On the transversality conditions and their genericity

In this note we review some results on the transversality conditions for a smooth Fredholm map f:×(0,T)→ between two Banach spaces ,. These conditions are well-known in the realm of bifurcation theory and commonly accepted as “generic”. Here we show that under the transversality assumptions the sections C(t)={x:f(x,t)=0} of the zero set of f are discrete for every t∈(0,T) and we discuss a somehow explicit family of perturbations of f along which transversality holds up to a residual set. The application of these results to the case when f is the -differential of a time-dependent energy functional ℰ:×(0,T)→ℝ and C(t) is the set of the critical points of ℰ provides the motivation and the main example of this paper