摘要
设β>1为实数,T_β为[0,1]的β变换.攀援集的任何两个点随着时间的转移会越来越接近但同时又总能在任意长时间后保持一定的距离.证明了在Lebesgue测度意义下关于T_β的攀援集是一个零测集.Distal点对的两个点表示随着时间的转移始终保持着一定的距离.如果固定其中一个点x_0,所有满足x∈[0,1)且lim inf n→∞|T_β~n(x)-T_β~n(x_0)|>0的点称为关于x_0的distal集,如果把这个集合记为R_β(x_0),根据Borel-Cantelli引理得到R_β(x_0)的Lebesgue测度为零.
Let β1 be a real number, Tβ is the β transformation defined on [0,1）. A scrambled set contains points which have the property that they can become nearer and nearer but meanwhile they can keep a positive distance within any long time. In this paper, we prove that the scrambled sets with respect to Tβ is a set with Lebesgue measure 0. A pair of points is called distal if they keep a positive distance with the time passing away. If we fix a point x0 in [0,1）, the points x ∈ [0, 1）which satisfies that lim inf n→∞｜Tβn（x）-Tβn（x0）｜〉0 is said to be distal with respect to x0, we denote the set containing all the distal point for xo as Rβ（x0）. We can get Rβ（x0） is a null set which means its Lebesgue measure is zero by Borel-Cantelli Lemma.
作者
张颖
ZHANG Ying(College of Mathematics, South China University of Technology, Guangzhou 510641, China)
出处
《数学的实践与认识》
北大核心
2017年第5期247-250,共4页
Mathematics in Practice and Theory
