Assouad and quasi-Assouad dimensions of Moran sets

TL;DR

This paper investigates the quasi-Assouad and Assouad dimensions of Moran sets, providing formulas and bounds under various conditions, including the introduction of quasi-normal and normal Moran sets.

Contribution

It offers new dimension formulas for Moran sets, especially when the minimal contraction ratio condition is relaxed, expanding understanding of their fractal geometry.

Findings

Derived quasi-Assouad dimension formulas for Moran sets with positive minimal contraction ratio.

Established bounds for quasi-Assouad dimensions without the positive ratio assumption.

Introduced quasi-normal and normal Moran sets to obtain exact dimension formulas.

Abstract

Moran sets are a non-autonomous generalization of self-similar sets. In this paper, we study the quasi-Assouad and Assouad dimensions of Moran sets in $\mathbb{R}^{d}$. First we provide quasi-Assouad dimension formulae for Moran sets satisfying $c_*>0$. Then, we provide the upper and lower bounds for quasi-Assouad dimension formulae for Moran sets without assuming $c_*>0$. To obtain the exact dimension formulae in this case, we define quasi-normal and normal Moran sets, and provide quasi-Assouad dimension formulae for these sets.