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\TITLE ON IRWIN'S PROOF OF THE PSEUDOSTABLE MANIFOLD THEOREM
\footnote{${}^{\rm 1}$}{{\rm This preprint is available from the
math-physics electronic preprints archive. Send e-mail to
{\tt mp\_arc@math.utexas.edu} for instructions}}
\AUTHOR Rafael de la Llave
\footnote{${}^2$}{Supported in part by National Science
Foundation Grants}
\footnote{${}^3$}{ e-mail address: {\tt llave@math.utexas.edu}}
\FROM Department of Mathematics
Univ. of Texas at Austin
Austin TX 78712
\AUTHOR C. Eugene Wayne
${}^2$
\footnote{${}^4$}{ e-mail address: {\tt wayne@math.psu.edu}}
\FROM Department of Mathematics
Pennsylvania State University
University Park, PA 16802
\ENDTITLE
\ABSTRACT
We simplify and extend Irwin's proof of the pseudostable manifold theorem.
\SECTION Introduction
In [Ir1], Irwin introduced a very clever method to prove the
stable manifold theorem near hyperbolic points.
The proof was then, streamlined in [W].
Compared to previous proofs of the stable manifold theorem, the proof
was technically quite simple since it only required the
use of the implicit function theorem in Banach spaces.
The Banach spaces considered had a very natural interpretation
as spaces whose elements were orbits. This made the method
very natural in the study of partially hyperbolic systems
(Pesin theory) for
which individual orbits are hyperbolic but there is
little global hyperbolicity in the system. (See e.g. [FHY].)
Later, in [Ir2] Irwin proposed a new method to prove the
pseudo-stable manifold theorem that also used spaces of sequences.
(A corollary of the pseudo-stable manifold theorem
is the center or the center stable manifold theorem.)
Unfortunately, the resulting proof was somewhat complicated
because it required the use of specialized
implicit function theorems that only worked in Banach
spaces of sequences.
The proof in [W] does not use
specialized implicit function theorems
but only claims Lipschitz regularity for the manifold.
Given a $C^r$ function with $r$ an integer, the proof in [Ir2]
can only conclude that the invariant manifold is
$C^{r-1}$ and that the $r-1^{\rm st}$ derivative is bounded.
The goal of this paper is to present a proof of the
pseudo-stable manifold theorem that
is based on the consideration of
spaces of orbits but nevertheless only uses the
contraction mapping theorem in Banach spaces.
The proof that we present here also produces sharp regularity results
with respect to the regularity of the mapping. In particular, it
can deal with the case that the map is $C^r$, $r\in \integer$
We also present some explicit examples that
show that the regularity claimed in the theorem
is sharp. In [Ir2], Irwin mentions the existence of such
examples and attributes them to Van Strien.
We believe that Irwin's method has several advantages.
Let us just mention that the idea of considering spaces of
sequences with the right type of long term behavior
and on which the requirement of being an orbit is imposed as
an equation (which can be solved using
{\it e.g.}\ fixed point theorems or variational methods) is
finding an increasing number of applications.
Also, since the spaces of sequences that
enter into the proof are characterized by their long term behavior,
they are invariant under changes of variables that respect rates of growth.
For example, for maps of the torus, the spaces of sequences
in the universal cover -- Euclidean space -- with growth
slower than a prescribed rate are invariant under
homeomorphisms of the torus. It follows immediately that the
pseudostable manifolds constructed by Irwin's method are invariant
under topological changes of variables in the torus.
The argument can also be adapted to prove
pseudo-stable invariant foliation theorems for Anosov systems on tori.
Some technical advantages are that it makes it
somewhat easier to deal with functions with H\"older regularity
than the graph transform method and that one can discuss directly
dependence on parameters.
The usual graph transform method uses operators which are
based on composition operators, which are badly behaved,
as operators between spaces of H\"older functions. Even
if it is, by now well known how to cope with these problems,
Irwin's method does it in a straightforward way.
For the pseudostable manifold theorem, dependence on parameters
can be proved easily by replacing the system with another
in which the parameter is included
and assigned trivial dynamics.
(See e.g [RT], [La].) Nevertheless, this trick does not work
with the stable manifold. Since Irwin's method only
uses the soft implicit function theorem, dependence on parameters is
automatic and it is possible to compute the derivatives rather explicitly.
Let us finally remark that the construction we present here is not
the only one which is possible. Even if the theorems we will
present include some uniqueness conclusions given
rates of growth of the orbits,
there are other constructions that also give rise to
invariant manifolds that could be called pseudostable
since they are also tangent to the pseudostable subspace $\SSS$.
In general they will not coincide with teh manifolds
we construct. At the end of the paper
we present a discussion of the possible pseudostable manifolds
that one can define and examples that illustrate that they
often differ. We will also discuss their regularity properties.
\SECTION Notation and statement of results.
{\bf Notation.}
Let $\XX$
be a Banach space . If $a$ is a real number bigger than $1$,
we define the space
$\SS^a$
as the space of sequences
$\{ \rho_n\}_{n=0}^{\infty}$
in
$\XX$
such that $ || \rho ||_a \equiv
\sup_n || a^{-n} \rho_n ||_{\XX} < \infty $. Equipped
with the $||\ ||_a$ norm,
$\SS^a$
is a Banach space. We will also observe that, when $ 0 < a \le a'$,
$\SS^a \subset \SS^{a'}$. We will denote the natural
immersion of $\SS^a$ into
$\SS^{a^\prime}$ by
$\imath_{a,{a'}}$
If $\XX$ and ${\YY}$
are Banach spaces and $r$ is not an integer,
we will define $C^r( \XX , \YY)$ -- or just $C^r$
if the context makes it clear which spaces we are referring to --
as the space of functions which can be differentiated $[r]$
times at every point and for which the
norm:
$$
\eqalign{
&|| \phi||_{C^r} \equiv
\sup_{x \in \XX} ||\phi(x)|| +
\dots +
\sup_{x \in \XX} ||D^{[r]} \phi(x)|| \cr
&+ \sup_{x,\Delta \in \XX} ||\phi(x+\Delta)-\phi(x)- D\phi(x)\Delta -
\dots - D^{[r]} \phi(x) \Delta^{\otimes [r]}|| / || \Delta ||^r
}
$$
is finite.
