Non-degeneracy conditions in KAM theory

Persistence of invariant tori in a perturbed dynamical system requires two kinds of conditions to be met. A strong non-resonance condition ensures a dense quasi-periodic orbit on both the unperturbed and the perturbed torus. A non-degeneracy condition enforces a sufficiently large subset of the unperturbed tori to be non-resonant and thus yields persistence. In the past 60 years various such conditions have been formulated and a number of them are reviewed here.