【摘要】 设([0,1),T_β)是β-变换动力系统.对x∈[0,1),记τ_n~β(x)为x首次返回到包含x的n阶柱集的时间.本文研究集合{x ∈ [0, 1) : lim inf n→∞(log τ_n~β(x))/log n= α, lim sup n→∞(log τ_n~β(x))/logn= γ}的Hausdorff维数,其中0αγ+∞,并证明了当α<1时,这个集合的Hausdorff维数是0;当α1时,这个集合的Hausdorff维数就是1.随后,本文还考虑了首次返回到球的维数谱,得到了类似的0-1律.更多还原

【Abstract】 Let([0, 1), T_β) be the β-transformation dynamical system. We consider the first recurrence time with the polynomial rate. Let τ_n~β(x) be the first return time of x ∈ [0, 1) to the cylinder of order n containing x. We prove that the Hausdorff dimension of the set{x ∈ [0, 1) : lim inf n→∞(log τ_n~β(x))/log n= α, lim sup n→∞(log τ_n~β(x))/logn= γ}is zero when α < 1 and one when α 1, where 0 α γ +∞. We also consider the first return time to the ball, and obtain the similar zero-one law as above.更多还原