Winning property of distal set for β-transformations. Issue 1 (2nd January 2021) - Yang, Qianqian; Wang, Shuailing
Abstract : 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.