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Abstract. 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.

In this paper, we establish Basmajian’s identity for certain (1,1,2)-hyperconvex Anosov representations from a free group into [Formula: see text]. We then study our series identities on holomorphic families of Cantor non-conformal repellers associated to complex (1,1,2)-hyperconvex Anosov representations. We show that the series is absolutely summable 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.

We prove an inequality that must be satisfied by displacement of generators of free Fuchsian groups, which is the two-dimensional version of the log(2k − 1) Theorem for Kleinian groups due to Anderson-Canary-Culler-Shalen ([1]). As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the two-dimensional Margulis constant and lengths of a pair of based loops, which improves a result of Buser's.