On the transversality conditions and their genericity

Abstract

In this note we review some results on the transversality conditions for a smooth Fredholm map $f: \mathsf X\times (0,T) \rightarrow \mathsf Y$ between two Banach spaces $\mathsf X,\mathsf Y$. 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 ${\mathcal C}(t)=\{x:f(x,t)=0\}$ of the zero set of $f$ are discrete for every $t\in (0,T)$ 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 $\mathsf X$-differential of a time-dependent energy functional $\fancyscript{E}:\mathsf X\times (0,T)\rightarrow \mathbb {R}$ and ${\mathcal C}(t)$ is the set of the critical points of $\fancyscript{E}$ provides the motivation and the main example of this paper.