# Multidimensional hyperbolic billiards

## TL;DR

This paper surveys the current state of multidimensional hyperbolic billiards, highlighting key results in higher dimensions and identifying central open problems in the field.

## Contribution

It provides a comprehensive overview of hyperbolic billiards beyond two dimensions and emphasizes the need for further research in this less understood area.

## Key findings

- Summarizes key results in multidimensional hyperbolic billiards
- Identifies central open problems in higher-dimensional billiard theory
- Highlights the contrast with well-developed planar billiard theory

## Abstract

The theory of planar hyperbolic billiards is already quite well developed by having also achieved spectacular successes. In addition there also exists an excellent monograph by Chernov and Markarian on the topic. In contrast, apart from a series of works culminating in Sim\'anyi's remarkable result on the ergodicity of hard ball systems and other sporadic successes, the theory of hyperbolic billiards in dimension 3 or more is much less understood. The goal of this work is to survey the key results of their theory and highlight some central problems which deserve particular attention and efforts.