Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions

TL;DR

The paper demonstrates how symmetry reduction in holomorphic iterated function schemes leads to a factorization of Selberg zeta functions, simplifying resonance calculations and revealing a spectral gap on Schottky surfaces.

Contribution

It introduces a symmetry-based factorization of dynamical and Selberg zeta functions for symmetric surfaces, enabling more efficient numerical analysis.

Findings

Factorization of zeta functions via symmetry reduces computational complexity.

First observation of a macroscopic spectral gap on Schottky surfaces.

Enhanced understanding of spectral properties of symmetric hyperbolic surfaces.

Abstract

Given a holomorphic iterated function scheme with a finite symmetry group $G$, we show that the associated dynamical zeta function factorizes into symmetry-reduced analytic zeta functions that are parametrized by the unitary irreducible representations of $G$. We show that this factorization implies a factorization of the Selberg zeta function on symmetric $n$-funneled surfaces and that the symmetry factorization simplifies the numerical calculations of the resonances by several orders of magnitude. As an application this allows us to provide a detailed study of the spectral gap and we observe for the first time the existence of a macroscopic spectral gap on Schottky surfaces.