$, then
$W^S\subset W^{S'}$.
\PROOF
If we consider the derivative at zero of
$f|_{W^{S'}}$ we see that its spectrum is precisely
$S'$ and, clearly $S$ satisfies the non-resonance
assumptions. Hence, we can find a $C^r$ manifold
associated to it in the restriction to $W^{S'}$.
Now, this manifold can be considered as a submanifold of
$X$, and it fulfills the assumptions of the uniqueness
theorem. So it is $W^S$, hence $W^S$ is contained in
$W^{S^1}$.
\REMARK
When $X$ is finite dimensional the conclusion $b)$
of \clm(I.1) can be
improved to $C^r$ using the same proof.
The idea of the proof is showing that
the sequence of functions
$\{D^r\Tau^n[w](x)\}_{n=0}^\infty$ are
equicontinuous and equibounded in $n$. The proof is not
very difficult given the bounds we have already developed
and a similar calculation is in \cite{La1}.
Unfortunately, this argument does not
work in infinite dimensional spaces
because uniform continuity on
the unit ball does not follow from continuity.
So, another argument is needed. For the stable manifold
case, $C^r$
regularity even in infinite
dimensional Banach spaces is
proved by a different argument in \cite{HP}.
It seems that this argument can be adapted to our case,
but since this borderline regularity in
infinite dimensional spaces seems specialized, we postpone the discussion
of this point.
The following proposition -- whose proof is well known,
will show that \clm(I.1) follows from \clm(I.2).
\CLAIM Proposition(normalform)
Given $C^\infty$ function $f$ satisfying assumptions $ii), iii), v), vi)$
of \clm(I.1)
, there is a $C^\infty$ map $\phi$ with a $C^\infty$ local inverse
such that
\item{$i)$}$ \phi(0) = 0 $
\item{$ii)$} $D\phi(0) = \Id $
\item{$iii)$} $\phi^{-1}\circ f \circ \phi$
verifies the assumptions of \clm(I.2).
\REMARK
We emphasize that \clm(normalform) does not require
assumption $iv)$ of \clm(I.1). That is, we do not require
that $\sigma(A_S)$ is contained inside the unit disk.
This will become crucial when we discuss pseudostable
non-resonant sets.
\PROOF
(See any of \cite{La}, \cite{St}, \cite{Ne},
\cite{BM} among many others for very similar
computations.)
We try to write $\phi=\phi^2\circ\dots\circ\phi^L$ where
each of the $\phi^i$ can be written as
$\phi^i=Id+\phi_U^i$ and $\phi_U^i(x,y)$ only depends on
the first argument and is multilinear of order $i$.
The implicit function theorem shows that $\phi^i$ has a
local inverse $(\phi^i)^{-1}(x,y) = (x,y) -
\phi_U^i(x)+\OO(\Vert x\Vert^{i+1})$.
Our goal is to determine $\phi_i^i$ so that
$\Pi_U(\phi^i)^{-1}\circ\dots\circ(\phi^i)^{-1}
\circ f \circ\phi^i\dots\phi^1(x,y)= \Pi_Uf(y) +
\OO(\Vert x\Vert^{i+1})$.
This can be achieved by
recursively
finding $\phi_U^i$
recursively
If we assume that
$\phi_U^l, l *L]}\, (x)
$$
and, matching powers derive equations for the
$D^iw(0)$, which can be solved because of the non-resonance
conditions.
It is easy to see that if the $D^iw(0)$ solve these
equations
$$
\Tau (\sum^L\, \frac 1{i!}\, D^iw(0)x^{\otimes i} +
w^{[>L]}\, (x) =\sum^L\, \frac 1i\, D^iw(0) x^{\otimes i}
+ \widetilde{\Tau }(w^{[>L]}\, (x)).
$$
The operator $\widetilde{\Tau }$ can be
studied by the same methods $\Tau$ was studied here.
But the computations are much more cumbersome.
