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Trace formula for billiard flow on surfaces with concave boundary
Abstract
We study grazing closed trajectories of the billiard flow φt on a compact surface (Σ,g) with concave boundary. Under a non-degeneracy condition, which is satisfied for example when (Σ,g) has negative sectional curvature, we compute the trace of a regularized pullback Eε(φt)* Eε near a closed trajectory grazing once, which is an analogue of the Atiyah--Bott--Guillemin trace formula for closed billiard trajectories.
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BibTeX
@misc{kangabire2026trace,
title = {{Trace formula for billiard flow on surfaces with concave boundary}},
author = {Alain Kangabire},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22571},
eprint = {2609.22571},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22571},
}
RIS
TY - UNPB
TI - Trace formula for billiard flow on surfaces with concave boundary
AU - Alain Kangabire
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22571
UR - https://arxiv.org/abs/2609.22571
AB - We study grazing closed trajectories of the billiard flow φ^t on a compact surface (Σ,g) with concave boundary. Under a non-degeneracy condition, which is satisfied for example when (Σ,g) has negative sectional curvature, we compute the trace of a regularized pullback E_ε(φ^t)^* E_ε near a closed trajectory grazing once, which is an analogue of the Atiyah--Bott--Guillemin trace formula for closed billiard trajectories.
ER -