Dimensions of a class of self-affine Moran sets and measures in $\R^2$

TL;DR

This paper investigates the fractal dimensions of a class of self-affine Moran sets and measures in , providing formulas for various dimensions and exploring their properties.

Contribution

It introduces a comprehensive analysis of the dimensions of self-affine Moran sets and measures, including explicit formulas and properties, extending prior work in fractal geometry.

Findings

Computed Hausdorff, packing, box-counting, and Assouad dimensions of the sets.

Derived Hausdorff, packing, and entropy dimension formulas for the measures.

Established relationships between different fractal dimensions of the measures.

Abstract

For each integer $k>0$, let $n_k$ and $m_k$ be integers such that $n_k\geq 2, m_k\geq 2$, and let $\mathcal{D}_k$ be a subset of $\{0,\dots,n_k-1\}\times \{0,\dots,m_k-1\}$. For each $w=(i,j)\in \mathcal{D}_k$, we define an affine transformation on~$\R^2$ by $$ \Phi_w(x)=T_k(x+w), \qquad w\in\mathcal{D}_k, $$ where $T_k=\operatorname{diag}(n_k^{-1},m_k^{-1})$. The non-empty compact set $$ E=\bigcap\nolimits_{k=1}^{\infty}\bigcup\nolimits_{(w_1w_2\ldots w_k)\in \prod_{i=1}^k\mathcal{D}_i} \Phi_{w_1}\circ \Phi_{w_2}\circ \ldots\circ \Phi_{w_k} $$ is called a \textit{self-affine Moran set}. In the paper, we provide the lower, packing, box-counting and Assouad dimensions of the self-affine Moran set $E$. We also explore the dimension properties of self-affine Moran measure $\mu$ supported on $E$, and we provide Hausdorff, packing and entropy dimension formulas of $\mu$.