Weak separation condition, Assouad dimension, and Furstenberg homogeneity (Q2790834)

In this paper, the authors investigate and extend Moran constructions and dimensional properties of their limit sets from the Euclidean spaces to more general settings, namely metric spaces. While the Hausdorff dimension and the similarity dimension of an iterated function system (IFS) consisting of similiarity contractions coincide on the Euclidean space under the open set condition, in the present paper it is proved that an IFS satisfying a weak separation condition possesses a Moran construction with the finite clustering property. Under such finite clustering property, it is proved that the Hausdorff dimension and the Asuoad dimension of such a limit set coincide. The associated shift space is investigated and finally, the Furstenberg homogeneous sets.