Prime number theorems for Basmajian-type identities

Abstract

We obtain asymptotic counting results with error terms for complex orthospectrum for Schottky groups and orbit counting function for quadratic polynomials. Moreover, we prove equidistribution of holonomy associated to these dynamical systems. Our results are obtained by considering generalized $L$-functions coming from the Basmajian-type identities introduced by the author in \cite{He}. We study the associated summatory functions using tools from analytic number theory and Thermodynamic Formalism, namely the Perron's formula and a Dolgopyat-type estimate on the spectrum of transfer operators.