\SECTION Dependence of the manifolds on the map and uniqueness results.
We can think of \clm(I.2) as defining a mapping that,
given any $N$ produces $w$. Since our point of view was
thinking of these results as perturbations of $N\equiv
0$, it is quite natural to investigate the dependence of
$w$ on $N$.
\CLAIM Theorem(I.3)
Assume the conditions of \clm(I.2) as well
as smallness
assumptions in $\Vert N\Vert_{C^r}$.
If we give the $w$'s the $C^{L+k}$ norm and the
$N$'s the $C^r$ norm the mapping $N \longrightarrow
w(N)$ is $C^{r-(L+k+3)}$ provided $r>L+k+3$.
An analogous result holds for the invariant manifolds
constructed in \clm(I.1).
\PROOF
If we were going to prove that
the mapping was differentiable the most natural to thing to
do would be
to write down
explicitly the $N$ dependence in $\Tau$ and apply the
implicit function theorem to the functional equation
$\Tau(w,N)=w$.
Unfortunately, this fails because in order to compute
$D_w\Tau$, the first derivative of $w$ enters. (See
\equ(I.2) )
However, to prove differentiability on parameters it is
possible to follow the strategy of the proof
of the implicit function theorem observing
that at the solutions, the mapping $\Tau$ is
differentiable with respect to $w$ and, moreover
$$
D\Tau:C^{L+k}\longrightarrow C^{L+k}
$$
is a contraction.
(This follow from $C^{L+k+1}$ smallness assumptions in
$N$ and in $w$, the later are implied by $C^{L+k+2}$
smallness assumptions in $N$.)
Consequently, the mapping $N \mapsto w(N)$, which,in principle, is only
continuous has a candidate $W'$ for a derivative at the
solutions of $\Tau(w)=w)$
$$
W'(N) = \sum_k^\infty\,
[D_w\Tau]^k\, D_N\Tau
$$
when $w$ is a solution.
(This follows by repeating the argument that lead to conclude
that
$D_w\Tau$ is a contraction in $C^L\longrightarrow
C^L$.
The assumptions needed to repeat the argument are
$C^{L+k+1}$ smallness in $N$ and in $w$, but the later are
implied by $C^{L+k+2}$ smallness in $N$.)
Since the sum $\sum\limits_k\, [D_w\Tau]^k$ converges, this map
is well defined and is a continuous function.
The fact that this is the true derivative comes from the
fact that it is continuous and, if we integrate back,
$$
\widetilde w(N_\lambda) = w(N_0 + \int_0^\lambda\
w'(N_0+t(N-N_0))\,dt
$$
satisfies
$$
\eqalign{
&\Tau(\widetilde w(N_0),N_0)-\widetilde w(N_0)=0 \cr
&\frac {d} {d\lambda} [\Tau(\widetilde w(N_\lambda
),N_\lambda ) - \widetilde w(N_\lambda)] = 0
}
$$
so that, by the uniqueness properties established
in theorem 2, we have that $\widetilde w\equiv w$ or that
$w'$ is a bona fide derivative of $w$.
Once we have that $w$ is differentiable, the expression
for the derivative we just need to observe that, working on
a segment, we have estimates for the remainder of the first order
Taylor formula, which are uniform in the segment -- here we use one
derivative more in $f$ -- so that indeed $w'$ is a true derivative.
The existence of higher derivatives can be established in the same
way or invoking the `` tangent functor trick'' of \cite{AR}.
We leave the details to the reader.
This is, of course the crucial step in the proof of
smooth dependence of the manifolds on the map.
To finish the proof we only have
to verify that the other step in the reduction -- choosing coordinates
along the spectral subspaces -- depends smoothly on the map.
But the smooth dependence of the spectral spaces on the linear map
is an standard result in functional analysis (see e.g. \cite{Ka}.)
The uniqueness part of theorem 2 can be considerably
strengthened. Since this may be useful for other
developments, we will formulate it precisely.
