The {\beta}-function describes the minimal average action associated with invariant measures of prescribed rotation number. In this talk, I will present rigidity phenomena of the {\beta}-function for generalized standard maps defined by analytic even potentials exhibiting KAM phenomena on a fixed set of Diophantine rotation numbers. I will present a KAM theorem depending analytically on the potential and on a suitable complex extension of the rotation number. This will allow us to obtain global rigidity results by local analysis near the zero potential of the {\beta}-function: in finite-dimensional spaces of perturbations, the {\beta}-Mather function is generically injective. For instance, nontrivial polynomial deformations of a generic potential are not {\beta}-preserving.