Heterogeneous ubiquitous systems in ℝd and Hausdorff dimension
Heterogeneous ubiquitous systems in ℝd and Hausdorff dimension
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DOI：
10.1007/s00574-007-0056-z
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发表时间：
2007-09
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影响因子：
--
通讯作者：
J. Barral;S. Seuret
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文献类型：
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作者：
J. Barral;S. Seuret
设{xnn}∈＿1为[0,1]d中的一个序列，λ{nn}∈＿1为收敛于0的正实数序列，δ bbb＿1。经典的泛在性结果涉及形式的limsup＿sets的Hausdorff维数的计算，let μ是在[0,1]d上，ρ2（0,1）和α>上的一个正Borel测度。考虑更精细的极限集$$ S_{\mu } {\left( {p,\delta ,\alpha } \right)} = {\bigcap\limits_{N \in \mathbb{N}} \; {{\bigcup\limits_{n \geqslant N:\mu {\left( {B{\left( {x_{n} ,\lambda ^{p}_{n} } \right)}} \right)} \sim \lambda ^{{p\alpha }}_{n} } {B{\left( {x_{n} ,\lambda ^{\delta }_{n} } \right)}} }} }. $$我们证明，在适当的假设下，我们可以计算出集合的豪斯多夫维数（ρ,δ,α）。此外，当ρ< 1时，在计算s （ρ,δ,α）的豪斯多夫维数时会出现未知的饱和现象。我们的结果适用于几种多重分形测度，andS（δ）对应于μ是像勒贝格测度一样的单分形测度的特殊情况。这些集合的维数的计算为研究一些新的对象和现象开辟了道路。本文给出了由（或与）b进展开性质、一些Birkhoff和的平均值和分支随机漫步以及随机覆盖数的渐近行为所决定的Diophantine近似的应用。
Let {xn}n∈ℕbe a sequence in [0, 1]d, {λn}n∈ℕ a sequence of positive real numbers converging to 0, andδ> 1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsup-sets of the formLetμbe a positive Borel measure on [0, 1]d,ρ2 (0, 1] andα> 0. Consider the finer limsup-set $$ S_{\mu } {\left( {p,\delta ,\alpha } \right)} = {\bigcap\limits_{N \in \mathbb{N}} \; {{\bigcup\limits_{n \geqslant N:\mu {\left( {B{\left( {x_{n} ,\lambda ^{p}_{n} } \right)}} \right)} \sim \lambda ^{{p\alpha }}_{n} } {B{\left( {x_{n} ,\lambda ^{\delta }_{n} } \right)}} }} }. $$We show that, under suitable assumptions on the measureμ, the Hausdorff dimension of the setsSμ(ρ,δ,α) can be computed. Moreover, whenρ< 1, a yet unknown saturation phenomenon appears in the computation of the Hausdorff dimension ofSμ(ρ,δ,α). Our results apply to several classes of multifractal measures, andS(δ) corresponds to the special case whereμis a monofractal measure like the Lebesgue measure.The computation of the dimensions of such sets opens the way to the study of several new objects and phenomena. Applications are given for the Diophantine approximation conditioned by (or combined with)b-adic expansion properties, by averages of some Birkhoff sums and branching randomwalks, as well as by asymptotic behavior of random covering numbers.