Typicalness of stable stability with respect to first approximation (Q1063770)

Let V be a closed differential manifold of class \(C^ 3\). Let S be the set of all diffeomorphisms of class \(C^ 1\) from V to V endowed with the \(C^ 1\)-topology. For \(f\in S\) consider the differential equation \(x'=f(x)\). Result: The space \(S\times V\) has a dense \(G_{\delta}\)- subset A that has the following property: if \(\limsup_{m\to +\infty}(1/m)\ell n\| d(f^ m)_ x\| <0\) for a certain (f,x)\(\in A\), then the set of those (g,y)\(\in S\times V\), for which the point y is exponentially stable with respect to the diffeomorphism g is a neighbourhood of (f,d). (Here \(df_ x\) denotes the derivative, \(\| \cdot \|\) the usual norm in the tangent space.)