  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.
