Assume that $C$ is a closed convex subset of a reflexive Banach space $E$ and $\varphi=\{T_i\}_{i\in I}$ is a family of self-mappings on $C$ of type $(\gamma)$ such that $F(\varphi)$, the common fixed point set of $\varphi$, is nonempty. From our results in this paper, it can be derived that: (a) If $\cup_{i\in I}F(T_i)$ is contained in a $3$-dimensional subspace of $E$ then $F(\varphi)$ is a nonexpansive retract of $C$; (b) If $\varphi$ is commutative, there exists a retraction $R$ of type $(\gamma)$ from $C$ onto $F(\varphi)$, such that $RT_i=T_i R=R$ $(\forall i)$, and every closed convex $\varphi$-invariant subset of $C$ is $R$-invariant; the same result holds for a non-commutative right amenable semigroup $\varphi$, under some additional assumptions. Moreover, the existence of a $(T_i)$-ergodic retraction $R$ of type $(\gamma)$ from $\widetilde{C}=\{(x_i)\in l^{\infty}(E): x_i\in C, \forall i\in I \}$ onto $F(\varphi)$ in $l^{\infty}(E)$ for the family $\varphi$ is discussed. We also apply some of our results to find ergodic retractions for nonexpansive affine mappings.
