Abstract:We introduce a new family of fractal dimensions by restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension. Among others, we prove that this family contains continuum many distinct dimensions, and they share most of the properties of the Hausdorff dimension, which answers negatively a question of Fraser. On the other hand, we also prove that among these new dimensions only the Hausdorff dimension behaves nicely with respect to Hölder functions.
We also consider the supremum of these new dimensions, which turns out to be another interesting notion of fractal dimension.
We prove that among those bilipschitz invariant, monotone dimensions on the compact subsets of $\mathbb{R}^n$ that agree with the similarity dimension for the simplest self-similar sets, the modified lower dimension is the smallest and when $n=1$ the Assouad dimension is the greatest, and this latter statement is false for $n>1$. This answers a question of Rutar.
From: Richard Balka [view email]
[v1]
Mon, 11 Dec 2023 15:45:15 UTC (23 KB)
[v2]
Wed, 20 Dec 2023 17:07:54 UTC (24 KB)
[v3]
Mon, 25 May 2026 03:08:34 UTC (25 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。