  We consider zero sum stochastic games. For every discount factor $\lambda$, a
time normalization allows to represent the game as being played on the interval
[0, 1]. We introduce the trajectories of cumulated expected payoff and of
cumulated occupation measure up to time t $\in$ [0, 1], under
$\epsilon$-optimal strategies. A limit optimal trajectory is defined as an
accumulation point as the discount factor tends to 0. We study existence,
uniqueness and characterization of these limit optimal trajectories for
absorbing games.
