AI 中文总结
本文探究小随机扰动下随机台球系统的李雅普诺夫指数与双曲性，发现非圆形椭圆、柠檬台球经扰动后兼具遍历性与双曲性，圆形台球则仅遍历且保持零李雅普诺夫指数，相关结果为受噪动力学系统研究提供新视角。
AI 中文摘要
本文研究小随机扰动下随机台球系统的李雅普诺夫指数与双曲性。研究首先回顾矩阵平稳序列和马尔可夫过程李雅普诺夫指数的必要理论背景，随后考察若干经典台球系统，证明随机扰动可显著改变其动力学性质：非圆形椭圆台球与柠檬台球在随机扰动下同时呈现遍历性与双曲性，而圆形台球虽具有遍历性，却保持零李雅普诺夫指数。本文还确立了最大李雅普诺夫指数变为正的条件，表明这些系统中出现双曲行为。上述结果为受随机噪声作用的动力学系统的稳定性与混沌性质提供了新见解，对光滑动力学系统的更广泛研究具有启示意义。
英文摘要
In this paper, we investigate the Lyapunov exponents and hyperbolicity of random billiard systems under small stochastic perturbations. Our study begins with a review of the necessary theoretical background on Lyapunov exponents for stationary sequences of matrices and Markov processes. We then consider several classical billiard systems and demonstrate that random perturbations can lead to significant changes in their dynamical properties. Specifically, we show that non-circular elliptic and lemon billiards exhibit both ergodicity and hyperbolicity under random perturbations, whereas circular billiards, while ergodic, maintain zero Lyapunov exponents. We also establish conditions under which the largest Lyapunov exponent becomes positive, indicating the emergence of hyperbolic behavior in these systems. These results provide new insights into the stability and chaotic properties of dynamical systems subjected to random noise, with implications for the broader study of smooth dynamical systems.
Journal refPure and Applied Functional Analysis, Volume 11, Number 1, 109-126, 2026
