Suppose I have an Liouville-Arnold integrable area preserving map $(\bar{x}, \bar{y}) = F(x,y)$ with a first integral $I(x,y)$.
How does one seek appropriate action-angle variables such that the map becomes a rigid rotation on each invariant curve? And is it possible to take the first integral $I$ as the action? In this case, how do I find the angle $\theta$ (as a function of $(x,y)$) ? Obviously for the case of a Hamiltonian $H(p,q)$ , one takes $\theta$ as as the integral around the closed path and $I = \int p dq$, but how does this change in case of maps?