Denote by $r_0$ a number -- not necessarily an integer -- in this
section such that
$$
\Vert A_U^{-1} \Vert \, \Vert A_S \Vert^{r_0} < 1.
$$
(That is, $r_0 < \log \Vert A_U^{-1}\Vert/\Vert A_S \Vert $)
Then set:
$$ \eqalign {
\chi^\delta = \bigl\{ w:B^S(1)
\longrightarrow U|w(0) &= 0; \qquad w\text{ Lipschitz };\cr
&\frac{\Lip(w|_{B^S(r)})}{r^{(r_0-1)}} \le
\delta\ r > 0 \bigr\}
}$$
where $\Lip$ denotes the Lipschitz constant
and, for the sake of typography are suppressing the
dependence of $\chi^\delta$ on $r_0$.
We observe that then, for all $w$ in this
space
$$\big\Vert |w| \big\Vert = \sup_{x\ne 0}\, \frac {\Vert
w(x)\Vert} {\Vert x\Vert ^{r_0}}
$$
is finite and, if we topologize $\chi^\sigma$
with this norm it is complete.
\CLAIM Theorem(I.4)
In the assumptions of \clm(I.2), there exist a $\delta$
such that $\Tau(\chi^\sigma)\subset \chi^\sigma$ and
$\Tau$ is a contraction there.\qquad
Hence, the $w$ of theorem 2, which is actually smooth, is
the only one function in $\chi^\delta$ satisfying
$\Tau w=w$.
\PROOF
The proof is a quite straightforward calculation. We
first show $\Tau$ is a contraction
$$\eqalign{
[\Tau u](x)-[\Tau w](x) =&
A_U^{-1}\big[ (u(A_Sx+N_S(x,u(x)))-w(A_Sx+N_S(x,u(x))))\cr
&+ (w(A_Sx+N_S(x,u(x)))-w(A_Sx+N_S(x,w(x)))\cr
&+ (N_u(x,w(x)))-N_u(x,u(x)))\big].
}
$$
Taking norms and dividing by $\Vert
x\Vert^{r_0}$, the first term can be estimated by inserting in
the numerator and denominator $\Vert
A_Sx+N_S(x,u(x))\Vert^{r_0}$; the second one uses that $\Vert
N_S(x,u(x)) - N_S(x,w(x))\Vert \le (\Lip\ N_S(x,u))\, \Vert
u(x) - w(x)\Vert$ and that this factor can be made as
small as we wish by assumption and a similar argument
works for the last.
The proof that $\Tau (\chi^\delta) \subset \chi^\delta$
follows easily from the chain rule for Lipschitz constants
we have:
$$\Lip\ \Tau [w]\big|_{B^S(r)}
\le\ \Vert A_U^{-1}\Vert (\Vert A_S\Vert + \Lip\ N_S(1+\delta ))\Lip\ w|_{B^S(r^*)} + \Lip\ N_U({r_0}+\delta)$$
where $r^* \ge \sup\limits_{|x|>r} \Vert
A_Sx+N_S(x,w(x))\Vert$ which up to errors arbitrarily
small by suitable smallness assumptions is just $\Vert
A_S\Vert r$.
If we now divide both sides by $r^{{r_0}-1}$ we
get
$$
\eqalign{
\frac{\Lip\ \Tau [w]\big|_{B^S(r)}}
{r^{{r_0}-1}} \quad & \le \quad \Vert A_U^{-1}\Vert (\Vert
A_S\Vert + \gamma )
\frac{\Lip\ w|_{B^S(r^*)}} {{r^*}^{{r_0}-1}}
\quad \frac{{r^*}^{{r_0}-1}} {r^{{r_0}-1}} \cr
&\le \quad \Vert
A_U^{-1}\Vert (\Vert A_S\Vert + \gamma )^{r_0} \quad
\frac{\Lip\ w|_{B^S(r^*)}} {{r^*}^{{r_0}-1}}
}$$
where $\gamma$ denotes a number which can be
made arbitrarily small by assuming smallness conditions
on $\Lip\ N$.
