We consider convex billiards bounded by a simple, smooth, closed curve. These convex billiards are proven to have KAM islands, if the boundary is smooth enough. Even for non-smooth convex billiards, the hyperbolicity is rare. Some exceptions are the Bunimovich stadium, and certain Lemon billiards. Moreover, hyperbolicity is rather difficult to prove, mainly because of the focusing effect of the wavefront upon hitting the convex boundary.
In this work, we consider certain random perturbations of convex billiards, following a recent work by Roberto Markarian and collaborators, we are able to prove that a large number of billiards are hyperbolic. We also identify some billiards that can not be hyperbolic under such perturbations.
张宏坤，美国麻省大学阿默斯特分校教授，是研究双曲动力学系统（包括混沌台球）统计特性的专家。现代动力系统的一个主要趋势是研究混沌动力系统产生的随机过程的随机性质，包括相关衰减率、中心极限理论和其他概率极限定理。张宏坤教授在这一领域做出了许多有意义的工作，在《 Comm. Math. Phys.》(9篇)，《 Trans. Amer. Math. Soc.》等国际权威杂志上发表论文50余篇。近年来，张宏坤教授的研究兴趣集中在动力系统和机器学习的交叉领域，取得了一些有意义的研究成果，其中与我校师生合作发表在 《Appl. Comput. Harmon. Anal.》，《IEEE Trans. Signal Inform. Process. Netw.》上的研究论文，受到国内外一些知名学者的引用和关注。
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