Assouad and quasi-Assouad dimensions of Moran sets

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 $c_{\ast }>0$. Then, we provide the upper and lower bounds for quasi-Assouad dimension formulae for Moran sets without assuming $c_{\ast }>0$. 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.