Notice that this is a extremely strong norm.
It includes a very strong control of the derivatives at infinity and
also estimates on the behavior of the Taylor remainder.
This norm makes $C^r(\XX , \YY)$ a Banach space.
The definition can be changed in an obvious way to include
$C^{r+\hbox{\sevenrm Lipschitz}}$ when $r$ is an integer
and we also obtain a Banach space with the obvious norms.
For the case that $r$ is an integer, we will require that
$$\sup_x \| \phi (x+\Delta) - \phi (x) - D\phi (x)\Delta -\cdots -
D^r \phi (x)\Delta^{\otimes r}\| \le \eta_\phi (\|\Delta\|) \|\Delta\|^r$$
where $\eta_\phi : \real^+\to \real^+$ is decreasing $\eta_\phi (0)=0$,
$\sup_{x \in \real^+} \eta_\phi(x) < \infty$.
Unfortunately, there is no easy way to make this space into a Banach
space. The obvious choice of norm does not afford any control on
the uniformity of the error. In a non-compact space, it is possible to
have uniform limits of uniformly continuous functions which are not
uniformly continuous.
The proof of the invariant
manifold will also work in this case, but it will require special
considerations so we will relegate it to an special section.
Notice that in this paper when we refer to $C^r$, $r$, not an integer,
we include the case when $r= k+$ Lipschitz, $k$ an integer. If we claim
that a theorem is true for $C^r$, $r$ not an integer,
and $r$ in a certain range that includes
$k+1$, then it is valid for $C^{k+\hbox{\sevenrm Lipschitz}}$.
The main result of this paper is:
\CLAIM Theorem(Irwin)
Let $f$ be a $C^r$ mapping from
$\XX$ to itself and $\XX = \SSS \oplus \UU$
a direct sum decomposition of
$\XX$
into two closed subspaces
which are invariant under $Df(0)$.
We will denote the corresponding projectors by $\Pi^S$ , $\Pi^U$
and assume -- without any loss of generality --
that the norm satisfies:
$$ || x || = \sup\{ || \Pi^{\bf S} x ||, || \Pi^{\bf U} x ||\} $$
Assume that, for some $a > 1$:
\item{$(i)$} $f(0) = 0 $
\item{$(ii)$} $|| Df(0)|_{\SSS} || < a $
\item{$(iii)$} $|| Df^{-1}(0)|_{\UU} || < a^{-r} $
\item{$(iv)$} $|| \tilde f ||_{C^r} $ is sufficiently small, where
$\tilde{f}(x) = f(x) - Df(0) x$~~,
\item{}Then, the set $W^c = \{x \in \XX | \sup_{n \ge 0 } | f^n(x)|a^{-n} < \infty \}$
is a $C^r$ manifold.
Moreover $W^c$ is tangent to $\SSS$.
\REMARK
Notice that, by the characterization of the set $W^c$, it is
clear that it is invariant under $f$.
We will see later that, in general, the manifold produced by this theorem
is not the only invariant smooth manifold tangent to $\SSS$.
\REMARK
The theorem above implies an analogue statement for
differential equations defined by $C^r$ vector fields
just by taking the time-1 map.
We observe, however that there are partial differential equations
-- e.g. semilinear parabolic partial differential equations -- that define an
smooth time one map even if the vector field is not even continuous.
\REMARK
Notice that the conclusions of the theorem are
independent of the choice of norms we pick in
$\XX$ even though the hypotheses
are not since they include the conditions
that certain operators are contractions.
It is an standard result in functional analysis
that, provided that $\sigma( Df(0) )$,
the spectrum of $Df(0)$,
satisfies:
\item{$(ii')$} $\sigma( Df(0)|_\SSS ) \subset
\{ z \in \complex \big| | z | < a \} $
\item{$(iii')$} $\sigma( Df(0)|_\UU ) \subset
\{ z \in \complex \big| | z | > a^r \} $
\vskip1pt
\noindent where $\sigma$ denotes the spectrum,
then, we can choose a norm $||| \qquad ||||$ in $\XX$
equivalent to the original one and such that
with respect to this new norm the conditions
$(ii)$ and $(iii)$ are satisfied.
The remaining condition $(iv)$ will depend on the choice of norm we made.
It is possible to choose
the norm $ ||| \qquad |||$ in such a way that
it is also the supremum of the norms of the projections.
\REMARK
For Banach spaces that admit
smooth functions with bounded
support and identically
one in a neighborhood of the origin,
-- usually termed bump functions --
one can obtain a version of the
pseudostable manifold theorem that
does not involve condition $iv)$
even if it does not recover growth conditions.
In effect,
we can consider the
map ${\hat f} $
defined by
${\hat f} (x) = \Phi(\eta x) f (x)
+ ( 1 - \Phi(\eta x) ) Df(0) x. $
where $\Phi$ is such a bump function.
We observe that, by choosing $\eta$
large enough, $iv)$ will be satisfied.
Moreover, since in a neighborhood $U$ of the
origin ${\hat f } = f $
the manifold $W^c $
obtained applying \clm(Irwin) to
${\hat f}$ will be invariant
under $f$ in a small neighborhood of
the origin. It is, then possible to
extend it in such a way that it is invariant under $f$.
