Weak separation condition, Assouad dimension, and Furstenberg homogeneity

TL;DR

This paper investigates the dimensional properties of limit sets from Moran constructions with finite clustering, linking Assouad and Hausdorff dimensions, and explores conditions for Furstenberg homogeneity in self-similar sets.

Contribution

It establishes a relation between Assouad and Hausdorff dimensions for certain fractal sets and characterizes Furstenberg homogeneity in self-similar sets.

Findings

Assouad dimension can be expressed via Hausdorff dimension for these sets.

Furstenberg homogeneous self-similar sets satisfy the weak separation condition.

Some self-similar sets satisfy the open set condition but are not Furstenberg homogeneous.

Abstract

We consider dimensional properties of limit sets of Moran constructions satisfying the finite clustering property. Just to name a few, such limit sets include self-conformal sets satisfying the weak separation condition and certain sub-self-affine sets. In addition to dimension results for the limit set, we manage to express the Assouad dimension of any closed subset of a self-conformal set by means of the Hausdorff dimension. As an interesting consequence of this, we show that a Furstenberg homogeneous self-similar set in the real line satisfies the weak separation condition. We also exhibit a self-similar set which satisfies the open set condition but fails to be Furstenberg homogeneous.