Assouad and quasi-Assouad dimensions of Moran sets
Junjie Miao, Minghui Xu
TL;DR
This paper studies Assouad-type dimensions for Moran sets in $\mathbb{R}^d$, introducing quasi-normal and normal Moran structures to obtain exact quasi-Assouad dimension formulas. It shows that when the minimal contraction $c_* > 0$, the quasi-Assouad dimension satisfies $\dim_{qA} E = t_* = t = t^*$, and it derives additional equalities under weaker contractivity conditions (e.g., $\lim_{k\to\infty}\overline{M}_k = 0$) or when BBC and a positive lower ratio hold, linking $\dim_{qA} E$ to $\dim_A E$ and to limits of $s_{k+1,k+l}$. The authors establish a general lower bound for $\dim_{qA}$ via $t_*$ and show that club and BBC yield upper bounds and, in several regimes, full equality with the Assouad dimension; they also provide a comprehensive set of examples to illustrate sharpness and limitations. The results provide a nuanced picture of when Moran structures admit exact quasi-Assouad and Assouad dimension formulas and clarify how regularity conditions influence dimension computations in inhomogeneous fractal constructions.
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.