Of course, such manifolds are also tangent to $\SSS$ at the origin.
Nevertheless, the characterizations of the points by the growth
of the orbits holds only with respect to $\hat f$, not with respect
to $f$.
Unfortunately, there are examples that
show that the family we obtain may depend on
the choice of $\Phi$. (See the examples at the end of this paper.)
We also remark that there are examples ( [BF], [Dev] ) of infinite dimensional Banach
spaces in which there are no smooth functions with compact support
even if the concept of smoothness is considerably weaker than that
we have considered here. On the other hand,
there are many Banach spaces for which the norm,
when restricted to a ball not containing the origin
is smooth in the sense considered here -- {\it e.g.} Hilbert spaces--
For these spaces, cut off functions exist.
\REMARK
Notice that if a function
has uniformly continuous derivatives on a ball
around the origin, cutting off
as above will produce a uniformly differentiable function.
Notice that in locally compact Banach spaces
-- this is equivalent to finite dimensional --,
it is automatic that all continuously differentiable
functions are uniformly differentiable on balls.
\REMARK
Notice that the growth condition imposed on the
orbits is not local since the
behavior of the map outside a very large ball
could affect the rate of growth of iterates.
On the other hand we observe that the
growth condition is invariant under changes of
variables that are either globally Lipschitz
or ${\rm Id} + L^\infty$.
The later situation arises when considering
Anosov systems of the torus -- or
of any manifold compact manifold whose universal cover is
$R^d$.
The conjugating homeomorphisms given by structural stability
are in ${\rm Id} + L^\infty$ of the lift.
Hence we have the
following corollary which
has applications to
rigidity theorems.
\CLAIM Corollary (interchanges)
Let $f$, $g$ be Anosov diffeomorphisms of
$\torus^d$ sufficiently close to
a linear one.
Let $h$ be a homeomorphism
sufficiently $C^0$ close to the identity such that
$f\circ h = h \circ g$.
Assume that $g(p) = p$ and
that the derivatives of
$g$, $f$ at $p$, $h(p)$ respectively satisfy the
hypothesis of \clm(Irwin).
Let $W^{c,f}$, $W^{c,g}$ be the manifolds obtained
applying \clm(Irwin) to the lifts of $f$, $g$ to the universal
cover and projecting the invariant manifolds in the conclusions to the
manifold. Then $h(W^{c,g} ) = W^{c,f}$.
\SECTION Proof of \clm(Irwin) for $r \notin \natural$
Following Irwin and Wells , we consider the mapping
$\chi : \SSS \times \SS^a \mapsto \SS^b$, $b\ge a$
defined by:
$$
\chi( x , \rho )_n =
\cases{
\Pi^S f( \rho_{n-1}) + \left(\Pi^U Df(0) \Pi^U\right)^{-1}
\left[ \Pi^U \rho_{n+1} - \Pi^U {\tilde f}(\rho_n) \right] &
if $n > 0$; \cr
x + \left(\Pi^U Df(0) \Pi^U \right)^{-1}
\left[ \Pi^U \rho_{1} - \Pi^U {\tilde f}(\rho_0) \right]
&if $n = 0$. \cr
}
\EQ(chi)
$$
The point of this definition is that for
any $b\geq a$
the condition
$ f(\rho^*_n) = \rho_{n+1}^*$ when $n \geq 0$
and $\Pi^S \rho_0^* = x$,
can be expressed
as the fixed point equation:
$$
\chi (x , \rho^* )_n = \rho^*~~.
\EQ(fixedpoint)
$$
The invariant manifold will be the
range of the mapping $x \mapsto \rho_0^*$ , where $\rho^* \in \SS^a$
solves the equation above, hence it will be an orbit that
does not grow too fast under iteration.
Notice that since $\SS^b \subset \SS^{b'}$ in a natural way whenever
$b' \ge b$, we can consider this equation as an equation
in any $\SS^b$, $b\ge a$, space provided that we prove uniqueness
of solutions in $\SS^b$.
What we want to do is to apply the implicit function theorem
to \equ(fixedpoint) for a conveniently chosen
$b$ (which will turn out to be, roughly, $a^r$)
The reason why we do not want to take $b= a$
is that $\chi$ will not be differentiable in that case.
The conditions $(ii)$ and $(iii)$ in \clm(Irwin) will be used to show
that some auxiliary mappings in those spaces are contractions.
The last claim of the theorem will be proved by computing explicitly
the derivative with respect to $x$ of
$x \mapsto \rho^*(x)$ when $x = 0$.
The following propositions will make all this more precise.
\CLAIM Proposition(differentiable)
Let $b>a$.
If $k$ is any noninteger number
such $1 0$. By substituting the definition of $\chi$ and applying the
Taylor formula with remainder for $f$ we get:
$$
\eqalign{
&\chi( x, \rho + \gamma )_n =
\chi( x , \rho )_n + \cr
&\Pi^S Df( \rho_{n-1})\gamma_{n-1} +
\left(\Pi^U Df(0) \Pi^U\right)^{-1}
\left[ \Pi^U \gamma_{n+1} - \Pi^U D{\tilde f}(\rho_n)\gamma_n \right] + \cr
&\Pi^S D^2f( \rho_{n-1})\gamma_{n-1}^{\otimes 2}
-\left(\Pi^U Df(0) \Pi^U\right)^{-1}
\Pi^U D^2{\tilde f}(\rho_n)\gamma_n^{\otimes 2} + \cr
& \dots \cr
&\Pi^S D^{[k]}f( \rho_{n-1})\gamma_{n-1}^{\otimes [k]}
-\left(\Pi^U Df(0) \Pi^U\right)^{-1}
\Pi^U D^{[k]}{\tilde f}(\rho_n)\gamma_n^{\otimes [k]} + R_n ~~.
