Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary representations

TL;DR

This paper develops thermodynamic formalism and transfer operator techniques to analyze Selberg zeta functions with unitary representations on hyperbolic surfaces, extending to billiard flows and Hecke triangle groups.

Contribution

It introduces new methods for studying Selberg zeta functions with unitary representations and extends these techniques to billiard flows and sequences of Hecke triangle groups.

Findings

Techniques for thermodynamic formalism applied to Selberg zeta functions.

Transfer operator methods for automorphic cusp forms.

Convergence analysis of transfer operators on Hecke triangle groups.

Abstract

Using Hecke triangle surfaces of finite and infinite area as examples, we present techniques for thermodynamic formalism approaches to Selberg zeta functions with unitary finite-dimensional representations $(V,\chi)$ for hyperbolic surfaces (orbifolds) $\Gamma\backslash\mathbb{H}$ as well as transfer operator techniques to develop period-like functions for $(\Gamma,\chi)$-automorphic cusp forms. This leads to several natural conjectures. We further show how to extend these results to the billiard flow on the underlying triangle surfaces, and study the convergence of transfer operators along sequences of Hecke triangle groups.