In this paper, we study nonlinear ergodic properties for an amenable semigroup of mappings of type ($\gamma$) in a Banach space. We prove that if $S$ is an amenable semigroup and $\varphi= \{T_t: t\in S\}$ is a semigroup of mappings of type $(\gamma)$ on a weakly compact, convex subset $C$ of a Banach space such that $D(\overline{co} F_{\frac 1n} (T_t),F(T_t))\to 0$, as $n\to\infty$ for each $t\in S$, then $F(\varphi)$ of common fixed points of $\varphi$ is nonempty and there exists a nonexpansive retraction $P$ form $C$ onto $F(\varphi)$, such that $PT_t= T_t P= P$ for each $t\in S$, and $Px \in \overline{co}\{ T_t x: t\in S\},$ $\forall x\in C$.
