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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?

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up vote 2 down vote accepted

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 towards Google books). Here, the results are stated for diffeomorphisms but the arguments can be adapted to the case of vector fields.

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