[论文解读] Interpolating between Hausdorff and box dimension

Amlan Banaji|arXiv (Cornell University)|Oct 30, 2023

Advanced Numerical Analysis Techniques被引用 1

一句话总结

本论文引入了广义中间维数，通过参数 θ ∈ (0,1) 限制覆盖集的相对大小，实现豪斯多夫维数与盒维数之间的插值。论文利用上、下达布导数建立了函数可实现为中间维数的充要条件，证明了无限共形IFS极限集的中间维数由豪斯多夫维数与不动点集维数的最大值决定，并推导出贝德福德–麦克默伦地毯中间维数的精确公式，表明其表现出可数多个相变点，且双利普希茨等价性蕴含多重分形谱的相等。

ABSTRACT

Hausdorff and box dimension are two familiar notions of fractal dimension. Box dimension can be larger than Hausdorff dimension, because in the definition of box dimension, all sets in the cover have the same diameter, but for Hausdorff dimension there is no such restriction. This thesis focuses on a family of dimensions parameterised by θ ∈ (0,1), called the intermediate dimensions, which are defined by requiring that diam(U) ⩽ (diam(V))ᶿ for all sets U, V in the cover. We begin by generalising the intermediate dimensions to allow for greater refinement in how the relative sizes of the covering sets are restricted. These new dimensions can recover the interpolation between Hausdorff and box dimension for compact sets whose intermediate dimensions do not tend to the Hausdorff dimension as θ → 0. We also use a Moran set construction to prove a necessary and sufficient condition, in terms of Dini derivatives, for a given function to be realised as the intermediate dimensions of a set. We proceed to prove that the intermediate dimensions of limit sets of infinite conformal iterated function systems are given by the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This applies to sets defined using continued fraction expansions, and has applications to dimensions of projections, fractional Brownian images, and general Hölder images. Finally, we determine a formula for the intermediate dimensions of all self-affine Bedford–McMullen carpets. The functions display features not witnessed in previous examples, such as having countably many phase transitions. We deduce that two carpets have equal intermediate dimensions if and only if the multifractal spectra of the corresponding uniform Bernoulli measures coincide. This shows that if two carpets are bi-Lipschitz equivalent then the multifractal spectra are equal.

研究动机与目标

• 通过细化分形覆盖中覆盖集相对大小的限制，推广中间维数。
• 利用达布导数，建立给定函数可作为紧集中间维数的充要条件。
• 刻画无限共形迭代函数系统极限集的中间维数。
• 推导自仿射贝德福德–麦克默伦地毯中间维数的精确公式，并分析其相变行为。
• 证明两个贝德福德–麦克默伦地毯具有相等的中间维数，当且仅当其对应均匀伯努利测度的多重分形谱一致。

提出的方法

• 通过要求在覆盖中所有 U, V 满足 diam(U) ≤ (diam(V))θ，引入广义的中间维数族，实现对覆盖集大小比的灵活控制。
• 利用莫兰集构造，将维数函数的可达性与函数的达布导数联系起来。
• 应用热力学形式化与速率函数分析技术，研究无限共形IFS吸引子的中间维数。
• 通过均匀伯努利测度关联的速率函数的勒让德变换，推导出贝德福德–麦克默伦地毯中间维数的显式公式。
• 通过生成函数的微分比较两个地毯的结构，利用多重分形谱相等性推导出其中间维数相等。
• 通过维度公式与参数关系的代数运算，建立地毯之间双利普希茨等价性与多重分形谱相等性之间的等价关系。

实验结果

研究问题

• RQ1 哪些函数可作为紧集的中间维数出现？其可实现性的条件是什么？
• RQ2 中间维数在无限共形迭代函数系统极限集上的行为如何？其取值由什么决定？
• RQ3 自仿射贝德福德–麦克默伦地毯的中间维数的精确形式是什么？其随 θ 如何变化？
• RQ4 两个贝德福德–麦克默伦地毯具有相等中间维数，当且仅当其均匀伯努利测度的多重分形谱一致吗？
• RQ5 地毯之间的双利普希茨等价性与多重分形谱相等性之间存在何种关系？

主要发现

• 无限共形IFS极限集的中间维数等于极限集豪斯多夫维数与收缩映射不动点集维数的较大值。
• 贝德福德–麦克默伦地毯的中间维数在 (γ⁻¹, 1) 上实解析，表现出可数多个相变点，此时维数函数的导数不连续。
• 两个贝德福德–麦克默伦地毯具有相等的中间维数，当且仅当其对应均匀伯努利测度的多重分形谱一致。
• 两个贝德福德–麦克默伦地毯之间双利普希茨等价，蕴含其多重分形谱相等；反之，若多重分形谱相等，则其中间维数也相等。
• 地毯的中间维数函数完全由参数 M, M₀, Rᵢ, Nᵢ 及比值 M′/M 决定，相变结构由速率函数 I(t) 编码。
• 函数 f(θ) 作为紧集中间维数的充要条件是：f 连续、单调递增，且其达布导数满足与某测度速率函数相关的增长约束。