Yan-li Xu | Searcharxiv

arXiv2022

Box-counting measure of metric spaces

In this paper, we introduce a new notion called the \emph{box-counting measure} of a metric space. We show that for a doubling metric space, an Ahlfors regular measure is always a box-counting measure; consequently, if $E$ is a self-similar set satisfying the open set condition, then the Hausdorff measure restricted to $E$ is a box-counting measure. We show two classes of self-affine sets, the generalized Lalley-Gatzouras type self-affine sponges and Barański carpets, always admit box-counting measures; this also provides a very simple method to calculate the box-dimension of these fractals. Moreover, among others, we show that if two doubling metric spaces admit box-counting measures, then the multi-fractal spectra of the box-counting measures coincide provided the two spaces are Lipschitz equivalent.

math.MG↗

arXiv2021

Dimension drop of connected part Of slicing self-affine Sponges

The connected part of a metric space $E$ is defined to be the union of non-trivial connected components of $E$. We proved that for a class of self-affine sets called slicing self-affine sponges, the connected part of $E$ either coincides with $E$, or is essentially contained in the attractor of a proper sub-IFS of an iteration of the original IFS.This generalize an early result of Huang and Rao [L. Y. Huang, H. Rao. \emph{A dimension drop phenomenon of fractal cubes}, J. Math. Anal. Appl. \textbf{497} (2021), no. 2] on a class of self-similar sets called fractal cubes. Moreover, we show that the result is no longer valid if the slicing property is removed. Consequently, for a Barański carpet $E$, the Hausdorff dimension and the box dimension of the connected part of $E$ are strictly less than the Hausdorff dimension and the box dimension of $E$, respectively. For slicing self-affine sponges in $\mathbb R^d$ with $d\geq 3$, whether the attractor of a sub-IFS has strictly smaller dimensions is an open problem.

math.DS↗