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Journal of Spectral Theory


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Volume 6, Issue 2, 2016, pp. 267–329
DOI: 10.4171/JST/125

Published online: 2016-04-06

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

David Borthwick[1] and Tobias Weich[2]

(1) Emory University, Atlanta, USA
(2) Universit├Ąt Paderborn, Germany

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.

Keywords: Iterated function scheme, Selberg zeta function

Borthwick David, Weich Tobias: Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions. J. Spectr. Theory 6 (2016), 267-329. doi: 10.4171/JST/125