由Dvoretzky随机覆盖引起的集合的Hausdorff维数

唐军民

数学学报 ›› 2014, Vol. 57 ›› Issue (1) : 51-70.

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数学学报 ›› 2014, Vol. 57 ›› Issue (1) : 51-70. DOI: 10.12386/A2014sxxb0006
论文

由Dvoretzky随机覆盖引起的集合的Hausdorff维数

    唐军民
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Hausdorff Dimension of Sets Arising from Dvoretzky Random Covering

    Jun Min TANG
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摘要

考虑了单位圆T=R/Z上的随机区间In(ω)=ωn+(-ln/2,ln/2),(mod 1),其中{ln}n≥1为一列单调下降并趋于0的正实数,{ωn}n≥1为T上的一列独立同分布且具有Gibbs分布测度的随机变量. 借助于重分形分析中的工具,估计了被随机区间序列{In(ω)}有限次覆盖以及无穷多次覆盖的集合的Hausdorff维数.

Abstract

We consider the random intervals In(ω)=ωn+(-ln/2,ln/2) (mod 1), where {ln}n≥1 is a sequence of positive real numbers which is decreasing to zero and {ωn}n≥1 is an i.i.d. sequence with Gibbs distribution measure on the circle T= R/Z. Using the tools from multi-fractal analysis, we estimate the Hausdorff dimension of sets which are covered finitely or infinitely many times by {In(ω)}.

关键词

随机覆盖 / Gibbs测度 / 局部维数 / 首中时

Key words

random covering / Gibbs measure / local dimension / hitting time

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唐军民. 由Dvoretzky随机覆盖引起的集合的Hausdorff维数. 数学学报, 2014, 57(1): 51-70 https://doi.org/10.12386/A2014sxxb0006
Jun Min TANG. Hausdorff Dimension of Sets Arising from Dvoretzky Random Covering. Acta Mathematica Sinica, Chinese Series, 2014, 57(1): 51-70 https://doi.org/10.12386/A2014sxxb0006

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基金

国家自然科学基金资助项目（10971069）
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