\QED
The following characterization of invariant
manifolds is more restrictive in the conditions we impose
in the spectrum, but, on the other hand does not make any
regularity assumptions.
This characterization roughly says that if we only
consider orbits for which the components are bounded by a
power of the $S$ component, the $S$ component determines
all of them. In other words, restricted to these
\lq\lq parabolic region\rq\rq\ the set of points that
converge is a graph.
This is quite analogue to the usual proof of the fact
that the stable manifold (characterized only by
topological properties) is indeed the graph of a function.
\CLAIM Theorem(I.5)
Let $f,A,S,U$ be as in \clm(I.1). Call
$$
\Gamma _{\rho, C,\ell} = \{ x\in B^\ell\big|\Vert
\Pi_Ux\Vert \le C\Vert \Pi_Sx\Vert^\rho \}.
$$
If $\rho$ satisfies $\rho < \frac{\ln\Vert
A_U^{-1}\Vert^{-1}}{\ln\Vert A_S\Vert}$ we can find
$\ell^*(\rho)$ in such a way that if two points
$x_1,x_2$ satisfy $\Pi_Sx_1=\Pi_Sx_2$ and
$\{ f^n(x_i)\}_{n=0}^\infty \subset \Gamma_{\rho,
C,\ell}$ (any $C \ge \ell < \ell^*(\rho))$ then $x_1=x_2$.
\PROOF
We will denote by $\varepsilon = \sup\limits_{x\in
B^\ell} \Vert DM(x)\Vert$. We will show that if
$\varepsilon$ is small enough, which amounts to $\ell$
small enough, we get the conclusions of the theorem. We
have if $x,y\in B^\ell$
$$
\eqalign{
\Vert \Pi_Uf(x) - \Pi_Uf(y)\Vert &\ge \Vert
A_U^{-1}\Vert^{-1} \, \Vert \Pi_U(x-y)\Vert -
\varepsilon\Vert\Pi_S(x-y)\Vert \cr
\Vert \Pi_Sf(x) - \Pi_S(f_y)\Vert & \le \Vert A_S\Vert \,
\Vert \Pi_S(x-y)\Vert + \varepsilon\Vert \Pi_U(x-y)\Vert.
}
$$
If we consider
%$$\pmatrix \Vert A_S\Vert & \varepsilon \\ \vspace{3\jot}
%-\varepsilon & \Vert A_U^{-1}\Vert \endpmatrix ^n \quad
%\pmatrix \Vert \Pi_S(x-y)\Vert \\ \vspace{3\jot}
%\Vert\Pi_U(x-y)\Vert \endpmatrix $$
$$\left( \matrix {\Vert A_S\Vert & \varepsilon \cr
-\varepsilon & \Vert A_U^{-1}\Vert }\right)^n \quad
\left( \matrix {\Vert \Pi_S(x-y)\Vert \cr
\Vert\Pi_U(x-y)\Vert } \right)
$$
the first component gives an upper bound for
$\Vert \Pi_Sf^nx-f^n(y)\Vert$ and the second a lower
bound for $\Vert \Pi_Ui (f^n(x)-f^n(y))\Vert$.
If we diagonalize the matrix we can see that
$$
\Vert \Pi_Sf^n(x)-f^n(y)\Vert \le (\Vert A_S\Vert +
\varepsilon' )^n\ (\Vert \Pi_S(x-y)\Vert + \varepsilon'
\Vert \Pi_U(x-y)\Vert )
$$
$$
\Vert \Pi_Uf^n(x)-f^n(y)\Vert \ge (\Vert A_U^{-1}\Vert -
\varepsilon' )^n\ (\Vert \Pi_U(x-y)\Vert - \varepsilon'
\Vert \Pi_S(x-y)\Vert ).
$$
($\varepsilon'$ depends only on $\varepsilon$ and is as
small as we wish with $\varepsilon$.)
