In this paper, we study a two-parameter family of convex billiard tables, by taking the intersection of two round disks (with different radii) in the plane. These tables give a generalization of the one-parameter family of lemon-shaped billiards. Initially, there is only one ergodic table among all lemon tables. In our generalized family, we observe numerically the prevalence of ergodicity among the some perturbations of that table. Moreover, numerical estimates of the mixing rate of the billiard dynamics on some ergodic tables are also provided.

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We investigate superdiffusion for stochastic processes generated by nonuniformly hyperbolic system models, in terms of the convergence of rescaled distributions to the normal distribution following the abnormal central limit theorem, which differs from the usual requirement that the mean square displacement grow asymptotically linearly in time. We construct a martingale approximation that follows the idea of Doob's decomposition theorem. We obtain an explicity formula for the superdiffusion constant in terms of the fine structure that originates in the phase transitions as well as the geometry of the configuration domains of the systems. Models that satisfy our main assumptions include chaotic Lorentz gas, Bunimovich stadia, billiards with cusps, and can be apply to other nonuniformly hyperbolic systems with slow correlation decay rates of order O(1/n).