Let $v$ be a $C^r$ vector field on a Banach space $V$ such that $0$ is its hyperbolic fixed point, and let $X\_v\subset V$ be its local stable manifold. Does $X\_v$ depend on $v$ smoothly?



smooth dependence of stable manifold on parameters
The answer to your question is yes: please see the subsection "Differentiable dependence of invariant manifolds and foliations on diffeomorphisms" (page 164) of Appendix 1 of Palis-Takens book "Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations" (see, e.g., this link <http://books.google.fr/books/about/Hyperbolicity_and_Sensitive_Chaotic_Dyna.html?id=pwydPA23KVUC&redir_esc=y> towards Google books). Here, the results are stated for diffeomorphisms but the arguments can be adapted to the case of vector fields.


