  In this paper, we study Basmajian-type series identities on holomorphic
families of Cantor sets associated to one-dimensional complex dynamical
systems. We show that the series is absolutely summable if and only if the
Hausdorff dimension of the Cantor set is strictly less than one. Throughout the
domain of convergence, these identities can be analytically continued and they
exhibit nontrivial monodromy.
