We introduce a new approach to studying the asymptotic behaviour of the iterates of composition operators on various weighted spaces of holomorphic functions over the unit disc \({{\mathbb {D}}}\) . This approach, which is related to the geometric properties of these spaces, allows us to significantly extend known results in this area from classical function spaces - such as Bergman spaces, Dirichlet spaces, standard weighted Banach spaces with sup-norm, and Bloch spaces - to their corresponding weighted spaces induced by doubling weights and fast weights. Among these, the case of fast weights seems to be especially interesting, as it leads to several surprising phenomena not observed in the other settings.
