Lazutkin coordinates and the conjugation problem for billiard maps
Dynamical Systems
2026-03-17 v4
Abstract
The conjugation problem for billiard maps conjectures that if two strictly convex billiards have conjugated billiard maps, the billiard tables must be homothetic to each other. We show that if two billiard maps are conjugated, the conjugation diffeomorphism is tangent to a Lazutkin change of coordinates. We also recompute the coefficients in the billiard map expansion as the angle ttends to zero, correcting some incorrect expressions in Lazutkin's computations.
Keywords
Cite
@article{arxiv.2503.19550,
title = {Lazutkin coordinates and the conjugation problem for billiard maps},
author = {Corentin Fierobe},
journal= {arXiv preprint arXiv:2503.19550},
year = {2026}
}
Comments
The results are incorrect as they rely on erroneous computations in Lazutkin's book concerning the expansion of the billiard map in Lazutkin coordinates