PreprintCC BY 4.0arXiv
Some open billiards with distinct top Lyapunov exponents
Abstract
We study open billiard flows in ℝ3 generated by reflections in the exterior of at least three ellipsoids with C3 boundaries. Each obstacle is congruent to K̃ = {(x,y,z)∈ ℝ3 : x2+y2+(βz)2 ≤ r2}, with parameter β>1, small radii, and centers Oi lying in the plane Γ={z=0}. The configuration satisfies the no-eclipse condition (H). For any Gibbs measure on the non-wandering set of the billiard ball map, we prove that if β>1/c12, for some constant c1>0 determined by the geometry of the configuration, the two positive Lyapunov exponents are distinct and positive: λ1(β)>λ2(β)>0. Moreover, λ2 is independent of β and corresponds to the restriction of the billiard flow to the plane Γ.
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BibTeX
@misc{dowais2026some,
title = {{Some open billiards with distinct top Lyapunov exponents}},
author = {Amal Al Dowais and Luchezar Stoyanov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03004},
eprint = {2610.03004},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03004},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Some open billiards with distinct top Lyapunov exponents
AU - Amal Al Dowais
AU - Luchezar Stoyanov
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03004
UR - https://arxiv.org/abs/2610.03004
AB - We study open billiard flows in ℝ^3 generated by reflections in the exterior of at least three ellipsoids with C^3 boundaries. Each obstacle is congruent to K̃ = {(x,y,z)∈ ℝ^3 : x^2+y^2+(βz)^2 ≤ r^2}, with parameter β>1, small radii, and centers O_i lying in the plane Γ={z=0}. The configuration satisfies the no-eclipse condition (H). For any Gibbs measure on the non-wandering set of the billiard ball map, we prove that if β>1/c_1^2, for some constant c_1>0 determined by the geometry of the configuration, the two positive Lyapunov exponents are distinct and positive: λ_1(β)>λ_2(β)>0. Moreover, λ_2 is independent of β and corresponds to the restriction of the billiard flow to the plane Γ.
ER -