Billiard Dynamics: An Updated Survey with the Emphasis on Open Problems

TL;DR

This survey provides an updated overview of billiard dynamics, emphasizing open problems and recent advances across elliptic, hyperbolic, and parabolic cases, highlighting ongoing research challenges and conjectures.

Contribution

It offers a comprehensive, expanded survey of billiard dynamics, including new problems, conjectures, and recent developments since the previous publication.

Findings

Updated bibliography with recent research

New open problems and conjectures formulated

Summary of recent progress in billiard dynamics

Abstract

This is an updated and expanded version of our earlier survey article \cite{Gut5}. Section $\S 1$ introduces the subject matter. Sections $\S 2 - \S 4$ expose the basic material following the paradigm of elliptic, hyperbolic and parabolic billiard dynamics. In section $\S 5$ we report on the recent work pertaining to the problems and conjectures exposed in the survey \cite{Gut5}. Besides, in section $\S 5$ we formulate a few additional problems and conjectures. The bibliography has been updated and considerably expanded.