}
\EQ(derivatives)
$$
where $[k]$ denotes the integer part of $[k]$
By the uniformity of the
Taylor expansion of $f$ that was built
into the definition of $C^r$,
when $k$ is not an integer,
the norm of the remainder $R$ can be bounded
by:
$
(|| \gamma_n||_\XX )^k +
(|| \gamma_{n-1}||_\XX )^k +
(|| \gamma_{n+1}||_\XX )^k
$.
Since
$
|| \gamma_n||_{\XX} \leq ||\gamma||_b b^n
$
we have
$|| R_n||_{\XX} \leq
K ||\gamma ||_b^k b^{nk}$.
where $K$ depends only on $b$,$k$ and $||f||_{C^k}$
In other words, if we set the derivatives of
$\chi$ of
order up to $[k]$ to the expressions obtained by
termwise differentiation, $||R||_{b^k} \leq K ||\gamma ||_b^k $,
which establishes the claim in the proposition.
\QED
\CLAIM Proposition(easy)
$\chi$ is $C^\infty$
with respect to $x$.
\PROOF
$x$ only enters in one of the coefficients
and it appears linearly.
\QED
\CLAIM Proposition (chicontracts)
Given a strong enough smallness assumption in
the hypothesis of \clm(Irwin),
$\chi( x, .)$
maps $\SS^b$ into $\SS^b$
and is a contraction
for all $b$, $a \le b \le a^r$,
all $x \in \SSS$.
The contraction factor is independent of
$x$,$b$.
\PROOF
Using the definition of the
$n^{\rm th}$ component of $\chi$
we have:
$$
\eqalign{
|\Pi^S (\chi(x , \rho )_n &- \chi(x,\gamma)_n)| \le \cr
&||Df{(0)}|| | \rho_{n-1}-\gamma_{n-1}|+ \sup_x ||D{\tilde f}(x)||
~ |\rho_{n-1} - \gamma_{n-1}|
}
$$
if $n > 0$ and, obviously, equal to $0$ if $n = 0$.
Using the definition of
$||\quad||_{\SS^b}$,
we can bound $|\rho_n - \gamma_n| \le || \gamma - \rho||_{\SS^b} b^n$.
Therefore:
$$
| \Pi^S( \chi(x,\rho)_n - \chi(x,\gamma)_n )| \le
\left( ||Df(0)|_\SSS|| + \epsilon' \right)b^{n-1} || \rho - \gamma||_{\SS^b}
$$
where $\epsilon' = \sup_x || D\tilde f||$.
Analogously, we can bound:
$$
\eqalign{
|\Pi^U (\chi(x , \rho )_n &-\chi(x,\gamma)_n)| \le \cr
&||Df{(0)}|_\UU^{-1}||
\left( \Pi^U (\rho_{n+1} - \gamma_{n+1}) \right) | + \sup_x ||D\tildef(x)||
|\rho_n - \gamma_n|\quad {\rm if}~ n > 0 \cr
|\Pi^U (\chi(x , \rho )_n &-\chi(x,\gamma)_n)| \le \cr
&|| Df(0)|_\UU^{-1}||
\left[ |\Pi^U (\rho_{1} - \gamma_{1}) | -
|| \Pi^U || \sup_x ||D{\tilde f}(x)||\quad |\rho_0 - \gamma_0 | \right]
{\rm if} ~n = 0 \cr
}
$$
Using again the definition of
$||\quad||_{\SS^b}$, we can bound
$$
|\Pi^U( \chi(x,\rho)_n - \chi(x,\gamma)_n)| \le
(|| Df(0)|_\UU^{-1}|| + \epsilon'') b^{n+1}|| \rho -\gamma||_{\SS^b}
$$
where $\epsilon'' = b^{-1}|| Df(0)|_\UU^{-1}|| \sup_x ||D\tilde f (x)||$.
Notice that both $\epsilon'$,$\epsilon''$ can be made arbitrarily small
by assuming that the smallness assumptions in
\clm(Irwin) are strong enough.
Using the fact that the norm
of a vector in $\XX$ is the supremum of the
norm of the projections, we obtain that:
$$ |\chi(x,\rho)_n - \chi(x,\gamma)_n| \le b^n \max\left( b(||Df(0)|_\SSS|| +\epsilon'),
b^{-1}(||Df(0)|_\UU^{-1}|| + \epsilon'')\right) || \rho - \gamma ||_{\SS^b}$$
Using the definition of $||\quad||_{\SS^b}$,
this means that the Lipschitz constant of
$\chi(x,.)$ is less than \hfill \break
$\max\left( b(||Df(0)|_\SSS|| +\epsilon'),
b^{-1}(||Df(0)|_\UU^{-1}|| + \epsilon'')\right)$
which can be made strictly less than one, uniformly in $x$ and $b$
under the hypothesis of the lemma.
A simpler version of these estimates shows that
$\chi(x,.)$ maps $\SS^b$ onto itself. (It suffices to observe that
$\chi(x,0)$ is in $\SS^b$ and estimate as here $\chi(x,\rho)-\chi(x,0)$.)
\QED
\REMARK
Notice that $|D^k\tilde f(x) \gamma^{\otimes k}| \le \| \tilde f\|_{C^k}
|\gamma|^k$. A calculation similar to the one we performed to show that
$\chi(x,.) $ was a contraction in $S^b$ $a\le b\le a^r$ shows that $D_2^k
\chi (x,\rho)$ is a bounded operator from $S^b$ to $S^{b^k}$ $\alpha \le b\le
\beta$ where $\alpha$ and $\beta$ can be made arbitrarily small and
arbitrarily large respectively by assuming that $\| D^{[k]}{\tilde f}\|_{C^{k -[k]}}$ is
sufficiently small.