If we now apply this result taking $x=x_i$, $y=0$, we
get $\Vert \Pi_Sf^n(x_i)\Vert \le (\Vert A_S\Vert +
\varepsilon' )^n\, \Vert \Pi_Sx_i\Vert$ and, if we apply
it with $x=x_1$, $y=x_2$ we obtain:
$$\Vert \Pi_U(f^n(x_1)-f^n(x_2))\Vert \ge (\Vert
A_U^{-1}\Vert^{-1}-\varepsilon )^n\,
\Vert\Pi_U(x-y)\Vert .
$$
Unless $\Vert \Pi_U(x-y)\Vert = 0$, this is a
contradiction with the assumption about the orbits of
$x_1$, $x_2$ satisfying $\Vert \Pi_Sf(x_i)\Vert \le
C\Vert \Pi_Sf(x_i)\Vert^\rho$.
\REMARK
We note that if we have a map satisfying the
conditions of \clm(I.2) and it is
$C^{r_0+\epsilon}$, in a sufficiently small ball it has to be
be in $\chi^\delta$. The reason is because, by the
non-resonance argument that we had
before, all the derivatives up to
order $[r_0]$ have to vanish. Then, the
fact that the map is in $C^{r_0 + \epsilon}$
implies that the remainder of the Taylor
expansion of the derivative has to
make it be in $\chi^\delta$.
The existence and uniqueness results developed so far can
be counterpointed with the following examples:
{\bf Example 1}
The mapping $(x, y ) \mapsto ( 1/2 x, 1/4 y) $,
besides the spectral subspaces has
$(x,x^2)$ as an invariant manifold.
This manifold is clearly analytic.
Therefore, in general there could be other invariant manifolds besides
the ones we consider here; notice how this example
violates the assumptions of both of our uniqueness
theorems. This example shows that the parameters appearing in
our uniqueness theorems
cannot be lowered.
{\bf Example 2}
Let $A$ be diagonalizable.
If the relation between eigenvalues $\lambda_{i_1}
\dots \lambda_{i_r} = \lambda_1$ holds, the map
$$
x \rightarrow Ax + \underline\ell_1x_{i_1} \dots
x_{i_r}
$$
where $\underline \ell_1$ denotes the eigenvector corresponding to $\lambda_1$
and $x_{i_j}$ denote the coordinates
along the directions of
the eigenvectors corresponding to $\lambda_{i_j}$
does not have a $C^r$ invariant manifold
tangent to the invariant subspace spanned by the
eigenvectors of $\lambda_{i_{1}} \dots \lambda_{i_{r}}$.
(We are not assuming $\lambda_{i_{1}} \dots
\lambda_{i_{r}}$ different.)
\PROOF
Since in a sufficiently small neighborhood we would have
that the manifold would have to be a graph, it suffices
to show there is no $C^r$ solution of $\Tau w=w$ (as
in \clm(I.1)).
If this $w$ had a Taylor expansion of order $r$, we could
match powers. Substituting the definitions we see this is
impossible.
The construction can be easily modified to produce a
similar counterexample when the matrix is not
diagonalizable. So, the non-resonance assumptions of our
theorem are sharp.
\SECTION Partial linearizations and pseudo-stable manifolds.
The following result is proved in \cite{BLW}.
(Even if the statement of Theorem 1.1
is only for $\real^n$, the remarks along the proof
make it clear that the result is true also for a general Banach space
which admits smooth cut-off functions.
We recall that a cut-off function is a function that takes the value one
on a ball and the value zero outside a bigger ball. For finite
dimensional spaces, the existence of smooth cut-off functions
can be proved very easily. On the other hand, for infinite dimensional
Banach spaces, it is a non-trivial assumption on the space.
