Regularity and uniqueness of one dimensional invariant manifolds

X. Cabr\'e and  E. Fontich

Abstract.
In this work we give sufficient conditions for the existence of
differentiable or analytical one
dimensional manifolds associated to an eigenvalue $\lambda $ and
to a corresponding eigenvector $v$ of it.
We consider first the case of local diffeomorphisms, the case of
differential equations being obtained easily from it. We look for the
invariant manifolds through a parametrization which gives the
linearization of the map along them, that is, we look for $\varphi $
defined on an interval of $\rr$ such that
$$
f(\varphi (t)) = \varphi (\lambda t).
$$
We restrict
ourselves to eigenvalues $\lambda $ of modulus different from one, the
others being arbitrary, but different from zero. First we
consider the differentiable case when the eigenvalue is non resonant.
In this case we get
that if the map is $C^m$ with $m$ bigger or equal than some value
$k$ , related to the structure
of the eigenvalues of $Df(0)$, there is an invariant manifold of
class $C^m$. If $m$ is strictly bigger that $k$ we get uniqueness.
Then, for the sake of completeness, the analytic non
resonant case is considered.
If the eigenvalue is resonant we construct the
bifurcation
equation from which we obtain the conditions for the existence of
solutions of class $C^m$. If we allow $m$ to be strictly bigger than
$k$, the bifurcation equation will
give the number of solutions of class $C^{m}$ (which may be zero). The
analytic case is also considered.