Using Hecke triangle surfaces of finite and infinite area as examples, we\npresent techniques for thermodynamic formalism approaches to Selberg zeta\nfunctions with unitary finite-dimensional representations $(V,\\chi)$ for\nhyperbolic surfaces (orbifolds) $\\Gamma\\backslash\\mathbb{H}$ as well as\ntransfer operator techniques to develop period-like functions for\n$(\\Gamma,\\chi)$-automorphic cusp forms. This leads to several natural\nconjectures. We further show how to extend these results to the billiard flow\…