  In this paper, we extend the scope of symbolic dynamics to encompass a
specific class of ideal polyhedrons in the 3-dimensional hyperbolic space,
marking an important step forward in the exploration of dynamical systems in
non-Euclidean spaces. Within the context of billiard dynamics, we construct a
novel coding system for these ideal polyhedrons, thereby discretizing their
state and time space into symbolic representations. This paper distinguishes
itself through the establishment of a conjugacy between the space of pointed
billiard trajectories and the associated shift space of codes. A crucial
finding herein is the observation that the closure of the related shift space
emerges as a subshift of finite type (SFT), elucidating the structural aspects
and asymptotic behaviour of these systems.
