Winning property of distal set for β-transformations

摘要

For any β > 1, let ([0,1), T_β) be the β-transformation dynamical system. For any y∈ [0,1), define the distal set of a given point y as D_β(y) ={ x ∈ [0,1): lim inf_n→∞ | T_β~n(x) - T_β~n(y) |> 0} Yang, Li, et al. [15] proved that the Hausdorff dimension of the distal set of any point is one for any β > 1. In this paper, we study the winning property of the distal set of a given point y. We prove that the distal set of a given point y is α-winning for any β > 1 and y ∈ [0,1), where α <1/64 is a constant. By the definition of winning set, it's obvious that the distal set of a given point y is a dense set.