Winning property of distal set for β-transformations

Abstract

For any , let be the β-transformation dynamical system. For any , define the distal set of a given point y as 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 and , where 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.