  Ergodic optimization and discrete weak KAM theory are two parallel theories
with several results in common. For instance, the Mather set is the locus of
orbits which minimize the ergodic averages of a given observable. In the
favorable cases, the observable is cohomologous to its ergodic minimizing value
on the Mather set, and the discrete weak KAM solution plays the role of the
transfer function. One possibility of construction of such a coboundary is by
using the non linear Lax-Oleinik operator. The other possibility is by using a
discounted cohomological equation. It is known that the discounted discrete
weak KAM solution converges to some selected weak KAM solution. We show that,
in the ergodic optimization case for a coboundary observable over a minimal
system, the discounted transfer function converges if and only if the
observable is balanced.
