X. Cabre, E. Fontich, R. de la Llave
The parameterization method for invariant manifolds I: manifolds
associated to non-resonant subspaces
(606K, ps)
ABSTRACT. We introduce a method to prove existence of
invariant manifolds and, at the same time
to find simple polynomial maps which are conjugated to the dynamics on them.
As a first application,
we consider the dynamical system given by a $C^r$ map
$F$ in a Banach space $X$ close to
a fixed point: $F(x) = Ax + N(x)$, $A$ linear, $N(0)=0$, $DN(0)=0$.
We show that if $X_1$ is an invariant subspace of~$A$
and $A$ satisfies certain spectral properties,
then there exists a unique
$C^r$ manifold which is invariant under $F$ and tangent to $X_1$.
When $X_1$ corresponds to spectral subspaces associated to
sets of the spectrum contained in disks around the origin
or their complement,
we recover the classical (strong) (un)stable manifold theorems.
Our theorems, however,
apply to other invariant spaces.
Indeed, we do not require $X_1$ to be an spectral subspace or even
to have a complement invariant under~$A$.