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

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.