Basmajian-type identities and Hausdorff dimension of limit sets

TL;DR

This paper explores series identities related to Cantor sets in complex dynamics, revealing their connection to Hausdorff dimension and demonstrating properties like analytic continuation and monodromy.

Contribution

It establishes a criterion linking the absolute summability of series to the Hausdorff dimension being less than one and analyzes their analytic properties.

Findings

Series are absolutely summable iff Hausdorff dimension < 1

Identities can be analytically continued with nontrivial monodromy

Provides new insights into complex dynamical Cantor sets

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.