发表机构
Department of Mathematics, College of Science and Arts, Najran University; Department of Mathematics and Statistics, School of Physics, Mathematics and Computing, University of Western Australia（纳季兰大学科学与艺术学院数学系; 西澳大学物理、数学与计算学院数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究三维空间中多个椭球外部的开放台球流，证明在特定几何条件下，两个正李雅普诺夫指数不同且为正，其中第二个指数独立于参数β。
AI 中文摘要
我们研究由至少三个具有$C^3$边界的椭球外部反射生成的$\nmathbb{R}^3$中的开放台球流。每个障碍物与$$\nilde{K} = \{(x,y,z)\in \mathbb{R}^3: x^2+y^2+(\beta z)^2 \leq r^2\}$$全等，其中参数$\beta>1$，半径较小，中心$O_i$位于平面$\Gamma=\{z=0\}$内。该构型满足无日食条件(H)。对于台球映射非游荡集上的任意吉布斯测度，我们证明：若$\beta>1/c_1^2$，其中$c_1>0$是由构型几何决定的常数，则两个正李雅普诺夫指数不同且为正：$\lambda_1(\beta)>\lambda_2(\beta)>0$。此外，$\lambda_2$与$\beta$无关，对应于台球流在平面$\Gamma$上的限制。
英文摘要
We study open billiard flows in $\mathbb{R}^3$ generated by reflections in the exterior of at least three ellipsoids with $C^3$ boundaries. Each obstacle is congruent to $$\tilde{K} = \{(x,y,z)\in \mathbb{R}^3 : x^2+y^2+(βz)^2 \leq r^2\},$$ 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 $Γ$.
Comments10 pages
