Coding of billiards in hyperbolic 3-space

TL;DR

This paper develops a symbolic coding system for billiard trajectories within ideal polyhedra in hyperbolic 3-space, establishing a conjugacy with shift spaces and revealing their subshift of finite type structure.

Contribution

It introduces a new coding method for hyperbolic billiards and proves the conjugacy with shift spaces, advancing the understanding of dynamical systems in non-Euclidean geometry.

Findings

Shift space closure forms a subshift of finite type

Established conjugacy between billiard trajectories and symbolic codes

Enhanced understanding of hyperbolic billiard dynamics

Abstract

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.