If $b$ is such that $a \le b \le b^r$,
Applying the contraction mapping theorem, whose hypothesis
are verified because of \clm(chicontracts), we obtain for every $x$ there
exist a $\rho^{*}(x)\in S^b$ which solves \equ(fixedpoint). Moreover such
$\rho^{*}(x)$ is the only solution in $S^b$.
Since $S^b\subset S^{b'}$ if $b'>b$, the existence part of the
conclusions is stronger the smaller the $b$ is, while the uniqueness
part of the conclusion is stronger the larger $b$ is.
Notice also that the elementary contraction mapping principle shows that
the map $x\to \rho^{*} (x)$ is $C^r$ $r\le$ Lipschitz if $\chi$ is. Hence
we have established \clm(Irwin) except for the regularities
greater than Lipschitz.
The following result completes the proof.
\CLAIM Lemma(differentiability)
The mapping $\rho^*: \SS \to S^{a^s}$ defined by requiring that
$\rho^*$ solves \equ(fixedpoint) is $C^s$ when $s\le r$, s not an integer.
\PROOF
We have already established the result for $s\le$ Lipschitz.
To prove the existence of higher derivatives we will derive
heuristically a formula for the derivatives and then, show that they
indeed satisfy the estimates that establish that they are derivatives.
If we take derivatives formally in \equ(fixedpoint), we obtain
$$D_1 \chi (x,\rho^*) + D_2\chi (x,\rho^*) D_x \rho^* = D_x \rho^*
\EQ(formalderivative)$$
Hence, we guess that the derivative of $\rho^*$ should be
$$\Delta \equiv - \bigl( D_2 \chi (x,\rho^*) -Id\bigr)^{-1}
D_1 \chi (x,\rho^*)
\EQ(derivativeguess)$$
Notice that since $D_2\chi : S^a\to S^a$ is a contraction, this is
an element of $S^a$.
To prove that \equ(derivativeguess) is indeed a derivative and that
$\rho^*$ is $C^{1+\beta}$ it suffices to show that:
$$\| \rho^* (x+y) - \rho^* (x) - \Delta y\|_{S^{a^{1+\beta}}}
\le C|y|^{1+\beta}
\EQ(goodbounds)$$
Since $\rho^* (x+y)$ is by definition the solution of $\chi (x+y,\rho^*)
= \rho^*$, and $\chi$ is a uniform contraction, \equ(goodbounds) follows from
$$
\| \chi (x+y, \rho^* (x + y) - \rho^* (x)
-\Delta y\|_{S^{a^{1+\beta}}} \le C|y|^{1+\beta} ~~,
$$
which can established by remembering that, by \clm(differentiable) we have:
$$\| \chi (x+y,\rho^*(x)+\Delta y) -
\chi (x,\rho^*(x)) -
D_1 \chi(x,\rho^* (x))y - D_2 \chi (x,\rho^*(x))\Delta y\|_{S^{a^{1+\beta}}}
\le C|y|^{1+\beta}$$
and using the definition of $\Delta$.
This establishes \clm(differentiability) for $s\le 1+$ Lipschitz.
Higher derivatives can be obtained by induction. If we have proved
the theorem for $s\le 1=1+$ Lipschitz we can obtain a guess for the
$i^{th}$ derivative by taking $i$ derivatives of \equ(fixedpoint).
A simple calculation shows that:
$$D_2 \chi (x,\rho^*) D_x^i\rho^* + R_{i-1} = D_x^i\rho^*
\EQ(formuli)$$
Where $R_{i-1}$ is a symmetric multilinear operator from $\SSS^{\otimes i}$
to $S^a$ whose expression involves tensor products of derivatives of $\rho$
up to order $i-1$.
Hence, it is natural to guess that the $i^{th}$ derivative will be
$$\Delta_i = - (D_2 \chi (x,\rho^*) - Id)^{-1} R_{i-1}
\EQ(deltai)$$
We now we interpret $D_2\chi (x,\rho^*)$ as a bounded operator
from $S^{a^i}$ to itself. By \clm(chicontracts) this operator is
a contraction and, hence $(D_2\chi (x,\rho^*) -Id)^{-1}$ exists as
a bounded operator from $S^{a^i}$ to itself.
As before, to show that this is indeed a bona fide derivative
it suffices to obtain estimates
$$\eqalign{
& \| \chi (x+y,\rho^*(x) + D\rho^* (x) y+\cdots + D^{i-1} \rho^* (x)
y^{\otimes i-1} +\Delta_i y^{\otimes i})\cr
&\qquad
- (\rho^* (x) +D\rho^* (x) y+\cdots + D^{i-1} \rho^* (x) y^{\otimes i-1}
+ \Delta_i y^{\otimes i})\|_{S^{a^{i+\beta}}}\cr
&\le C|y|^{i+\beta}\cr}
\EQ(toestimate)$$
We recall that by the construction of $R_{i-1}$ if $\chi$ is $C^{i+\beta}$
we have
$$\eqalign{
&\| \chi (x+y,\rho^*(x) +D\rho^* (x) y+\cdots + D^{i-1}\rho^* (x)
y^{\otimes i-1} +\Delta_i y^{\otimes i})\cr
&\qquad - [D_2 \chi (x,\rho^*) \Delta_i y^{\otimes i} + R_{i-1} y^{\otimes i} ]
\|_{S^{a^{1+\beta}}} \cr
&\le C|y|^{i+\beta}\cr}
\EQ(derivative)$$
If we substitute the expression \equ(deltai), for $\Delta_i$, into
\equ(derivative) we obtain
\equ(toestimate) and the theorem is established.
