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arXiv · 2609.22571

Trace formula for billiard flow on surfaces with concave boundary

Alain Kangabire
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Abstract

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.

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Alain Kangabire. 2026-09-18. Trace formula for billiard flow on surfaces with concave boundary. https://arxiv.org/abs/2609.22571

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