  We study a skew product IFS on the cylinder defined by Baker-like maps
associated to a finite family of potential functions and the doubling map. We
show that there exist a compact invariant set with attractive behavior and a
random SRB measure whose support is in that set. We also study the IFS ergodic
optimization problem for that finite family of potential functions and
characterize the maximizing measures and the critical value through a
discounting limit. This shows the connection between this maximization problem
and the superior boundary of the compact invariant set, which is described as a
graph of the solution of the Bellman equation.
