### Abstract

We consider quantized cat maps on the 2-dimensional torus, as well as their nonlinear perturbations. We first analyze the evolution up to the Ehrenfest time of states localized around a periodic point, showing a transition to equidistribution. Using this transition, we obtain constraints on the localization properties of eigenstates around periodic orbits. The analysis is much simpler in the unperturbed case, where one uses the algebraic properties of the map. Besides, the constraints we obtain are known to be sharp only for the unperturbed case.