Abstract: Considering a generic potential A and a dynamics given by f, we are interestes on the study of the existence and the principal properties of the f-invariant measures that maximizes the action given by S(\mu)=\int Ad\mu. The cases that are already studied consider hyperbolic dynamics and the main tool is the existence of a sub-action function depending on A and f. We will resume some results and show how to deal with a dynamics on the circle with a critical point.