We investigate the relationship between the Hecke spectrum and periodic orbits on arithmetic quotients of the Bruhat–Tits building for PGL₃ over a field of formal series. In this setting, closed geodesics correspond to conjugacy classes of hyperbolic elements, and their enumeration naturally leads to zeta functions that generalize Ihara’s zeta function from rank one. These zeta functions—defined in terms of edge and chamber structures—reflect both geometric and spectral data, although the precise nature of this correspondence in higher rank is not yet fully understood. We outline how Hecke operators act on the space of automorphic forms and how their spectral decomposition suggests deeper connections with the analytic properties of the zeta functions. The talk will conclude with a discussion of ongoing questions, such as the search for identities between different types of zeta functions, and the explicit construction of periodic orbits using tools from higher-dimensional continued fractions.