\QED
\REMARK
Notice that, in the formula for $\chi$,
$x$ enters only linearly, so it is very easy to compute
derivatives with respect to $x$.
The condition $(ii)$ of \clm(Irwin) (together with
smallness assumptions in $\tilde f$)
implies that ${\cal S}^a$ gets mapped into
itself.
This finishes the proof of \clm(Irwin) except
for the claim of the
manifold being tangent to the space $\SSS$.
Substituting the explicit formula for the derivative of the map with respect to $x$
we find that $D_x\rho^*(0)= 0$.
This finishes the proof of \clm(Irwin).
\QED
\REMARK
Out of this method of proof
it is very easy to conclude smooth dependence on parameters for
Irwin's manifolds.
If we consider that $f_\lambda$ is an smooth family
of $C^r$ maps, the
same arguments that we have used before to check differentiability
of $\chi$
with respect to $x$, can be used to establish differentiability
of $\chi_\lambda$ with repect to $\lambda$.
\SECTION Proof of \clm(Irwin) for $r \in \natural$
When $r \in \natural$, it is not true that $\chi \in C^r$, hence, the
previous argument cannot establish that $\rho^*: \SSS \mapsto \SS^{a^r}$
is $C^r$.
Nevertheless, we observe that the regularity of the
pseudostable manifold only requires regularity of the mapping
$\rho^*_0: \SSS \mapsto \XX$ obtained by taking the
zeroth component of the mapping $\rho^*$.
We will be able to establish this regularity by considering
topologies on the spaces of sequences weaker than
those induced by the $||\quad||_{\SS^a}$ norms
we considerd before. In particular, componentwise convergence will play a role.
We will assume in the rest of this section that the
hypotheses of \clm(Irwin) hold and that $r$ is an integer.
The following proposition is trivial.
\CLAIM Proposition (component)
For every $n \in \natural$, the map
$\chi_n:\SSS \times \SS^a \mapsto \XX$
that produces the $n^{\rm th}$ component of
$\chi(x,\rho)$ is $C^r$.
\PROOF
Just observe that each component depends only on a finite number of
components and that for a fixed number of components,
the $\SS^a$ norm is stronger than the norm in each of the components.
\QED
Notice that the derivative will depend only on a finite number of components.
An inmediate corollary of the previous result is:
\CLAIM Lemma (residual)
Given $N \in \natural $ and $\epsilon > 0$,
it is possible to find $\delta > 0$ such that
if $ 0 < ||\gamma||_{\SS^a} \le \delta$,
then,
$$
\sup_{n \le N}\left| \chi_n (x,\rho +\gamma) - D\chi_n(x,\rho)\gamma - \cdots -
D^r \chi_n(x,\rho)\gamma^{\otimes r} \right| \le \epsilon||\gamma||_{\SS^a}
\EQ(smallremainder)
$$
We also have:
$$
\sup_{n \le N}\left| \chi_n (x,\rho +\gamma) - D\chi_n(x,\rho)\gamma - \cdots -
D^r \chi_n(x,\rho)\gamma^{\otimes r} \right|{ 1 \over ||\gamma||_{\SS^a}} \le K < \infty
\EQ(uniformbounds)
$$
\PROOF
The last inequality is a consequence of the
hypothesis $f \in C^r$ with our definition of
$C^r$, which included the hypothesis that the
modulus of continuity $\eta_f$ was bounded.
Notice that applying the results of the previous section, we can
conclude that $\rho^*$ is $C^{r-1}$.
We also can obtain a guess for what the $r^{\rm th}$ derivative
should be in the same way that we did before, that is, taking
$r$ formal derivatives of
\equ(fixedpoint) and solving the resulting equation that the
$r^{\rm th}$ derivative should solve as in \equ(deltai).
To check that this procedure is well defined, notice that
the computation of $R_{i-1}$ in \equ(formuli) only requires the existence
of $r-1$ derivatives. The existence of the inverse in \equ(deltai)
can be guaranteed provided that $D\chi$ is a contraction from
$\SS^{a^r}$ to $\SS^{a^r}$. The later is the case if
we assume that
$||Df(0)|_\SSS|| + || D\tilde f|| < a$
$||Df(0)|_\UU^{-1} || + || D\tilde f|| < a^r$, which
we can assume by if we assume that
$||D \tilde f ||$ is small enough.
Unfortunately, this guess fails to be a derivative in the
sense we considered before because if
we substitute it as before in
\equ(fixedpoint), we do not obtain a residual which is small
in the $\SS^{a^r}$ sense. This is due to the fact that
$\chi$ is not $C^r$ as a mapping into $\SS^{a^r}$.
Nevertheless, using the fact that $\chi$ is differentiable
componentwise, we can obtain that the residual has an arbitrarily large number of
arbitrrily small components.
\CLAIM Lemma (residual)
Denote by $\rhoguess(x,y) =
\rho^*(x) + D\rho^*(x)y + \cdots \Delta_r y^{\otimes r}$,
where $\Delta_r$, is obtained as in \equ(deltai).
Then, it is possible to find $K < \infty$ such that
$$
\sup_n\left| \chi_n(x+y,\rhoguess(x,y)) - \rhoguess_n(x,y) \right| \le K |y|^r
\EQ(uniformguess)
$$
Moreover, given $\epsilon > 0$, $N \in \natural$, it
is possible to find $\delta > 0$ such that
$|y| \le \delta$ implies:
$$
\sup_{n \le N} \left| \chi_n(x+y,\rhoguess(x,y)) - \rhoguess_n(x,y) \right|
\le \epsilon |y|^r
\EQ(smallguess)
$$
\PROOF
We recall that $\Delta_r$ was chosen precisely in such a way that
if we expand in powers of $y$
$\chi(x+y, \rhoguess(x,y)) - \rhoguess(x,y) $, the
Taylor expansion up to order $r$ vanishes.
