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We consider some discrete and continuous dynamics in a Banach space involving a non expansive operator and a corresponding family of strictly contracting operators (λ, ): = λ ($\frac{1-\lambda }{\lambda }$ ) for λ  ] 0,1] . Our motivation comes from the study of two-player zero-sum repeated games, where the value of the -stage game (resp. the value of the λ-discounted game) satisfies the relation = ($\frac{1}{n}$, $v_{n-1}$) (resp. $v_{\lambda }$ = (λ, $v_{\lambda }$)) where is the Shapley operator of the game. We study the evolution...