This may be considered a followup question to smooth dependence of stable manifold on parameters. In particular, I would like to know in more detail how smooth parameter dependence of a (generalized) stable manifold depends on the choice of manifold topology.
Consider the ODE $\dot{x} = f(x,\nu)$ with $f \in C^k$ with $k \ge 1$, jointly in $x \in \mathbb{R}^n$ and a parameter $\nu \in \mathbb{R}^p$. Assume that for $\nu = 0$ the origin is a fixed point, $f(0,0) = 0$, and that it admits a splitting $$ \mathbb{R}^n = E^s \oplus E^u $$ with real numbers $a < b$ such that $\textrm{spec}\big(Df(0)|_{E^s}\big) < a$ and $\textrm{spec}\big(Df(0)|_{E^u}\big) > b$. Then the generalized stable manifold theorem says that there exists an invariant manifold $W^s(0) \in C^r$ tangent to $E^s$ at $x = 0$ for any $r \ge 1$ such that $r a < b$ and $r \le k$.
Question: how smoothly does $W^s(0)$ depend on $\nu$?
The original question has an answer that refers to the book "Hyperbolicity & sensitive chaotic dynamics at homoclinic bifurcations" by Palis and Takens, appendix 1. However, that only seems to state continuous dependence on parameters, in $C^k$ topology.
First of all, I expect the answer to depend on the choice of topology on the stable manifolds. Since this is essentially a local question, using the compact-open $C^l$-topology seems to make sense. However, I expect the answer to depend on the choice of $l$. If we prove smooth dependence on $\nu$ using a phase space expansion trick by adding $\dot{\nu} = 0$ (if $a \ge 0$), or by adding a parameter in the Perron/Irwin method, then either way, we only find $W^s(0) = \{ x_u = h(x_s,\nu) \}$ with $h \in C^r(E^s \times \mathbb{R}^p;E^u)$, so for any $l+m \le r$, I only see that $C^m$-smooth dependence on $\nu$ in the compact-open $C^l$-topology follows.
My feeling is that this result is indeed optimal, but I am not sure about this. Does anyone have a reference to a statement and/or counterexample(s) to support this?
EDIT: I should have read more carefully. Palis and Takens actually do make the claim that there is differentiable dependence as I stated above for the function $h$. This leaves the question whether this result is optimal.