Even if it is impossible to estimate the remainder in the
sense of $\SS^{a^r}$, it is possible to estimate it component by
component. In each component, we obtain that the
remainder can be bounded by the
remainder of the Taylor expansion of $f$. Hence, using the uniformity of the
modulus of continuity of the derivative, we obtain \equ(uniformguess).
Moreover, \equ(smallguess) follows because it is a bound in the
Taylor remainder of a finite number of components.
\QED
We observe that in our situation,
\clm(chicontracts) still applies and, hence
we have
$\rho^*(x,y) = \lim_{i \to \infty} \chi^i(x+y, \rhoguess(x,y) )$
where we denote by $\chi^i$ the application of $\chi(x+y, .)$ $i$ times
and the limit is understood in the $\SS^{a^r}$ sense
--{\it i. e.} componentwise.
Hence, we can estimate
$$
\left| \rho^*_n(x+y) - \rhoguess_n(x,y)\right| \le
\sum_{i=0}^\infty |\chi^{i+1}_n (x+y, \rhoguess)
-\chi^{i}_n (x+y, \rhoguess) |
\EQ(estimatetodo)
$$
\CLAIM Lemma (Iterativeestimate)
Let $\alpha = \max( ||Df(0)|_\SSS,
|| Df(0)|_UU^{-1}||) + ||D\tilde f||_{C^0}$,
which, we will assume according to the hypothesis of \clm(Irwin)
is strictly less than $1$.
Let $\rho, \gamma \in \SS^{a^r}$ be such that
for some $ 0 < \epsilon < K$, $N \in \natural$ we have:
$$
\eqalign{
&\sup_{n \le N} |\rho_n - \gamma_n | \le \epsilon \cr
&\sup_{n } |\rho_n - \gamma_n | \le K \cr
}
$$
Then
$$
\eqalign{
&\sup_{n \le N-1} |\chi_n(x,\rho) - \chi_n(x, \gamma) |
\le \epsilon \alpha \cr
&\sup_{n } |\chi_n(x,\rho) - \chi_n(x, \gamma) | \le K \alpha \cr
}
$$
\PROOF
If $n > 0$, we estimate
$$
\eqalign{
&\left|\Pi^\SSS( \chi_n(x,\rho) - \chi_n(x, \gamma) )\right| \le \cr
&\left| \Pi^\SSS Df(0)\Pi^\SSS(\rho_{n-1} - \gamma_{n-1})
+\Pi^\SSS({\tilde f}( \rho_{n-1} ) -
{\tilde f}( \gamma_{n-1} ) ) \right| \cr
&\le \alpha | \rho_{n-1} - \gamma_{n-1}|
}
\EQ(stableest)
$$
And
$$
\eqalign{
&\left|\Pi^\UU( \chi_n(x,\rho) - \chi_n(x, \gamma) )\right| \le \cr
&|| Df(0)|_\UU^{-1} || |\Pi^\UU(\rho_{n+1} - \gamma_{n+1}) | +
|\Pi^\UU( {\tilde f}( \rho_{n} ) - {\tilde f}( \gamma_{n} ) )| \le \cr
& | Pi^\UU(\rho_{n+1} - \gamma_{n+1})| + || D {\tilde f}||_{C^0} |\rho_n - \gamma_n|
}
\EQ(unstableest)
$$
If $0< n \le N-1$ both \equ(stableest), \equ(unstableest) can be bounded by $\alpha \epsilon$.
If $n \ge N$ both terms can be estimated by $\alpha K$.
The case $n=0$ is easier and is left to the reader.
\QED
By applying repeatedly the
lemma we derive the following corollary.
\CLAIM Lemma (summedestimates)
Let $\alpha$, $\epsilon$, $N$, $\rho$, $\gamma$
be as in \clm(Iterativeestimate). Let $n \in \natural$
be such that $n < N$. Then
$$
\sum_{i=0}^\infty |\chi^i_n(x, \rho) - \chi^i_n(x,\gamma)| \le
{\epsilon \over 1 - \alpha} + { K \alpha^{N-n} \over 1 - \alpha}
$$
Now we can prove that $\rho^*_0(x)$ is $C^r$
and that $[\Delta_r]_0$ is a bona-fide derivative.
(Where we denote by $[\Delta_r]_0 y^{\otimes r} = ( \Delta_r y^{\otimes r} )_0$ .)
This amounts to showing that
given $\epsilon > 0 $ we can find
$\delta > 0$ such that if
$|y| \le \delta$, then
$$
|\rho^*_0(x+y) -\rho^*_0(x) -
D\rho^*_0(x)y -\cdots - [\Delta_r]_0 y^{\otimes r} |
\le \epsilon |y|^r
\EQ(finalestimate)
$$
( The fact that we can find a $K$ such that
$|\rho^*_0(x+y) -\rho^*_0(x) - D\rho^*_0(x)y -\cdots -
[\Delta_r]_0 y^{\otimes r} |
\le K |y|^r$ that we included in the definition of $C^r$
follows very easily by observing that $[\Delta_r]_0$ is uniformly bounded.)
To prove that we can satisfy \equ(finalestimate),
we observe that by \clm(residual), we can find
$K$
in such a way that
$ | \chi_n(x+y, \rhoguess(x,y)) - \rhoguess_n(x,y) | \le K|y|^r$.
Choose $N$ big enough that
$\alpha^N(1-\alpha)^{-1}K \le \epsilon/2$
and take the $\delta$ provided that \clm(residual)
that guarantees that for
$|y| \le \delta$ we have:
$$
\sup_{n\le N} |\chi_n(x+y,\rhoguess(x,y)) - \rhoguess_n(x,y) | \le |y|^r (1-\alpha)\epsilon/2
$$
Applying \clm(summedestimates) with $\rho = \rhoguess(x,y)$,
$\gamma = \chi(x+y,\rhoguess(x,y))$,
we obtain \equ(finalestimate) and, hence, the theorem is proved.
