We consider an operator $\Ps$ defined on a set of real valued functions and satisfying two properties of monotonicity and additive homogeneity. This is motivated by the case of zero sum stochastic games, for which the Shapley operator is monotone and additively homogeneous. We study the asymptotic of the trajectories defined by $v_n=\frac{\Ps^n(0)}{n}$ ($n\in N , n \rightarrow \infty$) and $v_\lambda=\lambda\Ps\left(\frac{1-\lambda}{\lambda}v_\lambda\right)$ ($\lambda \in (0,1], \lambda \rightarrow 0$). Examining the iterates of $\Ps$, we exhibit analytical conditions on the operator that imply that $v_n$ and $v_\lambda$ have at most one accumulation point for the uniform norm. In particular this establishes the uniform convergence of $v_n$ and $v_\lambda$ to the same limit for a large subclass of the class of games where only one player control the transitions. We also study the general case of two players controlling the transitions, giving a sufficient condition for convergence.