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

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.