\QED
\SECTION Examples
The following examples shows that in general
the manifolds constructed are not more
differentiable than the claim of \clm(Irwin).
The two examples are quite instructive since they show that there are
different obstructions to differentiablility.
\CLAIM Example(notmorediff)
Consider the mapping $f$ of $\real^2 \mapsto \real^2$
defined by:
$$
(x,y) \mapsto( 2x , 3 y + \varphi(x))
$$
where $\varphi$ is a $C^\infty$ function with support
in the interval $[0.9,1]$.
If $\varphi$ is not identically $0$, for any $\epsilon > 0$, the
pseudo stable manifold obtained by taking $a = 2 + \epsilon$
and $r$ such that
$a^r = 3 -\epsilon$
in \clm(Irwin) is not $C^{{\log3 \over \log2 } + \epsilon}$.
\PROOF
In this case, we can construct the manifold
almost explicitly.
Since for sufficiently large $x$ , the mapping
$f$ agrees with the linear transformation,
if $x > 1$, the only point $p$ of the form $p = (x,y)$
such that $f^n(p)(3 - \epsilon)^{-n}$ remains bounded
is precisely $p = (x,0)$.
We can construct the
whole invariant manifold
by iterating backwards
this manifold that we have
so, we can see
it will be the graph of the
function:
$$
\sum_{n=0}^\infty 3^{-n} \varphi( 2^n x)
$$
\QED
Notice, however that the map $f$ in
\clm(notmorediff) has invariant manifolds tangent
to the $x$ axis which are $C^\infty$.
Such manifolds can be readily constructed by
declaring that $[-0.8, 0.8 ]$ is in the manifold and determining the
rest of the manifold in such a way that the invariance property holds.
It will be the graph of the function
$$
\sum_{n=0}^\infty 3^{n} \varphi( 2^{-n} x)
$$
However most of
the the points of this smooth invariant manifold
have orbits that grow asymptotically with the largest
eigenvalue.
Notice further that if we cut-off the function as indicated
in the remarks after \clm(Irwin) we could obtain just the
linear map $(x,y) \to (2x, 3y)$ for which the invariant
manifold is just the coordinate axis. The invariant manifold produced
for the original map would then be the one produced in the previous
remark, and not the one produced by direct application of the theorem.
Unfortunately, it is not true that all maps have
smooth pseudo-stable manifolds as the following example shows:
\CLAIM Example(nolocalsmooth)
The map $f:\real^2 \mapsto \real^2$
$f(x,y) = (2 x , 4 y +x^2)$
does not leave invariant any $C^2$ manifold tangent to the $x$ axis
in any neighborhood of the origin.
\PROOF
If such a manifold existed it would be possible to write it as
the graph of a mapping $w$ from the $x$ axis to the $y$ axis.
The invariance of the manifold is equivalent to
the map $w$ satisfying:
$w(2 x) = 4 w(x) + x^2$.
If the map $w$ were twice differentiable, we would have:
$ 4 w''(2x) = 4 w''(x) + 2$,
which evaluated at $x =0$ produces a contradiction.
\QED
Notice that the local obstruction
studied in \clm(nolocalsmooth)
would not have worked for the
$f$ in \clm(notmorediff).
We used crucially that $4 = 2^2$.
It turns out that it is possible to show that
given non--resonance conditions, one gets
locally invariant manifolds which are smooth.
\SECTION References
\ref
\no{AM}
\by{R. Abraham, J. Marsden}
\book{Foundations of mechanics}
\publisher{Benjamin}
\yr{1978}
\endref
\ref
\no{BF}
\by{R. Bonic, J. Frampton}
\paper{Smooth functions on Banach manifolds}
\jour{Jour. Math. Mech.}
\vol{16}
\pages{877--898}
\yr{1966}
\endref
\ref
\no{Dev}
\by{R. Deville}
\paper{Geometric implications of the existence of very smooth bump functions in Banach spaces}
\jour{Isr. Jour. Math.}
\vol{67}
\pages{1--22}
\yr{1989}
\endref
\ref
\no{FHY}
\by{A.Fathi , M. Herman , J--C. Yoccoz}
\paper{A proof of Pesin's stable manifold theorem}
\inbook{Lec. Notes in Math. }
\vol{1007}
\yr{1983}
\publisher{Springer}
\endref
\ref
\no{Ir1}
\by{M.C. Irwin}
\paper{On the stable manifold theorem}
\jour{Bull. London Math. Soc.}
\vol{2}
\pages{196--198}
\yr{1970}
\endref
\ref
\no{Ir2}
\by{M.C. Irwin}
\paper{A new proof of the pseudostable manifold theorem}
\jour{Jour. London Math. Soc.}
\vol{21}
\pages{557--566}
\yr{1980}
\endref
\ref
\no{La}
\by{O. E. Lanford III}
\paper{Bifurcation of periodic solutions into invariant tori: the work of Ruelle and Takens}
\inbook{ Lect. Notes in Math. }
\vol {322}
\publisher{Springer}
\yr{1973}
\endref
\ref
\no{RT}
\by{D. Ruelle, F. Takens}
\paper{On the nature of Turbulence}
\jour{Comm. Math. Phys.}
\vol{20}
\pages{167-192}
\yr{1971}
\endref
\ref
\no{W}
\by{J. C. Wells}
\paper{Invariant manifolds of non-linear operators}
\jour{Pac. Jour. Math}
\vol{62}
\pages{285-293}
\yr{1976}
\endref
\end