  Given a symbol $\varphi,$ i.e., a holomorphic endomorphism of the unit disc,
we consider the composition operator $C_{\varphi}(f)=f\circ\varphi$ defined on
the Banach spaces of holomorphic functions $A(\mathbb{D})$ and
$H^{\infty}(\mathbb{D})$. We obtain different conditions on the symbol
$\varphi$ which characterize when the composition operator is mean ergodic and
uniformly mean ergodic in the corresponding spaces. These conditions are
related to the asymptotic behaviour of the iterates of the symbol. As an
appendix, we deal with some particular case in the setting of weighted Banach
spaces of holomorphic functions.