For example, the space of continuous functions on the
interval does not admit a $C^1$ cut-off function \cite{K}.
More generally,
a separable Banach space admits a Frechet differentiable cut-off function
if and only if its dual is separable \cite{LeW}.
Any Hilbert space admits smooth cut-off functions.
\CLAIM {Theorem}(blw)
Let $f,g$ be $C^r$ $r \in \natural$ diffeomorphisms of a Banach space,
admitting smooth cut-off functions.
$f(0)=g(0)=0$, and let
$A$ and $B$ be numbers computed explicitly in the proof
which depend only on the spectrum of $Df(0)$.
\vskip1pt
Assume
\smallskip
\item{i)} $D^i f(0) = D^i g(0) $ \quad $i=0,\ldots,k1\ ,\qquad
|\lambda^p|\ |\mu|^q <1$$
To avoid unnecessary complications, we will also assume that the map
$f$ is $C^\infty$.
We will furthermore assume that the unstable manifold of $Q$ is contained
in the stable manifold of $P$.
Such situations arise after a Hopf bifurcation.
In this case,
we have
$$\eqalign{
&\lambda^p = 1-c_1\sqrt{\ep} +o(\ep)\ ;\quad
\lambda^q = 1-c_2\sqrt{\ep} +o(\ep)\cr
&|\mu^{p,q} -1| = o(\ep^{N/2})\cr}$$
where $\ep$ is the bifurcation parameter
$N$ is the order of the smallest resonance in the bifurcation
(it is at least 4) and $c_1$, $c_2$ are positive constants.
Hence, the assumption $|\lambda^p|\,|\mu^q|<1$ is certainly true for
small values of the perturbation parameter.
The standard theory of normally hyperbolic manifolds \cite{HPS}, \cite{Fe},
\cite{Wi} shows that this situation is structurally stable. In particular,
the invariant circles continue to exist.
Moreover, the invariant circle described before is
$C^{r_0-\delta}\ \forall\ \delta>0$, $r_0 = \log |\lambda^p|/\log |\mu^p|$.
(This can be seen by observing that, by taking sufficiently high iterates
we can get the exponent of normal contraction to be not bigger than
$\lambda^p|+\delta$ outside of a neighborhood of the point $Q$.
Also the exponent of contraction outside this neighborhood is not smaller
than $\log |\mu^p|-\delta$.
We can arrange that the circle minus this neighborhood gets mapped into
itself.
The standard theory gives us $C^{r_0-\delta}$ in this set.
However, the theory of unstable manifolds shows the circle near the orbit
$Q$ --- it is a piece of the unstable manifold --- is $C^\infty$.
We want to show how the theory of non-resonant manifolds developed in this
paper allows us to obtain computable conditions that show that this
manifold is not $C^{r_0+\delta}$.
Since our goal is to exclude that the circle produced by Hopf bifurcation
is $C^{r_0+\ep}$ in concrete cases, we proceed by contradiction.
We assume that it is $C^{r_0}$ and then derive numerical facts that
should hold.
Given a concrete map, these numerical facts can be refuted by a finite
precision calculation.
We can also show that they hold only in sets of infinite codimension.
We will distinguish two cases
$\ln |\lambda^p|/\ln |\mu^p| \notin \natural$ and
$\ln |\lambda^p|/\ln |\mu^p|\in \natural$.
The first case is the generic one.
We will refer to it as the non-resonant case.
The linear map $Df^n (p_1)$ has exactly two invariant subspaces.
One of them associated to $\lambda^p$ and another one to $\mu^p$.
In the non-resonant case, there are two one dimensional invariant
subspace for $Df^n$.
Each of them has a one dimensional non-resonant invariant manifold.
Of course the manifold associated to $\lambda^p$ is the well known strong
stable manifold.
We also note that in the circles that appear after the Hopf bifurcation,
the unstable manifold of $Q$ does not agree with the strong stable
manifold of $P$.
