When the weak separation condition implies the generalize finite type in $\mathbb{R}^d$

TL;DR

This paper proves that for certain iterated function systems in Euclidean space, the weak separation condition is equivalent to the generalized finite-type condition, and extends related concepts to higher dimensions.

Contribution

It establishes the equivalence between weak separation and generalized finite-type conditions in $\,\mathbb{R}^d$ and extends the notion of net intervals to higher dimensions.

Findings

Weak separation condition iff generalized finite-type condition in $\,\mathbb{R}^d$

Extension of net intervals to higher dimensions

Calculation of local dimension for self-similar measures

Abstract

Let $\mathcal{S}$ be an iterated function system in $\mathbb{R}^d$, with full support and some restrictions on the allowable rotations. We show that $\mathcal{S}$ satisfies the weak separation condition if and only if it satisfies the generalized finite-type condition. With this in mind, we extend the notion of net intervals from $\mathbb{R}$ to $\mathbb{R}^d$. We also use net intervals to calculate the local dimension of a self-similar measure with the finite-type condition and full support.