As we argued in \clm(I.1) these are the only $C^{L}$ invariant
manifolds in a neighborhood of $P$.
To show that the circle is not $C^{L}$ we just need to show
that these two non-resonant manifolds of $P$ do not agree with the
unstable manifold for $Q$.
For the point of view of numerical applications we just point out that
the non-resonant manifolds and the unstable ones are numerically computable
with high accuracy with a finite calculation and it is possible to show
that they do not agree with a finite calculation.
>From the theoretical point of view we compute the derivatives with
respect to parameters and it is easy to show that,
these derivatives depend on the values of the perturbation --- and its
derivatives --- evaluated at different points
(roughly the perturbation of the unstable manifold of $Q$ depends on the values
of the perturbations near $Q$ and that of the non-resonant manifold
at $P$ on the values of the perturbation at $P$)
hence, their agreement is an infinite codimension phenomenon.
Even if the manifolds happen to agree at one point a generic perturbation
would destroy the agreement.
The resonant case can be handled similarly,
but one needs to distinguish different possibilities.
The strong stable manifold of $P$ still exists and is smooth but it is
possible to check that it does not agree with the unstable manifold of $Q$.
Unless there is certain
combination of derivatives that vanish, there is no $C^{r_0}$ intermediate
invariant manifold and, of course we are done showing that the circle
is not $C^{r_0+\delta}$.
If this combination of derivatives vanishes, then,
as we showed, there is an intermediate
non-resonant manifold but, the same arguments as in the non-resonant
case may be used to
exclude that it agrees with the unstable manifold of $Q$ and, as before,
this leads to the circle not being $C^{r_0+\delta}$.
\SUBSECTION Non-resonant invariant foliations.
As a third application we discuss the possibility of extending these
results to invariant foliations.
Given a diffeomorphism $f$ on a compact manifold it an standard construction
\cite{HP},
\cite{HPS}, \cite{Sh} to consider the operator $\tilde f$ acting on
$C^0$ vector fields by
$$(\tilde f v) (x) = \exp_x^{-1} f(\exp_{f^{-1}(x)} (v(f^{-1}(x)))$$
where $\exp_x v(x)$ denotes the differential geometry exponential map
obtained by flowing a unit of time the geodesic with initial conditions
$x$, $v(x)$.
(It is useful to think of $\exp_x v(x)$ as $x+v(x)$.
Indeed this is what it amounts to in Euclidean space.)
It is not difficult to check that if $\|v\|_{C^0}$ is sufficiently small
$\tilde f$ is well defined.
Moreover,
$$[D\tilde f(0)] =f^*$$
where $f^*$ is the push forward
$$[f^*v] (x) = Df(f^{-1}x) v(f^{-1}(x))$$
The spectrum of $f^*$ in the complexification of $C^0$ vector fields (to
study spectral properties, it is much better to have a complex space)
has been intensively studied since \cite{Ma}.
In that paper it is shown that if $f$ is an Anosov system the spectrum
consists of annuli.
Moreover, quite remarkably, the spectral projections associated to each
of the annuli correspond to projections over a subbundle.
As a corollary of this last property, we obtain that the number of annuli
is at most the dimension of the space.
Hence, if a subset of these annuli satisfies the non-resonance conditions
of \clm(I.1) we can obtain non-resonant invariant manifolds for $\tilde f$.
For the case of an annulus, this non-resonant invariant manifold is
constructed directly in \cite{Pe}.
In the case where the non-resonant set is the whole stable component
(resp. the annuli within a ball of radius $\rho<1$) this is the way
that stable (resp. $\rho$ stable) foliations are constructed in
\cite{HP}, \cite{HPS}.
Unfortunately, the geometric interpretation of the non-resonant invariant
manifolds for $\tilde f$ is more complicated than that of the stable
(or $\rho$ stable ones).
The non-resonant invariant manifolds for $\tilde f$
correspond to ``{\sl invariant leaf fields}''.
That is, maps that to each point $x$ associate a leaf $L_x$
-- a diffeomorphic image of a disk -- in such a way that
$$\eqalign{
f(L_x) &\subset L_f(x)\cr
T_x L_x & = E_x^s\cr}
\EQ(leafs)$$
(The proof consists in walking through the proof of the non-resonant manifold
checking that all the steps are bundle maps.
Fuller details can be found in \cite{Pe}
for the one annulus case or in \cite{JLP} or in
lecture notes by the author. In any case, these details
can be more or less found in \cite{Sh})
The regularity and uniqueness
statements in \clm(I.1) carry through to show that the
leaf fields are characterized uniquely by \equ(leafs) and having $C^L$ leaves.
The result in \clm(I.1) implies
that the leaves $L_x$ are $C^{r-1 + {\rm Lip}}$ if the map is $C^r$.
(It can be improved to $C^r$.)
In the case of the stable manifold, ($\rho$-stable manifold) it is
possible to show that these leafs are a foliation because
$$y \in\bigcup_{n\ge0} f^{-n} (L_x) \Leftrightarrow d(f^n (x),f^n(y))
\le K_{x,y} \lambda^n$$
(resp. $\le K_{x,y}\rho^n$) and this is clearly an equivalence relation.
Unfortunately, this argument does not carry through for general
non-resonant manifolds and indeed the conclusions are false.
In \cite{JLP} there are examples where these leaf fields fail to be a
foliation in a very strong sense. For a generic
map $f$, any neighborhood contains intersections of
leaves.
There is another twist to the discussion of
invariant foliations corresponding to these invariant sets.
There is another
construction of invariant manifolds that correspond to spectral subsets.
For example \cite{Ir} (A more modern version with several
extensions is \cite{LW})
describes method to construct invariant manifolds
(usually called pseudo-stable) associated to spectral sets of the form
$\{z\in\complex \mid |z| \le\rho\}$ with $\rho >1$.
By taking intersections of these manifolds with strong stable manifolds
of the inverse it is possible to obtain invariant manifolds associated
to spectral subsets of the form
$\{z\in\complex \mid \rho_- \le |z| \le \rho_+\}$.
We emphasize that the construction of Irwin manifolds does not
involve non-resonance conditions.
The somewhat surprising fact is that these Irwin manifolds are not the same
as those constructed in this paper even in the case where the manifolds
in these paper can be defined.
In \cite{LW} one can find examples where these Irwin manifold are not smooth
and, therefore do not coincide with the smooth non-resonant manifolds constructed
in this paper.
The Irwin construction can be lifted to maps on manifolds.
This seems to require extra properties of the manifold such as having
$\real^n$ as universal cover
and that the map is globally close to
linear, in contrast with the situation described here,
which can be carried out in any manifold and for any map that
whose Mather spectrum satisfies the non-resonance conditions. An example
when all these conditions occur is perturbations of linear automorphism of the tori.
When the Irwin construction
can be carried out for maps on a manifold, it leads to foliations
(It turns out that $y \in W^{s,{\rm Irwin}}_x \iff d(f^n(x) - f^n(y)) \le
C_{x,y} \rho_+^n$, $n \ge 0$, with $f$ in the universal cover
which is an equivalence relation,
so that it indeed is a foliation.) but the leaves may
be significantly less smooth than the map -- the degree of
differentiability is related to the gaps of the Mather
spectrum --.
There are also uniqueness statements for these Irwin foliations based on
asymptotic behavior or, in the fact that they are foliations.
We refer to \cite{LW} for proofs of the results on Irwin pseudostable
manifolds.
In summary, the construction of non-resonant invariant manifolds in
this paper can be lifted to all manifolds, it produces leaf fields of
smooth leaves that, in a generic case, fail to be foliations.
Moreover there is another natural construction (Irwin's) that only works
on certain manifolds and for certain maps but which produces foliations with leaves that are
not smooth.
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